From 4327e8780c99c1afa7abe365bc2d08dad76ce896 Mon Sep 17 00:00:00 2001 From: michelevantaggi02 Date: Fri, 11 Aug 2023 12:55:04 +0200 Subject: [PATCH 01/37] import appunti da obsidian --- .../Anno 2/Calcolo Numerico/CalcNum-merged.md | 1665 +++++++++++++++++ .../Calcolo Numerico/CalcNum-merged.pdf | Bin 0 -> 431444 bytes .../argomenti/0. Introduzione.md | 67 + .../1. Stabilit\303\240 numerica.md" | 31 + .../argomenti/2. Error Analysis.md | 142 ++ .../3. Errore Per Le Funzioni Razionali.md | 92 + .../argomenti/4. Problemi Ben Posti.md | 32 + .../5. Derivata di pi\303\271 variabili.md" | 42 + .../6. Valutazione di un polinomio.md | 71 + .../Calcolo Numerico/argomenti/7. Vettori.md | 46 + .../Calcolo Numerico/argomenti/8. Matrici.md | 21 + .../argomenti/9. Altre Strutture.md | 110 ++ .../argomenti/a0. Sistemi lineari.md | 130 ++ .../argomenti/a1. Algoritmo di Gauss.md | 145 ++ .../argomenti/a2. Matrici tridiagonali.md | 55 + .../argomenti/a3. 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Introduzione + +Branca della matematica che si occupa di dare strumenti per il calcolo numerico di quantità continue (soluzioni di un'equazione) + +```ad-example + +$x^2=2$ soluzione $\alpha > 0$ + +$\alpha = \sqrt 2$ (soluzione simbolica) + +$\alpha$ limite dell'iterazione + +$$x_{k+1} = \frac {x^2_k + 2 }{2x_k}$$ + +``` + +# Calcolo Simbolico VS Calcolo Numerico + +```ad-example +collapse: open + +$P(x) = 0$ con $p(x) = x^3 + x^2 -x + 1$ ha un'unica soluzione $\alpha$ + +- $\alpha = \frac 13(-1 - \frac 4{\sqrt[3]{19 - 3\sqrt{33}}} - \sqrt[3]{19 - 3\sqrt{33}})$ soluzione simbolica +- $\alpha$ è il limite dell'iterazione:$$x_{k+1} = x_k - \frac {p(x_k)}{p'(x_k)}, \quad x_0 = 2$$ + +``` + +```ad-example +collapse: open + +Eq: $x^5 + x^4 - x + 1 = 0$ unica soluzione $\alpha$ + +- $\alpha$ non esprimibile in forma chiusa con simboli noti (soluzione simbolica) +- $\alpha$ è il limite dell'iterazione:$$x_{k+1} = x_k - \frac {p(x_k)}{p'(x_k)}, \quad x_0 = -2$$ + +``` + +Quasi tutte le civiltà hanno una formula per le equazioni di 2° grado. + +Nel XVI secolo sono state trovate per le equazioni di 3° e 4° grado (Tartaglia, Cardano, Ferrari). + +```ad-important +Teorema di Ruffini-Abel: + +Non esiste una formula per le equazioni di 5° grado + +``` + +La materia si occupa di dare strumenti teorici (teorie matematiche) per il calcolo numerico di quantità continue. + +Si cerca di trovare una soluzione ai problemi continui che sia utilizzabile. Non basta una formula, ma serve un algoritmo ce fornisca una soluzione in un tempo ragionevole tenendo conto che il calcolatore lavora in quantità finite. Vogliamo quindi trattare __quantità infinite__ con __strumenti finiti__. + +# Quantità Continue + +- Una o più equazioni non lineari, radici di polinomi +- Calcolo quantità algebra lineare, soluzione di sistemi lineari +- Determinanti, autovalori/vettori, rango, nucleo, immagine +- Calcolo degli integrali +- Approssimazione di funzioni + +Per ogni problema esistono una o più soluzioni. + +## Efficienza + +Gli algoritmi devono avere importanti requisiti: +- Basso costo computazionale +- [[1. Stabilità numerica|Stabilità numerica]] + + + +# 1. Stabilità numerica + + +```ad-example +title: Esempio + +## Sviluppo in serie di Taylor di $e^x$ con resto di Lagrange +$$e^x=1+ \frac x{1!} + \frac {x^2}{2!} + \frac {x^3}{3!} + ... = \sum_{k=0}^n {\frac {x^k}{k!}} = e^{\epsilon} \frac{x^n+1}{n+1}!$$ + +Si hanno due algoritmi per il calcolo di $e^x$ +$$e^x \approx \sum_{k=0}^n {\frac {x^k}{k!}}$$ +$$e^x \approx \frac 1{e^{-x}} \approx \frac 1{\sum_{k=0}^n {\frac {(-x)^k}{k!}}}$$ + +Il primo funziona meglio con esponente positivo, il secondo con esponente negativo + +``` + +# Relazioni Con Altre Discipline + +## Matematica Discreta + +Si cerca di utilizzare una sola formula per la risoluzione di problemi, mentre qua ci servono più formule + +# Algoritmi E Strutture Dati + +Si occupa dello studio di problemi discreti, mentre qui ci si occupa di problemi continui + +# Relazione Con Il Mondo Reale + +I problemi del mondo reale possono essere __modellizzati__ in teorie matematiche. + +Le soluzioni matematiche vengono __applicate__ nel mondo reale. + +![[Modellizzazione.png]] + + +# 2. Error Analysis + + + +I numeri reali per la maggior parte contengono infinite informazioni, a parte i pochi numeri razionali che ne contengono di limitate. + +# Numeri Razionali + +Sono dei numeri che rispettano la descrizione di un determinato teorema, e che non rispettano altri teoremi come, ad esempio, il teorema degli zeri o quello di Lagrange. + +I numeri razionali sono infiniti, ma noi possiamo lavorare solo con insiemi finiti di numeri. + +```ad-question +title: Problema + +Selezionare un numero finito di valori che possano rappresentare i numeri reali. + +## Distribuzione uniforme + +Scegliere $\varepsilon > 0$, N pari: +$$\varepsilon = \{\varepsilon k: - \frac N2 < k \leq \frac N2, k \in Z\}$$ + +Possiamo rappresentare i numeri reali con l'insieme: +$$ S = \{-\varepsilon\frac N20$ di cifre della mantissa, $m,M$ numeri positivi, definiamo un insieme di __floating point numbers__ + +$$F(\beta, t, m, M) = \{0\} \cup \{\pm \beta^p\sum_{i=1}^t {\beta^{-i}d_i}, \; -m \leq p \leq M, \; 0 \leq d_i < \beta \text{ intero per } i=1,...,t, \; d_1 \neq 0\}$$ + +```ad-example +title: Esempio +collapse: open + +$F := F(10, 2, 2, 3)$ + +Cardinalità: $1 + 2(m+M+1)(\beta - 1)\beta^{t-1}$ + +Max(F): $\Omega=\beta^M\sum_{i=1}^t{\beta^{-i}(\beta - 1)} = \beta^M(1- \beta^{-t})$ + +MinPositivo(F): $\omega = \beta^{-m}\beta^{-1} = \beta^{-m-1}$ + +Per $x \geq \Omega$ i numeri non possono essere rappresentati, mentre per $0 < x < \omega$ c'è un grande errore relativo + +``` + +```ad-example +title: Esempio 2 (Utile per dopo) +collapse: open + +$F(2,2,1,1)$ è composto da 13 numeri + +Questi numeri __non sono uniformi__. Tra $\frac 12$ e $\frac 14$ e tra $\frac 12$ e 1 c'è lo stesso numero di elementi di $F$. + +![[Pasted image 20230125111411.png]] + +Data $S = \{x \in R : \omega \leq x \leq \Omega\}$ costruiamo una __funzione di rappresentazione__ +$$fl : R \to F \cup\{\pm \infty\}$$ + +Con una delle due regole, dato $x=\beta^p \sum_{i=1}^\infty {\beta^{-i}d_i} \in S$: + +- Troncamento: $\tilde x = fl(x) = \beta^p \sum_{i=1}^t {\beta^{-i}d_i}$ +- Arrotondamento: $\tilde x = fl(x)$ il troncamento di $x + \frac{\beta^{p-t-1}}2$ + +Se $x > \Omega$ impostiamo $fl(x) = \infty$ (__overflow__), se $0\leq x \leq \omega$ impostiamo $fl(x) = 0$ (__underflow__), stesso ragionamento per i numeri negativi. + +``` + +^59010a + +Definiamo la __precisione macchina__ come $u=\beta^{-t+1}$ per il troncamento e $u=\frac{\beta^{-t+1}}2$ per l'arrotondamento. + +```ad-important +title: Teorema + +Dato $x \in S$ abbiamo il seguente limite per l'errore relativo: +$$|\frac{fl(x) - x}x| < u$$ + +``` + +__$F$ deve seguire determinate proprietà algebriche__: + +$$x,y \in F \centernot \implies x + y \in F$$ + +Dobbiamo definire delle __operazioni__ con i numeri a virgola mobile. + +Assumendo che esiste una somma a virgola mobile $\oplus$ tale che se $x,y \in F$ (se non avviene overflow): + +$$x\oplus y \in F, \quad x \oplus y = (x+y)(1 + \varepsilon), \quad |\varepsilon| < u$$ + +Similmente definiamo $\otimes, \ominus, \oslash$. + +Un'idea è quella di definire $x \oplus y = fl(x+y)$ ma i dettagli sono più complicati. + +Infatti le varie operazioni seguono solo alcune delle proprietà delle operazioni elementari: + +- commutatività della somma +- commutatività del prodotto +- $x \oslash x = 1$ + +Non seguono infatti: + +- associatività di somma e prodotto +- proprietà distributiva +- semplificazione +- potrebbe succedere che $x \otimes y = z \otimes y, \quad y \neq 0, \quad x \neq z$ + +[[3. Errore Per Le Funzioni Razionali]] + +# 3. Errore Per Le Funzioni Razionali + + + + +Data una funzione razionale $f: R^2 \to R$ che sia $f = \frac pq$ con p e q polinomi. + +Dall'analisi matematica sappiamo che $f$ è definita e differenziabile per $q \neq 0$ (assumendo che p e q siano primi). + +Ci sono due tipi di errori nella valutazione di f con i valori in virgola mobile + +## Errore Inerente + +Non valutiamo f(x) ma valutiamo $f(\tilde x)$ dove $\tilde x = fl(x)$ [[2. Error Analysis#^59010a|funzione approssimata]] + +$$\varepsilon_{IN} = \frac{f(\tilde x) -f(x)}{f(x)}, \quad f(x) \neq 0$$ + +Se l'errore inerente è relativamente piccolo possiamo dire che il problema è __ben condizionato__, altrimenti che è __mal condizionato__. + +L'errore inerente può essere definito anche per funzioni __non razionali__ + +## Errore Algoritmico + +Non si valuta $f(\tilde x)$ ma si valuta $\tilde f (\tilde x)$ + +$$\varepsilon_{ALG} = \frac{\tilde f (\tilde x ) - f (\tilde x )}{f(\tilde x )}, \quad f(\tilde x ) \neq 0$$ + +Se l'errore algoritmico è relativamente piccolo possiamo dire che l'algoritmo $f$ è __numericamente stabile__, altrimenti che è __numericamente instabile__. + +L'errore algoritmico può essere definito anche per le funzioni elementari che vengono trattate come operazioni. + +## Errore Totale + +$$\varepsilon_{TOT} = \frac {\tilde f(\tilde x ) - f(x)}{f(x)}, \quad f(x) \neq 0$$ +Dà una misura genuina dell'errore nella valutazione. + +La situazione ideale è $|\varepsilon_{tot}| < u$, ma in pratica è sufficiente $|\varepsilon_{tot}| < Mu$ con M costante. + +```ad-important +title: Teorema + +Dato $x \in R^n \backslash \{0\}$ e $f : R^n \to R$ razionale con $f(x) \neq 0$, dove $\tilde x = fl(x)$, allora + +$$\varepsilon_{TOT} = \varepsilon_{IN} + \varepsilon_{ALG} + \varepsilon_{IN}\varepsilon_{ALG}$$ + +``` + +Se $\varepsilon_{IN}$ e $\varepsilon_{ALG}$ tendono a 0 con $u \to 0$, abbiamo +$$\varepsilon_{TOT} = \varepsilon_{IN} + \varepsilon_{ALG} + o(u) = \varepsilon_{IN} + \varepsilon_{ALG}$$ + +``````ad-example +collapse: open + +Sia $\tilde x = fl(x) = x(1 + \varepsilon_1), \quad |\varepsilon_1| < u$ ottenuto come +$$\frac {\tilde x - x}x = \varepsilon_1 \iff \tilde x - x = x\varepsilon_1 \iff \tilde x = x(1 + \varepsilon_1)$$ + +Abbiamo per $x \neq 0$: + +$$\varepsilon_{IN} = \frac {\tilde x^2 - x^2}{x^2} = \frac{[x(1+\varepsilon_1)]^2 - x ^2}{x^2} = \frac{x^2(1 + \varepsilon_1)^2 - x^2}{x^2} = \frac{x^2[(1 + \varepsilon_1)^2 - 1]}{x^2} = $$ +$$ = 2 \varepsilon + \varepsilon^2$$ + +Dato che a noi interessa quello che succede con $u \to 0$, possiamo considerare solo i termini più lenti. + +$$ |\varepsilon_{IN}| = |2\varepsilon + \varepsilon^2| \leq 2|\varepsilon| + |\varepsilon^2| < 2u + u^2 = 2u, \quad u \to 0$$ + +è [[4. Problemi Ben Posti#^3243f8|ben condizionato]]. + +__Esiste un'altra formula__ per calcolare l'errore inerente: + +$$\varepsilon_{IN} = \frac x{f(x)}f'(\xi)\varepsilon_x$$ + +Dove $\varepsilon_x = \varepsilon_1$ è la rappresentazione dell'errore in x se $f \in C^2(conv(x, \tilde x))$ allora: +$$\varepsilon_{IN} = \frac x{f(x)} f'(x)\varepsilon_x + o(u) = \frac {x \cdot 2x}{x^2}\varepsilon_x = 2\varepsilon_x$$ + +```ad-done +title: Dimostrazione + +Uso il teorema di Lagrange applicato alla funzione in un intorno di 0 + +$$f(x) = f(0) + f'(0)x + o(x) \doteq f(0) + f'(0)x$$ + + +``` + +```ad-important + +Il termine $\varepsilon_{IN} = \frac{f(\tilde x) - f(x)}{f(x)} \doteq \frac x{f(x)}f'(x)\varepsilon_x$ si chiama __fattore di amplificazione__ e misura l'amplificazione dell'errore + +``` + + +`````` + +^f13037 + + + +# 4. Problemi Ben Posti + + +Un problema è composto di tre parti: + +- Dati +- Incognite +- Condizioni +```ad-example + +Sistemi lineari + +Data una matrice A (coefficienti) e un vettore b (lato destro), trovare tutti i vettori x (incognite) tali che $Ax = b$ + +``` + +Le incognite sono funzioni dei dati e nella pratica, possono essere presentate solo delle approssimazioni. + +```ad-important +title: Problemi ben posti + +- Hanno una soluzione +- La soluzione è unica +- La soluzione dipende continuamente dai dati + +La dipendenza continua dai dati è meno ovviamente importante: +- I dati nei problemi reali sono affetti da errori +- I calcoli vengono eseguiti da un'aritmetica finita e ci sono errori di arrotondamento + +``` + +^3243f8 + +# Dipendenza Continua Dai Dati + + + +# 5. Derivata di più variabili + + +# Derivata Di Fréchet + +$$f:R^n \to R$$ +$$Df(x_0) : R^2 \to R$$ +Se $x = (x_1,...,x_n), \quad x_i \neq 0, \quad f(x) \neq 0$ l'[[3. Errore Per Le Funzioni Razionali#Errore Inerente|errore inerente]] è: +$$\varepsilon_{IN} \doteq c_1\varepsilon_1 + ... + c_n\varepsilon_n, \quad c_i = \frac {x_i}{f(x)}\frac {\partial f} +{\partial x_i}(x), \quad \varepsilon_i = \frac {\tilde x_i - x_i}{x_i}$$ + +dove $\varepsilon_i$ è l'[[3. Errore Per Le Funzioni Razionali|errore di rappresentazione]] e $c_i$ il [[3. Errore Per Le Funzioni Razionali#^f13037|coefficiente di amplificazione]] + +$$\varepsilon_{IN} \doteq \frac {x_1}{f(x)}\frac {\partial f} +{\partial x_1}(x)\varepsilon_1 + ... + \frac {x_n}{f(x)}\frac {\partial f} +{\partial x_n}(x)\varepsilon_n$$ + +```ad-example +collapse: open + +Data $f(x,y) = xy$, la moltiplicazione. Allora $\varepsilon_{IN} \doteq c_x\varepsilon_x + c_y\varepsilon_y$, dove + +$$c_x = \frac x{f(x,y)} \frac {\partial f}{\partial x}(x,y) = \frac x{xy} y = 1,\quad c_y = \frac y{f(x,y)}\frac{\partial f}{\partial y}(x,y) = \frac y{xy}x = 1$$ + +Abbiamo quindi +$$c_x = 1, \quad c_y = 1$$ +E il problema è [[4. Problemi Ben Posti#^3243f8|ben posto]]. +``` + +```ad-example +collapse: open + +Data $f(x,y) = x + y$ la somma. Allora $\varepsilon_{IN} \doteq c_x\varepsilon_x + c_y\varepsilon_y$, dove +$$c_x = \frac x{f(x,y)} \frac {\partial f}{\partial x}(x,y) = \frac x{x + y},\quad c_y = \frac y{f(x,y)}\frac{\partial f}{\partial y}(x,y) = \frac y{x + y}$$ + +Abbiamo quindi +$$c_x = \frac x{x+y}, \quad c_y = \frac y{x + y}$$ +E il problema diventa mal posto quando $x \approx -y$ + +``` + +^27d1db + +[[#^27d1db|Questo fenomeno]] si chiama __cancellazione numerica__ +$$|\varepsilon_{IN}| = |\frac{x\varepsilon_x + y\varepsilon_y}{x + y}| \leq \frac{|x\varepsilon_x| + |y\varepsilon_y|}{|x+y|}$$ + + +# 6. Valutazione di un polinomio + + + +Dati $p$ e $x$, calcolare $p(x)$ + +- $R^n(K^n)$ vettori con $n$ componenti in $R(K)$ +- $R_n(x)(K_n(x))$ polinomi di grado al più n con coefficienti reali +- $R^{nxm}(K^{nxm})$ matrici n x m con elementi in R +- $R^{n_1 x n_2 x ... x n_l}$ tensore di $l$ dimensioni +- $p(x) = a_0 + a_1 x + ... + a_n x^n$ + +In computer grafica quasi tutto dipende da delle curve, le superfici possono essere curve, ecc... + +Il calcolatore per disegnare il grafico di una funzione conta numerosi puntini all'interno del grafico della funzione e ne valuta il valore. + +$$\sum_{i=0}^n{a_ix^i}$$ + + s = 0 + for i = 0:n + s = s + a(i) * x^i + end + +ho $\frac{(n+1)(n+2)}2$ operazioni + + s = 0 + p = 1 + for i = 0:n + s = s + a(i) * p + p = p * x + end + +il numero totale è $3(n+1)$ operazioni + +# Algoritmo Di Ruffini-Horner + +$$a_0 + a_1 x + a_2 x^2 + a_3 x^3$$ + +$$((a_3 x + a_2)x + a_1) x + a_0$$ + +$$((...((a_n x + a_{n - 1})x + a_{n-2})x + ...)x + a_1)x + a_0$$ + +$$s_n = a_ns_i =s_{i + 1}x + a_i$$ + +$$\begin{cases} s_n = a_n \\ s_i = s_{i+1}x+a_i\end{cases}\quad \text{metodo di Ruffini-Horner}$$ + + s = a(n) + for i = n-1:0 + s = s * x + a(i) + end + +$2n$ operazioni + +```ad-check +title: Dimostrazione per induzione + +Passo base: + +Per $n=0$ il polinomio è $p(x) = a_0$ per ogni $x$, il passo base quindi è verificato. + +Passo induttivo: + +La proposizione è vera per polinomi di grado $n-1 \implies$ vera per polinomi di grado $n$ + +$$p(x) = a_0 + x(a_1 + a_2 x + ... + a_{n-1} x^{n-2} + a_n x^{n-1}) = a_0 + xq(x)$$ +Dove il polinomio $q$ ha grado $n-1$ e può essere scomposto con il teorema di Ruffini-Horner + +``` + +# Funzioni Razionali con Ruffini-Horner + +Il teorema di Ruffini-Horner può essere usato anche per una funzione razionale del tipo:$$r(x)=\frac{p(x)}{q(x)}$$ +Dove $p$ ha grado $m$ e $q$ ha grado $n$, è sufficiente applicare l'algoritmo a $p(x)$ e $q(x)$ separatamente e poi dividere i risultati, richiedendo $2(m+n) + 1$ operazioni aritmetiche. + + + +# 7. Vettori + + +$$w,v \in K^n$$ + +$$v=$$ + +$$w = $$ + +$$v + w = $$ + +# Prodotto Riga per Colonna Tra Due Vettori + +$$v^t \cdot w = [v_1,...,v_n]\begin{bmatrix} +w_1 \\ \vdots\\w_n +\end{bmatrix} = v_1w_1 + v_2 w_2 +... + v_nw_n$$ + +# Prodotto Matrice per Vettore + +$$A \in K^{mxn}$$ +$$b \in K^n$$ +$$Ab = c \in K^m$$ +$$A = $$ + +Faccio il prodotto riga per colonna tra le righe della matrice e il vettore. + +Ci sono anche altre interpretazioni, come quello impiegato nel deep learning + +# Prodotto Tra Matrici Di Strassen + +Algoritmo per semplificare il prodotto tra due matrici 2x2. + +La moltiplicazione riga per colonna impiega 8 moltiplicazioni e 4 addizioni. + +L'algoritmo di Strassen impiega 7 moltiplicazioni e 18 addizioni. + +Può sembrare che impieghi un tempo peggiore rispetto alla riga per colonna, ma la moltiplicazione tra due vettori impiega molto di più che l'addizione, rendendo quindi l'algoritmo più veloce. + +L'algoritmo viene poi trasformato in moltiplicazioni in blocchi. Rendendo qualsiasi moltiplicazione tra matrici $2^nx2^n$ come 7 moltiplicazioni e alcune addizioni tra 4 matrici $2^{(n-1)}x2^{(n-1)}$ ricorsivamente. + +Questo ha fatto scaturire una corsa all'algoritmo più veloce per risoluzione di prodotti tra matrici. + +Ma questi algoritmi non vengono usati. + +# Somme Di Elementi in Parallelo + +Ho un vettore di $2^n$ elementi che devono essere sommati tra loro. + +Potrei scorrere tutto il vettore e sommare il primo elemento con il secondo poi con il terzo e così via, però sarebbe troppo lento. Posso quindi sommare in parallelo gli elementi a due a due, creando così un secondo vettore di $2^{n-1}$ elementi e ripeto l'operazione finché non arrivo a un vettore di 1 elemento. Il risultato è lo stesso ma eseguendo le operazioni parallelamente con vettori molto grandi l'algoritmo è molto più veloce. + +# 8. Matrici + + +Il [[6. Valutazione di un polinomio#Algoritmo Di Ruffini-Horner|metodo di Ruffini-Horner]] può essere interpretato come moltiplicazioni di matrici, che quindi può essere eseguito in parallelo impiegando $2 \log_2(n)$ cicli macchina. +$$\begin{bmatrix} +x & a_0\\0&1 +\end{bmatrix}\begin{bmatrix} +x & a_1\\0&1 +\end{bmatrix} ... \begin{bmatrix} +x & a_n\\0&1 +\end{bmatrix}\begin{bmatrix} +0\\1 +\end{bmatrix}$$ +Queste matrici hanno una struttura particolare per cui si possono effettuare meno operazioni, aggiungendo che queste operazioni possono essere eseguite in parallelo non si ottiene un considerevole vantaggio. +# Matrice Sparsa + +Una matrice che ha molti 0: + +$$A \in K^{nxn}$$ + +$$numZeri(A) << n^2$$ + +si riesce a implementare il [[7. Vettori#Prodotto Matrice per Vettore|prodotto matrice-vettore]] con $2l$ operazione anziché con $2n^2$ + + + +# 9. Altre Strutture + + + +- Sottospazi vettoriali +- Matrici diagonali (prodotto matrice-vettore, prodotto righe per colonne) + +# Matrici Diagonali + +$$\begin{bmatrix} +1 & 0 & 0\\ +0 & 1 & 0\\ +0 & 0 & 1 +\end{bmatrix}$$ + +$$D_n = \{A \in K^{nxn},, a_{ij} = 0, i \neq j\}$$ + +``````ad-question +title: Le matrici diagonali sono sparse? + +$$ numZeri(A) \leq n$$ + +$$\lim\limits_{n \to \infty}{\frac {numZeri(A)}{n^2}} = 0$$ + +```ad-check + +Sono strutture sparse, sono sufficienti $2p + +``` +`````` + +``````ad-question +title: è un sottospazio vettoriale? + +$$A,B \in D_n, \quad \alpha \in K \implies A + B \in D_n, \quad \alpha A \in D_n$$ + +```ad-check +title: Somma tra matrici + +Per dimostrare la somma basta dimostrare che $c_{ij} = 0, i \neq j$: +$$c_{ij} = a_{ij} + b_{ij} = 0 + 0 = 0, i \neq j$$ + + +``` +```ad-check +title: Moltiplicazione per uno scalare + +Stessa cosa della somma: +$$c_{ij} = \alpha \cdot a_{ij} = \alpha \cdot 0 = 0, i \neq j$$ + +``` + +`````` + +``````ad-question +title: è una sottoalgebra di $K^{nxn}$? + +Un'algebra è un sottospazio vettoriale in cui è definito un prodotto compatibile con le operazioni dello spazio vettoriale. + +V è un gruppo se dati $A,B,C \in V$ e $\alpha \in K$: +- $A(B + C) = AB + AC$ +- $(A + B)C = AC + BC$ +- $\alpha(AB) = (\alpha A)B = A(\alpha B)$ + +Le matrici nxn sono un'algebra con il prodotto riga per colonna + +```ad-check +title: Sottoalgebra + +$$A,B \in D_n \implies AB \in D_n$$ +$$C := AB$$ +$$c_{ij} = \sum_{l = 1}^n{a_{il}b_{lj}} = a_{ii}b_{ij}$$ + + + +``` + + +`````` + +## Algoritmi per Il Prodotto Tra Matrici Diagonali + +### Algoritmo Non Strutturato + + for i = 1:n + for j = 1:n + c(i,j) = 0 + for l = 1:n + c(i,j) = c(i,j) + a(i,l) * b(l,j) + + +L'algoritmo strutturato si può ottenere spesso partendo dal codice di quello non strutturato + +## Algoritmo Strutturato + + c = 0 + for i = 1:n + c(i,i) = a(i,i) * b(i,i) + +# Strutture E Algoritmi Strutturati + +Strutture: sottospazi vettoriali, sottoalgebre + +Algoritmo strutturato: algoritmo che sfrutta la struttura + +```ad-example +collapse: open + +- Matrici diagonali: sottoalgebra +- Matrici triangolari: sottoalgebra +- Matrici tridiagonali: sottospazio vettoriale + +``` + + +# a0. Sistemi lineari + + +$$Ax=b$$ + +$$A \in K^{nxn}, x \in K^n, b \in K^n$$ + +$$\begin{cases}a_{1,1}x_1 +. .. a_{1,n}x_n=b_1 \\ a_{2,1}x_1 +. .. a_{2,n}x_n=b_2 \\\vdots\\ a_{n,1}x_1 +. .. a_{n,n}x_n=b_n \end{cases}$$ + +# Teorema Di Rouché-Capelli + +Il sistema lineare Ax = b ammette soluzione se e solo se rango(A) = rango(A|b). + +Se esiste almeno una soluzione allora l'insieme delle soluzioni è un sottospazio affine di dimensione n-rango(A). + +$K^n$ spazio vettoriale + +v sottospazio vettoriale di dimensione l $o \leq l \leq n$ + +$v_0 \in K^n$ la traslazione di $v_0, t_0$ è l'applicazione + +$$t_{v_0}:K^n \to K^n$$ + +$$v \to v+v_0$$ + +L'insieme $W=\{v+v_0, v \in V, v_0 \in K^n\}$ è un sottospazio affine di $K^n$ di dimensione dim(V). + +```ad-info + +- I sottospazi affini di dimensione 1 sono detti rette +- I sottospazi affini di dimensione 2 sono detti piani +- I sottospazi affini di dimensione n-1 sono detti iperpiani +- I sottospazi affini di dimensione 0 sono detti punti (non sono sottospazi) +``` + +$Ax=b \iff c_1x_1+c_2x_2+...+c_nx_n=b \iff$ b è combinazione lineare delle colonne di A $\iff$ rango(A)=rango(A|b) + +$$|c_1|...|c_n|=\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}$$ + +# Problema Ben Posto + +- una soluzione esiste +- la soluzione è unica +- la soluzione dipende in modo continuo dai dati +la soluzione è unica $\iff rk(A)=rk(A|b)=n \iff n-rk(A)=0$ +```ad-important +title: Teorema + +Il sistema lineare Ax=b è ben posto se e solo se A è quadrata e invertibile +``` + +## Caso 1 + +$mn \quad rk(A)=rk(A|b)=n \iff$ la soluzione non è unica, anche se il sistema ammette una soluzione unica per un dato b, esiste $c: \forall \varepsilon >0$ il sistema $Ax=c+\varepsilon c$ non ammette soluzioni $\implies$ la soluzione non può dipendere in modo continuo dai dati +$$Im(A) =\{v \in K^n, v=Ax, x\in R^n\} \subseteq K^n$$ +$$dim(Im(A))=rk(A)\leq n0$$ +per ogni $\varepsilon >0$ questo sistema non ha soluzione se esiste $y: Ay=b+c\varepsilon$ +$$Ay=Ax+c\varepsilon \quad Ay-Ax=c\varepsilon \quad A(\frac{y-x}{\varepsilon})=c$$ +è una contraddizione. + +## Caso 3 + +$m=n\quad det(A)=0 \implies$ la soluzione non esiste o non è unica $\implies$ non è ben posto + +## Caso 4 + +A quadrata, $m=n\quad det(A) \neq 0 \implies$ la soluzione è unica. +Dipende in modo continuo dai dati? +$$x=A^{-1}b\quad (A^{-1})_{ij}=\frac{det(\bar{A}^{ij})}{det(A)}(-1)^{i+j}$$ +$$x_i=\sum_{l=1}^n{(A^{-1})_{il}b_l}=\sum_{l=1}^n{\frac{det(\bar{A}^{il})}{det(A)}(-1)^{i+l}b_l},\quad i=1,..,n$$ +è una funzione continua di $a_{il}, b_l$ poiché è razionale + +## Algoritmi Di Risoluzione + +### Caso Facile (matrici triangolari) + +$$T_n=\{A\in K^{nxn}, a_{ij}=0, i>j\}$$ +$$\begin{bmatrix}a_{1,1}&...&a_{1,n}\\\vdots &\ddots &\vdots\\ 0 &...& a_{n,n} \end{bmatrix}$$ +Sistema lineare Tx=b dove T è triangolare superiore +$$\begin{cases}t_{1,1}x_1 + t_{1,2}x_2 + t_{1,3}x_3=b1 \\ \quad\quad\quad\quad t_{2,2}x_2+t_{2,3}x_3=b_2 \\ \quad\quad\quad\quad\quad\quad\quad\quad t_{3,3}x_3=b_3 \end{cases}$$ +Calcolo il termine i-esimo +$$t_{i,i}x_i+t_{i,i+1}x_{i+1}+...+t_{i,n}x_n=b_i$$ +$$x_i=\frac{b_i-t_{i,i+1}x_{i+1}-...-t_{i,n}x_n}{t_{i,i}}=\frac{b_i - \sum_{j=i+1}^n{t_{i,j}x_j}}{t_{i,j}}$$ +$$Tx=b \implies t_{i,1}x_1+t_{i,2}x_2+...+t_{i,i}x_{i}+...=b_i$$ +$$\sum_{j=i}^n{t_{i,j}x_j}=b_i$$ + +### Algoritmo Di Sostituzione All'indietro + + for i=n:-1:1 + s=b(i) + for j= i+1:n + s = s - t(i,j) * x(j) + x(i) = s / t(i,i) + +costo di un passo del ciclo esterno: 2(n-i)+1 ops +$$\sum_{i=1}^n{2(n-i)+1} = 2\sum_{i=1}^n{2n-2i+1}=2\sum_{i=1}^n{n}-2\sum_{i=1}^n{i}+\sum_{i=1}^n{1}=2n^2-n(n+1)+n=n^2 ops$$ + +```ad-note + +L'algoritmo fallisce se $t_{i,i}=0$ + +Si può dimostrare che $det(T)=t_{1,1}t_{2,2}...t_{n,n}$ + +$det \neq 0 \iff t_{i,i} \neq 0 \iff$ l'algoritmo è applicabile +``` + +### Algoritmo Di Sostituzione in Avanti + +Per sistemi triangolari inferiori +$$\begin{cases} +t_{1,1}x_1&&&&=b_1\\ +t_{2,1}x_1&+t_{2,2}x_2&&&=b_2\\ +t_{3,1}x_1&+t_{3,2}x_2&+t_{3,3}x_3&&=b_2\\ +t_{4,1}x_1&+t_{4,2}x_2&+t_{4,3}x_3&+t_{4,4}x_4&=b_2\\ +\end{cases}$$ +$$\begin{bmatrix} +*&0&0&0\\ +*&*&0&0\\ +*&*&*&0\\ +*&*&*&* +\end{bmatrix}$$ + + for i = 1:n + s = b(i) + for j = 1:i-1 + s = s - t(i,j) * x(j) + x(i) = s / t(i,i) + +# a1. Algoritmo di Gauss + + +22-11-2022 + +Ax=b, A quadrata e invertibile + + for h=1:n-1: + for i = h+1:n: + l(i,h)=a(i,h) / a(h,h) + a(i,h) = 0 + for j=h+1:n: + a(i,j) -= l(i,h) * a(h,j) + b(i) -= l(i,h) * b(h) + +L'algoritmo per come è implementato richiede + +$$\sum_{h=1}^{n-1}{2(n-h)^2} = \frac 23n^3$$ + + +$$\begin{bmatrix} +a_{11},&a_{12},&a_{13},&\dots \\ +a_{21},&a_{22},&a_{23},&\dots \\ +\vdots,&\vdots,&\vdots,&\ddots +\end{bmatrix}$$ +Il primo passo l'algoritmo ha come pivot $a_{11}$ +Non ce se la fa a staglie dietro mannaggia + +Dato $A\in K^{nxn}$ il minore principale di testa di ordine $j \leq h \leq n$ di A è la matrice $M_h \in K^{nxn}: (M_n)_{ij} = a_{ij}$ + +```ad-important +title: Teorema + +Sia $A \in K^{nxn}$ e siano $M_1,...M_n$ i suoi minori principali di testa. Il metodo di Gauss è applicabile se e solo se $det(M_h) \neq 0, h=1,...,n-1$ + +``` + +``````ad-attention +title: Dimostrazione + +Supponiamo il metodo di Gauss sia applicabile $\implies$ si può ottenere la matrice $$U=A_n=\begin{bmatrix} +a_{11} & * & \\ +& a_{22} & & \\ +&& a_{33} &\\ +& 0 && a_{44} +\end{bmatrix}$$ +Sia $N_h$ il minore principale di testa di U di ordine h, esso è una matrice triangolare e il suo determinante è dato dagli elementi sulla sua diagonale, quindi $det(N_h) = u_{11} u_{22} ... u_{hh}$ + +Siccome il metodo è applicabile allora $$a_{ii} \neq 0, i=1,...,n-1 \implies a_{11} ... a_{hh} \neq 0\implies det(N_h) \neq 0, h=1,...,n-1$$ + +```ad-done + +Si ha che $det(N_h) = det(M_h) \implies det(M_h) \neq 0, h=1,...,n-1$ poiché le operazioni (le combinazioni di righe) del metodo di Gauss non cambiano il determinante della matrice a cui si applica, ma non cambia neanche quello dei suoi minori principali di testa +``` +```ad-failure + +Se il metodo di Gauss fallisce al passo h allora $a_{hh}=0 \quad(a_{jj} \neq 0, j=1,..,h-1)$ e si può costruire la matrice +$$A_h = \begin{bmatrix} +a_{11} & & & &\\ +& \ddots & & & \\ +& & a_{hh}=0 & \\ +& & & & & +\end{bmatrix}$$ +Sia $N_h$ il minore principale di testa di ordine h di $A_h$, $det(N_h)=0$ + +Ma $det(M_h)=det(N_h)$ per quanto detto prima, quindi il metodo non è applicabile +``` +`````` + +Se A è una matrice definita positiva allora la condizione del teorema è verificata + +$A \in R^{nxn}$ è definita positiva se: +- $A^T=A$ (simmetrica) +- $v^T Av > 0$ se $v \in R^n \setminus \{0\}$ + +Se A è a dominanza diagonale allora la condizione del teorema è verificata + +A è a dominanza diagonale (stretta) se +- $|a_{ii}| > \sum_{j=1, j\neq i}^n{|a_{ij}|}$ + +A è a dominanza diagonale se: +- $|a_{ii}| \geq \sum_{j=1, j\neq i}^n{|a_{ij}|}$ +- vale il maggiore stretto per almeno un indice + +# Sistema lineare +Come si risolve un sistema lineare? +$$A_1=A\to A_2 \to A_3 \to ... \to A_n = U$$ +$$[A|b] = [A_1|b_1] \to [A_2|b_2]$$ + + for h=1:n-1: + for i = h+1:n: + l(i,h)=a(i,h) / a(h,h) + for j=h+1:n: + a(i,j) = a(i,j) - l(i,h) * a(h,j) + a(i,h) = 0 + b(i) = b(i) - l(i,h) * b(h) + +Si può interpretare il metodo di Gauss come operazioni sulle equazioni +$$Ax=b \iff A_1x=b_1 \iff ... \iff A_nx=b_n \iff Ux=\tilde{b}$$ +Applicando il metodo di Gauss si parte da Ax=b e si ottiene un sistema triangolare equivalente $Ux=\tilde{b}$ che si può risolvere con il metodo di [[a0. Sistemi lineari#Algoritmo Di Sostituzione All'indietro|sostituzione all'indietro]] + +Qual è il costo computazionale? +$\frac 23 n^3$ per l'eliminazione +$n^2$ per per la soluzione di $Ux=\tilde{b}$ +$\sum_{h=1}^n{2(n-h) \sim n^2}$ per il calcolo di $\tilde{b}$ + +```ad-note +$det(A_1) = det(A_2) = ... = det(U)$ perché le operazioni che eseguiamo non modificano il determinante. +è sufficiente quindi applicare il metodo di Gauss con $\frac 23 n^3$ operazioni (e n-1 moltiplicazioni che si possono trascurare) + +``` + +# Sistemi Lineari Con Membro Destro Multiplo + +$$Ax=b_1, Ax=b_2,...,Ax=b_l$$ +risolvendo indipendentemente i sistemi lineari sono richieste $\frac 23 n^3 l$ operazioni. + +Applico il metodo di Gauss ad A modificando consistentemente anche le colonne relative ai termini noti +$$[A|b_1b_2...b_l]=[A_1|b_1^{(1)}b_2^{(1)},...b_l^{(1)}] \to [A_2|b_1^{(2)}b_2^{(2)}...b_l^{(2)}] \to ... \to[A_n|b_1^{(n)}b_2^{(n)}...b_l^{(n)}]=[U|\tilde{b_1} \tilde{b_2}...\tilde{b_3}]$$ +$$Ux=\tilde{b_1},Ux=\tilde{b_2},...,Ux=\tilde{b_l}$$ +Risolvendo in "parallelo" $\frac 23 n^3l$ operazioni. + +Quante operazioni in più al primo passo? $2l(n-1)$ +al secondo passo? $2l( n-2)$ +al passo i? $2l(n-i)$ +in totale saranno $\sum_{i=1}^{n-1}{2l(n-i)}$ +$$\sum_{i=1}^{n-1}{2l(n-i)} = 2l \sum_{i=1}^{n-1}{(n-i)}\sim\frac {2ln^2}2 = ln^2$$ +$\frac 23 n^3 + ln^2$ operazioni anziché $\frac 23 n^3 l$ per trovare i sistemi $Ux=\tilde{b_1},...,Ux=\tilde{b_l}$ per risolverli sono necessarie l sostituzioni all'indietro e quindi altre $ln^2$ operazioni. + +In totale sono $\frac 23 n^3 + 2n^2 l$ operazioni per risolvere l sistemi lineari con sistemi coefficienti + + +## Calcolare l'inversa di una matrice +X è l'inversa di A (nxn) +se AX=I +$$A[X_1|X_2|...|X_n] = [F_1|F_2|...|F_n]$$ +$$[AX_1|AX_2|...|AX_n] = [F_1|F_2|...|F_n]$$ +$$AX_1=F_1,AX_2=F_2,...,AX_n=F_n$$ +```ad-note +title: Nota + +Il problema di calcolo dell'inversa si riconduce ad un problema di risolvere un sistema lineare con membro destro multiplo + +``` + +Utilizzando l'algoritmo precedente ci vogliono +$$\frac 23 n^3 + 2n^2n = \frac 83 n^3$$ +In realtà si può calcolare l'inversa con $2n^3$ operazioni + +# a2. Matrici tridiagonali + + +$$A=\begin{bmatrix} +*&*&&&&\\ +*&*&*&&&\\ +&*&*&*&&\\ +&&*&*&*&\\ +&&&*&*&*\\ +&&&&*&*&*\\ +&&&&&*&* +\end{bmatrix}$$ +$T_n=\{A\in K^{nxn}, a_{ij}=0, |i-j| > 1\}$ è un sottospazio vettoriale poiché è una def. dalla posizione degli elementi nulli se sommo o moltiplico per uno scalare elementi nulli rimangono nulli. + +```ad-fail +title: sono una sottoalgebra? + +è vero che $A,B \in T_n \implies AB \in T_n$? +$$ + +\begin{bmatrix} + +1&1&0\\ + +1&1&1\\ + +0&1&1 + +\end{bmatrix} +\begin{bmatrix} + +1&1&0\\ + +1&1&1\\ + +0&1&1 + +\end{bmatrix} += \begin{bmatrix} +2&2&1\\ +2&3&2\\ +1&2&2 +\end{bmatrix} + +$$ + +Come si vede dall'esempio, il prodotto tra due matrici tridiagonali non ha come risultato una matrice tridiagonale, quindi non è una sottoalgebra + +``` + +```ad-question +title: Qual è la dimensione vettoriale? + + +$$3n-2$$ +``` + +Trovare un algoritmo che calcola il prodotto di una matrice diagonale $A \in T_n$ per un vettore $b \in K^n$ e che non richieda più di 6n operazioni e darne un'implementazione in pseudocodice (non usare il prodotto matrice sparsa-vettore) + +Primo approccio: scrivo la matrice e il vettore e vedo cosa succede + +Secondo approccio: scrivo il componente $c_i$ + +algoritmo mio + + prodotto(A,b): + c[1] = A[1,1] * b[1] + for i=:2:n: + c[i-1] += A[i-1,i] * b[i] + c[i] = A[i,i-1] * b[i-1] + A[i, i] * b[i] + + +# a3. Fattorizzazione LU + + + +Algoritmo di Gauss $\iff$ fatt. LU + +$$A_1\to A_2 \to ...\to U=A_n = \begin{bmatrix} +* & * & * & * \\ +0&* &*&*\\ +0&0&*&*\\ +0&0&0&* +\end{bmatrix}$$ +costruiamo una matrice L triangolare inferiore con 1 sulle diagonali tale che $L_{ij}$ è definito dal metodo di Gauss $i>j$ +$$L=\begin{bmatrix} +1&0&0\\ +l_{ij}&1&0\\ +l_{ij}&l_{ij}&1 +\end{bmatrix}$$ +$$l_{ih}=\frac{a_{ih}}{a_{hh}} \quad l_{ij}=\frac{a_{ij}}{a_{jj}}$$ +$$a_{ij}^{(h+1)}=\begin{cases} +a_{ij}^{(h)}-l_{ih}a_{hj}^{(h)},& i>h,j\geq h\\ +a_{ij}^{(h)}&alternative +\end{cases}$$ + +``````ad-important +title: Teorema + +$A=LU$ (con le valutazioni viste sopra) + +```ad-check +title: Dimostrazione + +$$A=LU \iff a_{ij}=(LU)_{ij}\quad i,j=1,..,n$$ +$$(LU)_{ij} = \sum_{h=1}^n{l_{ih}u_{hj}}=\sum_{h=1}^{min\{i,j\}}{l_{ih}u_{hj}} = a_{ij}$$ + +$$i>j \quad a_{ij}=\sum_{h=1}^j{l_{ih}u_{hj}}=\sum_{h=1}^j{l_{ih}a_{hj}^{(h)}} = \sum_{h=1}^j{a_{ij}^{(h)} - a_{ij}^{(h+1)}} = a_{ij}^{(1)} - a_{ij}^{(2)} + $$ +$$u_{hj}=a_{hj}^{(h)}\quad a_{ij}^{(h+1)} = a_{ij}^{(h)} - l_{ih}a_{ij}^{(h)}$$ + +``` + + +`````` + +# Implementazione Del Metodo Di Gauss + +[[a1. Algoritmo di Gauss]] + +Si parte dalla matrice completa $[A_1|b_1]$ e si eliminano tutti gli elementi della prima colonna a parte il primo, per fare questo si usa la somma di una riga per il multiplo della prima. Il valore per cui si moltiplica è dato dal rapporto dell'elemento nella posizione da eliminare per l'elemento sulla diagonale. + +```ad-example +collapse: open + +Al passo 1, se voglio eliminare l'elemento nella riga 2 ($a_{21}$) devo eseguire l'operazione $r2 = r2 - \frac {a_{21}}{a_{11}}r1$ + +``` + +Bisogna ripetere questi passaggi per ogni colonna usando come pivot la diagonale della matrice + +Il rapporto è poi utilizzato per formare la matrice L, infatti ogni elemento di L: +$$l_{ij} = \frac{a_{ij}}{a_{jj}}$$ + +Il risultato saranno poi due matrici triangolari: +- U triangolare superiore +- L triangolare inferiore + + + + +# Come Si Risolve Un Sistema Lineare? + +$A=LU$ calcoliamo la fattorizzazione $LU$ di $A$ con il metodo di Gauss + +$$Ax=b \iff LUx=b \iff L(Ux) = b\implies \begin{cases} +Ly=b\\ +Ux=y +\end{cases}$$ +Risolviamo il primo sistema $Ly=b$ con l'[[a0. Sistemi lineari#Algoritmo Di Sostituzione in Avanti|algoritmo di sostituzione in avanti]] e poi $Ux=y$ con l'[[a0. Sistemi lineari#Algoritmo Di Sostituzione All'indietro|algoritmo di sostituzione all'indietro]] ottenendo x. + +## Quante Operazioni Richiede? +Il costo totale delle operazioni sulla matrice $A$ è di$$\sum_{k=1}^{n-1}{[2(n-k)^2+(n-k)]}$$ +Di cui: + +- $\frac 23 n^3$ per la fattorizzazione +- $n^2+n^2$ per le due sostituzioni di $\tilde b$ e la soluzione di $Ux = \tilde b$ (trascurabili) + +il costo computazionale è lo stesso ($y= \tilde b$) +$$\begin{bmatrix} +\varepsilon & 1\\ +1&0 +\end{bmatrix} +\begin{bmatrix} +x\\y +\end{bmatrix}= +\begin{bmatrix} +1+\varepsilon\\1 +\end{bmatrix}$$ +il pivot per $\varepsilon >0$ non è nullo e la soluzione è unica $x=\begin{bmatrix}1\\1\end{bmatrix}$ + + +# a3a. Esercizio + + +Descrivere una variante strutturata dell'algoritmo di Gauss che richieda non più di 10n operazioni per la soluzione del sistema lineare $Ax=b$ con A matrice tridiagonale. + +$$[A|b]\to ... \to [U|\tilde{b}]$$ + +$$Ux=\tilde{b}$$ + +$\frac 23n^3$ operazioni algoritmo di Gauss applicato ad A + +$n^2$ operazioni per aggiornare b + +$n^3$ operazioni per risolvere $Ux=\tilde b$ + +A è tridiagonale se $a_{ij} = 0, |i-j|>1$ + +Al primo passo del metodo di Gauss è necessario eliminare gli elementi + +$$a_{21},...,a_{n1}$$ + +Alcuni di essi sono già nulli e quindi alcune operazioni si possono evitare + +$$r^T_2 \gets r^T_2-l_{21}r^T_1$$ + +Nel caso tridiagonale la combinazione di righe modifica solo un elemento e quindi sono richieste solo 2 operazioni (e una divisione per calcolare $l_{21}$) il primo passo richiede 3 operazioni. + +Al secondo passo si osserva che la sottomatrice $B_2$ è tridiagonale, è sufficiente quindi la combinazione lineare + +$$r_3^T \gets r_3^T - l_{32} r_2^T$$ + +che richiede 3 operazioni. + +L'algoritmo richiede 3(n-1) passi in totale + +Implementazione: + +Algoritmo originale di gauss + + for h=1:n-1 + for i=h+1:n + l(i,h) = a(i,h) / a(h,h) + for j=h+1:n + a(i,j)=a(i,j)-l(i,h)*a(h,j) + +Algoritmo modificato + + for h=1:n-1 + l(h+1, h) = a(h+1, h) / a(h,h) + a(h+1,h+1) = a(h+1,h+1) - l(h+1,h)*a(h,h+1) + a(h+1,h=0)=0 + b(h+1) = b(h+1) - l(h+1, h)*b(h) + +Per calcolare U sono sufficienti 3(n-1) operazioni, per calcolare $\tilde b$ sono sufficienti 2(n-1) operazioni + +Risolvere $Ux = \tilde b$ sono sufficienti 3(n-1) + 1 operazioni + +$8n + +$$\begin{bmatrix} +a_{11}&a_{12}&0&...&...&0\\ +&a_{22}&a_{23}&0&...&0\\ +&&\ddots&\ddots&\ddots&\vdots\\ +&&&&&a_{nn} +\end{bmatrix} + +\begin{bmatrix} +x_1\\x_2\\\vdots\\x_n +\end{bmatrix}= +\begin{bmatrix} +\tilde{b_1}\\\tilde{b_2}\\\vdots\\\tilde{b_n}\\ +\end{bmatrix}$$ + +Si può risolvere con O(n) operazioni? + +Modo 1 + +$$i-esima\to\begin{cases} +U_{11}x_1+U_{12}x_2=\tilde{b_1}\\ +U_{22}x_2+U_{23}x_3=\tilde{b_2}\\ +\vdots\\ +U_{ii}x_i+U_{i i+1}x_{i+1}=\tilde{b_i}\\ +\vdots\\ +U_{nn}x_n=\tilde{b_n}\\ +\end{cases}$$ + x(n)=bt(n)/u(n,n) + for i=n-1:-1:1 + x(i) = (bt(i)- u(i,i+1) *x(i+1)) / u(i,i) + +3(n-1) + 1 operazioni $\sim$ 3n operazioni + +[[a3. Fattorizzazione LU]] + +# a4. Variante del Massimo Pivot Parziale (GEPP) + + +```ad-summary +title: Riassunto [[a1. Algoritmo di Gauss]] +- Implementazione +- Costo computazionale $\frac 23 n^3$ +- Risolve molti problemi + - sistemi lineari + - inversa + - determinante +- Equivalente alla fattorizzazione LU +- non è sempre applicabile ma ci sono dei criteri di applicabilità (veri per alcune matrici) + +``` + +$A=LU$ in aritmetica esatta + +$\tilde A = \tilde L \tilde U$ in aritmetica finita, ed esiste $|\tilde {A_{ij}} - A_{ij}| \leq \gamma(n)(|\tilde {A_{ij}} | + |\tilde{L_{ij}}||\tilde{U_{ij}}|)$ + +Quindi il metodo di Gauss è un __algoritmo instabile__, non è adeguato. + +# Implementazione + +Ad ogni passo prima dell'eliminazione si scambiano due righe per ottenere il pivot di massimo modulo. + +Al primo passo si cerca sulla riga/colonna il primo elemento di massimo modulo e se questo elemento non è $a_{11}$ allora si scambia la prima riga con quella dell'elemento di massimo modulo. + +$$A_1=\begin{bmatrix} +a_{11} &...\\ +a_{22} &...\\ +a_{33} &...\\ +\vdots & \vdots\\ +a_{nn} &...\\ +\end{bmatrix} \to |a_{p1}| = max\{|a_{11}|,...,|a_{nn}|\}\to B_1= \begin{bmatrix} +a_{p1} &...\\ +a_{22} &...\\ +a_{33} &...\\ +\vdots & \vdots\\ +a_{nn} &...\\ +\end{bmatrix} \to eliminazione \to $$ +$$\to A_2 = \begin{bmatrix} +a_{11} & a_{12}&...\\ +0 & a_{22}&...\\ +0 & a_{32}&...\\ +\vdots & \vdots & \vdots\\ +0 & a_{n2}&...\\ +\end{bmatrix}\to scambio \to B_2 = \begin{bmatrix} +a_{11} & a_{12}&...\\ +0 & a_{p2}&...\\ +0 & a_{32}&...\\ +\vdots & \vdots & \vdots\\ +0 & a_{n2}&...\\ +\end{bmatrix} \to A_3 = ...$$ + +$argmax\{|a_{11}|,...,|a_{nn}|\}$ è l'insieme degli indici in cui il massimo viene raggiunto + + for h=1: n-1 + p = 1; M = abs(a(h,h)); + for i = h+1: n + m = abs(a(i,h)); + if(m > M) + M = m; + p = i; + if p != h + for j=h:n + swap = a(h, j) + a(h, j) = a(p, j) + a(p, j) = swap + swap = b(h) + b(h) = b(p) + b(p) = swap + for i = k+1 : n + //metodo di gauss + +# Costo Computazionale + +I confronti al primo h sono +$$n-(h+1) + 1 = n-h$$ +E i valori assoluti da calcolare +$$n-h+1$$ +$2(n-h) + 1$ confronti al passo $h$ + +Il numero totale è +$$\sum_{h=1}^{n-1}{2(n-h)+1} \sim n^2$$ +$n^2$ è trascurabile rispetto al costo necessario per l'eliminazione ($\frac 23 n^3$) + +# Applicabilità + +$$\begin{bmatrix} +0 & \dots\\ +0 & \vdots\\ +\vdots & \vdots\\ +0 & \dots +\end{bmatrix}$$ +Al primo passo il metodo si arresta se la prima colonna è nulla $\implies$ A non è invertibile $\iff$ det(A)=0 + + +```ad-important +title: Teorema + +Sia $A \in K^{nxn}$ invertibile (condizione sufficiente). Il metodo di Gauss con la variante del massimo pivot parziale è applicabile ad A. + +``` +```ad-fail +title: Dimostrazione + +Se il metodo fallisce al passo $\ell$ allora A non è invertibile perché si avrà che $a_{ll}=0$, di conseguenza se l'elemento con il pivot massimo è 0, tutti gli altri elementi al disotto sono 0 (il GEPP funziona con i valori assoluti). + +Calcoliamo il det(A) sviluppandolo rispetto alla prima colonna, procedendo in questo modo anche per la colonna 2 e così via. + +Si arriva ad $A_l$ tramite operazioni che non modificano il determinante (combinazioni di righe) e scambi che non modificano il segno +$$|det(A)| = |det(A_l)| = 0 \implies det(A)=0$$ + +Il rango della matrice è quindi al più $k-1 < n$, e quindi non è invertibile, di conseguenza non si può utilizzare il GEPP + +``` + +# Matrici Di Permutazione + +Interpretiamo gli scambi di righe tramite operazioni tra matrici: + +$T \in \mathfrak{S}_n \quad t:\{1,...,n\} \to \{1,...,n\}$ biiettiva +$$\varrho:t \to \Pi$$ +$$f_1=\begin{bmatrix} +1\\0\\\vdots\\0 +\end{bmatrix},f_2=\begin{bmatrix} +0\\1\\\vdots\\0 +\end{bmatrix},...,f_n=\begin{bmatrix} +0\\0\\\vdots\\1 +\end{bmatrix}$$ +```ad-example + +$$\sigma(1)=2,\quad\sigma(2) = 1, \quad n=2$$ + +$$f_1=\begin{bmatrix} +1\\0 +\end{bmatrix},f_2=\begin{bmatrix} +0\\1 +\end{bmatrix}$$ + +$$p(\sigma)=[f_{\sigma(1)}|f_{\sigma(2)}] = [f_2|f_1] =\begin{bmatrix} +0&1\\1&0 +\end{bmatrix}$$ + +Date la matrice $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ e la matrice di permutazione $p$: +$$\begin{bmatrix} +0&1\\1&0 +\end{bmatrix}\begin{bmatrix} +a&b\\c&d +\end{bmatrix}=\begin{bmatrix} +c&d\\a&b +\end{bmatrix}$$ + + +``` + +``````ad-important +title: Teorema + +Sia data la [[a4. Variante del Massimo Pivot Parziale (GEPP)#Implementazione|fattorizzazione LU con GEPP]] esiste una matrice di permutazione $\Pi$ tale che$$\Pi A = LU$$ +La matrice $\Pi$ è ottenuta raccogliendo tutti gli scambi di riga effettuati durante [[a4. Variante del Massimo Pivot Parziale (GEPP)#Implementazione|l'algoritmo]] + +```ad-check +title: Sono invertibili? + +Se vogliamo risolvere il sistema $Ax=b$, le sue soluzioni coincideranno con $\Pi Ax = \Pi b \iff LU=\Pi b$ e quindi il problema si risolve permutando il vettore $b$ e risolvendo la [[a3. Fattorizzazione LU#Come Si Risolve Un Sistema Lineare?|fattorizzazione LU]]. + +``` + + +`````` + +# Calcolo Del Determinante +Oltre che a [[a1. Algoritmo di Gauss#Sistemi Lineari Con Membro Destro Multiplo|risolvere sistemi lineari]] l'Algoritmo di Gauss può essere usato per calcolare il determinante e l'inversa di una matrice . + +L'osservazione sta nel fatto che il metodo di gauss non cambia il determinante della matrice, ma solo il suo segno (Se avvengono degli scambi di righe). + +Quindi, con la variante del massimo pivot parziale, si hanno molti scambi di righe che devono essere contati. Si ottiene quindi che $det(A)=det(U)$ se gli scambi sono pari, e $det(A)=-det(U)$ se gli scambi sono dispari. + +Si può scrivere quindi $det(A)=(-1)^sdet(U)$, con $s$ il numero di scambi di righe. + +Ottenendo la triangolare superiore $U$, il determinante si trova solo sulla diagonale, e quindi $$det(U) = (-1)^s \prod_{i=1}^n{u_{ii}}$$ +Il costo del prodotto è di $n-1$ operazioni, quindi è trascurabile rispetto al metodo di gauss. + + +# a5. Curva di Bezier + + +Dati $P_0,P_1,...,P_n$ punti di controllo, la curva + +$$B(t)=\sum_{i=0}^n{(\substack{n\\i})t^i(1-t)^{n-i}P_i}$$ + +è detta curva di Bezier + +```ad-example +title: Esempi + +## 1 + +N=1 $$B_{0,1}(t)=\substack{\sum_i^1{(\substack{1\\i}) t^i( 1-t)^{1-i}P_i} =\\ +(\substack{1\\0}) t^0( 1-t)^{1} P_0 + (\substack{1\\1}) t^1( 1-t)^{0}P_1 = \\ +1 \cdot 1 (1-t)^1P_0 + 1t(1-t)^0P_1}= +(1-t)P_0 + tP_1$$ + + + +``` + + +# Inviluppo convesso di n+1 punti + +L'inviluppo convesso in n+1 punti in $R^n$ è il più piccolo insieme convesso che li contiene + +# Proprietà delle curve di bezier + +Il vettore tangente (derivata) su B in 0 è $n(v_1 - v_0)$ e ci permette di determinare la retta tangente di $\gamma$ in $\gamma (t_n)$ è $\gamma(t_0) + \gamma' (t_0) t$ + + + + +# a6. Modello della corda elastica + + +1. Trovare la posizione di una corda elastica soggetta a forza esterna. +2. Trovare una funzione che ne deriva la forma. +3. Trovare il valore di questa funzione in alcuni punti. + +n masse uguali e equidistanti collegate tra loro da molle. + +4. Scriviamo l'equazione dell'equilibrio per singola massa + - Ogni massa si può muovere solo in verticale + - L'equazione dell'equilibrio è $F_y=0$ + - Le forze con componenti verticali su $m_i$ sono + - La forza peso + - La forza elastica della massa $m_{i+1}$ + - La forza elastica della massa $m_{i-1}$ + +$$|F|=k(\ell - \ell_0)=k\ell,\quad \ell_0=0$$ + +Se $y_i$ è la posizione verticale di $m_i$ allora la forza che $m_{i+1}$ esercita su $m_i$ ha componente verticale $k(y_{i+1} - y_{i})$ + +La massa $m_{i-1}$ esercita una forza $k(y_{i-1} - y_i)$ + +L'equazione diventa + +$$\begin{cases} +-mg +k(y_{i+1} - y_i) + k(y_{i-1} - y_i) = 0 \\ +-mg + k(y_2 - y_1) + k(-y_1)= 0\\ +-mg+k(-y_n) + k(y_{n-1} - y_n) = 0 +\end{cases}$$ + +Le incognite sono $y_1,..,y_n$ le incognite delle molle +$$\frac {-mg}k + y_{i+1} - y_i + y_{i-1} - y_i = 0$$ +$$\begin{cases} +-y_{i-1} + 2y_i - y_{i+1} = -\frac{mg}k\\ +2y_1 - y_2 = - \frac{mg}k\\ +2y_n - y_{n-1} = - \frac{mg}k +\end{cases}$$ + +Il sistema ha n equazioni e n incognite, è un sistema lineare del tipo $Ax=b$, con A tridiagonale + +# a7. Interpolazione + + +```ad-hint +title: Problema dell'interpolazione polinomiale + +Dati $x_0,..,x_n \in [a,b]$ e dati $y_0,...,y_n \in R$ trovare un polinomio p(x) di grado al più n tale che: +$$p(x_0) = y_0,\quad p(x_1) = y_1,\quad p(x_n) = y_n$$ + +``` + +$$\begin{cases} +a_0+a_1x_0+a_2x_0^2+...+a_nx_0^n=y_0\\ +\vdots\\ +a_0+a_1x_n+a_2x_n^2+...+a_nx_n^n=y_n +\end{cases} \text{è un sistema lineare}$$ +$$V= +\begin{bmatrix} +1 &x_0&x_0^2&...&x_0^n\\ +1&x_1&x_1^2&...&x_1^n\\ +\vdots&\vdots&\vdots&\ddots&\vdots\\ +1&x_n&x_n^2&...&x_n^n +\end{bmatrix} \text{matrice di Vandermorde} +$$ +```ad-note + +Il problema è ben posto se e solo se V è invertibile:: +$$det(V) = \prod_{\substack{i>j\\i,j=0}}^2{(x_i - x_j)} = (x_1 - x_0)(x_2 - x_1) (x_2 - x_0)$$ + +``` + +``````ad-info +title: Corollario + +Il problema dell'interpolazione polinomiale è ben posto se e solo se i nodi sono distinti. + +```ad-success +title: Dimostrazione + +Siccome è un sistema lineare, deve essere $det(V) \neq 0$ ma: +$$det(v) \neq 0 \substack{\iff & x_i-x_j \neq 0 & i > j \\ +\iff& x_i \neq x_j & i \neq j\\ +\iff &\text{i nodi sono distinti}}$$ + +``` + + +`````` +```ad-example +title: Esempio + +$x_0,x_1 \in [a,b], \quad y_0,y_1 \in R$ il polinomio di interpolazione ha grado al più 1, quindi abbiamo soluzione unica + +$$ p(x) = a + bx$$ + +$$ +\begin{cases} +a+bx_0=y_0\\ +a+bx_1=y_1 +\end{cases} \implies +\begin{bmatrix} +1 & x_0\\ +1 & x_1 +\end{bmatrix}\begin{bmatrix} +a\\b +\end{bmatrix}=\begin{bmatrix} +y_0\\y_1 +\end{bmatrix} +$$ +$$ +\begin{bmatrix} +a\\b +\end{bmatrix}=\begin{bmatrix} +1 & x_0\\ +1 & x_1 +\end{bmatrix}^{-1}\begin{bmatrix} +y_0\\y_1 +\end{bmatrix} +$$ +$$ +\frac 1{x_1-x_0} +$$ +``` + +L'utilizzo principale del polinomio di interpolazione + +# Polinomi Di LAGRANGE Associati a $x_0,...,x_n$ Distinti + +- $L_0(x)= \frac{(x-x_1)(x-x_2)...(x-x_n)}{(x_0 - x_1) (x_0 - x_2) ... (x_0 - x_n)}$ +- ... +- $L_i(x) = \frac{(x-x_1)(x-x_2)...}{}$ + +$$L_i(x) = \prod_{\substack{j=0 \\ j\neq i}}^n \frac{x - x_j}{x_i - x_j} \quad i = 1,...,n$$ + +```ad-example +title: Esempio due nodi + +$x_0 = -1 \neq x_1 = 3$ + +$$L_0(x)=\frac{x - 3}{-1 -3} = \frac {x - 3}{-4} \quad\quad L_1(x) = \frac{x - (-1)}{3 - ( - 1)} = \frac{x + 1}{4}$$ + + +``` + +``````ad-tip +title: Teorema + +Siano $x_0,..,x_n \in [a,b]$ distinti e siano $L_0(x),...,L_n(x)$ i polinomi di LAGRANGE allora + +1. $L_i(x_i) = \delta_{ij} : \begin{cases} 1 & i=j \\ 0 & i \neq j \end{cases}$ +2. 2 polinomi di Lagrange sono una base di $R_n[x]$ +3. Dati $y_0,..,y_n \in R$, il polinomio di interpolazione rispetto a $\begin{bmatrix}x_0 \\ y_0\end{bmatrix},...,\begin{bmatrix}x_n \\ y_n\end{bmatrix}$ è + +$$p(x) = \sum_{i=1}^n {y_iL_i(x)}$$ + +```ad-done +title: Dimostrazione + + +1. $L_i(x_j) = \frac{(x_j-x_0) ... (x_j-x_j)...(x_j - x_{i+1})}{...} = 0, \quad i \neq j$ + +$L_i(x_i) = \frac{(x_i - x_0)...(x_i - x_{i-1})(x_i - x_{i+1})}{(x_i - x_0)...(x_i - x_{i-1})(x_i - x_{i+1})} = 1, \quad i = j$ + +2. Siccome $R_n[x]$ ha dimensione n+1 è sufficiente dimostrare che sono linearmente indipendenti +$$\sum {\alpha_i L_i(x)} = 0 \implies \alpha_i = 0\quad i=0,...,n$$ +valutiamo la combinazione lineare in $x_j \quad j = 0,..,n$ +$$0=\sum_{i=0}^n {\alpha_iL_i(x_j)} = \alpha_jL_j(x_j) = \alpha_j$$ + +3. $p(x) = \sum_{i=0}^n{y_iL_i(x)}$ dobbiamo dimostrare che $p(x_j) = y_j$ poiché p ha grado al più n e i nodi sono distinti +$$ p(x_j) = \sum_{i=0}^n {y_i L_i(x_j)} = y_jL_j(x_j) = y_j$$ +``` + + +`````` + +## Algoritmo per Il Calcolo Di p(x) E La Sua Valutazione + +$$ \sum_{i=0} ^ n {y_i\prod_{j \neq i}{\frac{x-x_j}{x_i - x_j}}} = \sum_{i=0} ^ n {y_i\frac{\prod_{j \neq i}{x-x_j}}{\prod_{j \neq i}{x_i - x_j}}}$$ + +# Interpolazione per Approssimare Funzioni + +Si può utilizzare l'interpolazione polinomiale per l'approssimazione di funzioni continue. + +Data $f \in C[a,b]$ e dati $x_0,...,x_n \in [a,b]$ il polinomio di interpolazione è l'unico polinomio di grado __al più $n$__ tale che $p(x_i) = f(x_i)=y_i$. + +Una volta approssimata una funzione $f(x)$ con un polinomio $p_n(x)$ si ha un errore:$$r(x)=f(x)-p_n(x)$$ +detto __resto dell'interpolazione__ + +```ad-important +title: Teorema + +Sia $f \in C^{k+1}[a,b]$ e sia $p_n(x)$ il polinomio di interpolazione relativo a $x_0,...,x_n \in [a,b]$ distinti. Allora $\forall x \in [a,b], \exists \xi (x) \in [a,b] :$ $$r(x)=f(x)-p_n(x)=\frac{f^{(k+1)}(\xi(x))}{(n+1)!}(x-x_0)...(x-x_n)$$ +In generale non è detto che aumentando il numero dei nodi l'approssimazione migliori (anche se la funzione da approssimare è regolare). + +``` diff --git a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.pdf b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3f1f17f8852d3b3dfa9a104c2b16fc7a63855b04 GIT binary patch literal 431444 zcmdSA1#l!wmMth|W>$(BQq0WE%ur%xW@cuVN?c+tF_oA~Tw>-DtAwlfrp>1NZ~s~Q zwqrKCX`UG#5jQ;CJv=?m^>Z$Tl9&VokdY0JV&Uv$2M!2e1~`~l!}0Mki8`AbyE-@n zsO25Jt!!G=L4INBetjw&8?U{g#tc>gcQA=kBJ98!=kd2XC;Z$!R)^_5fFegw>SH% zBUt~lLrjeU%y3L1@&G0U2WLBD+y69T`xhhNzc=Fg7b8X>Gt0lX1OC%9_`KymN)t17 zwlTN=&szdn{>oD z1^BBmDxbAsl5nth{p+j*JAnN^&Ps3qIR2621aST%#|_~AM-Ipg_)LFa{nzy(A`YGa z-MVLHC@7>RK0hrXBjqP0=jh)TyO}*flqyetZZsvd2 z!phe5b3iaj*nU=4%-qz$%p8tM&fMO@)e^wM#?CDu0B~{r9M5)ep1Bv=8}>w;NZkeH 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z?pKi3wVsKbnP&)ys%0Uos78uS^$#+)GX<&ME)RI568nNn&q-c2V$k9bi6f2sN=6P*qy;04Nh^KHCES)E`?l`=dmfF@mTo zu8R#S@OO{xcHuRm+rWPgy?=gx?C)SazFn`|X 0$ + +$\alpha = \sqrt 2$ (soluzione simbolica) + +$\alpha$ limite dell'iterazione + +$$x_{k+1} = \frac {x^2_k + 2 }{2x_k}$$ + +``` + +# Calcolo Simbolico VS Calcolo Numerico + +```ad-example +collapse: open + +$P(x) = 0$ con $p(x) = x^3 + x^2 -x + 1$ ha un'unica soluzione $\alpha$ + +- $\alpha = \frac 13(-1 - \frac 4{\sqrt[3]{19 - 3\sqrt{33}}} - \sqrt[3]{19 - 3\sqrt{33}})$ soluzione simbolica +- $\alpha$ è il limite dell'iterazione:$$x_{k+1} = x_k - \frac {p(x_k)}{p'(x_k)}, \quad x_0 = 2$$ + +``` + +```ad-example +collapse: open + +Eq: $x^5 + x^4 - x + 1 = 0$ unica soluzione $\alpha$ + +- $\alpha$ non esprimibile in forma chiusa con simboli noti (soluzione simbolica) +- $\alpha$ è il limite dell'iterazione:$$x_{k+1} = x_k - \frac {p(x_k)}{p'(x_k)}, \quad x_0 = -2$$ + +``` + +Quasi tutte le civiltà hanno una formula per le equazioni di 2° grado. + +Nel XVI secolo sono state trovate per le equazioni di 3° e 4° grado (Tartaglia, Cardano, Ferrari). + +```ad-important +Teorema di Ruffini-Abel: + +Non esiste una formula per le equazioni di 5° grado + +``` + +La materia si occupa di dare strumenti teorici (teorie matematiche) per il calcolo numerico di quantità continue. + +Si cerca di trovare una soluzione ai problemi continui che sia utilizzabile. Non basta una formula, ma serve un algoritmo ce fornisca una soluzione in un tempo ragionevole tenendo conto che il calcolatore lavora in quantità finite. Vogliamo quindi trattare __quantità infinite__ con __strumenti finiti__. + +# Quantità Continue + +- Una o più equazioni non lineari, radici di polinomi +- Calcolo quantità algebra lineare, soluzione di sistemi lineari +- Determinanti, autovalori/vettori, rango, nucleo, immagine +- Calcolo degli integrali +- Approssimazione di funzioni + +Per ogni problema esistono una o più soluzioni. + +## Efficienza + +Gli algoritmi devono avere importanti requisiti: +- Basso costo computazionale +- [[1. Stabilità numerica|Stabilità numerica]] + diff --git "a/triennale/Anno 2/Calcolo Numerico/argomenti/1. Stabilit\303\240 numerica.md" "b/triennale/Anno 2/Calcolo Numerico/argomenti/1. Stabilit\303\240 numerica.md" new file mode 100644 index 000000000..2225e62f2 --- /dev/null +++ "b/triennale/Anno 2/Calcolo Numerico/argomenti/1. Stabilit\303\240 numerica.md" @@ -0,0 +1,31 @@ +```ad-example +title: Esempio + +## Sviluppo in serie di Taylor di $e^x$ con resto di Lagrange +$$e^x=1+ \frac x{1!} + \frac {x^2}{2!} + \frac {x^3}{3!} + ... = \sum_{k=0}^n {\frac {x^k}{k!}} = e^{\epsilon} \frac{x^n+1}{n+1}!$$ + +Si hanno due algoritmi per il calcolo di $e^x$ +$$e^x \approx \sum_{k=0}^n {\frac {x^k}{k!}}$$ +$$e^x \approx \frac 1{e^{-x}} \approx \frac 1{\sum_{k=0}^n {\frac {(-x)^k}{k!}}}$$ + +Il primo funziona meglio con esponente positivo, il secondo con esponente negativo + +``` + +# Relazioni Con Altre Discipline + +## Matematica Discreta + +Si cerca di utilizzare una sola formula per la risoluzione di problemi, mentre qua ci servono più formule + +# Algoritmi E Strutture Dati + +Si occupa dello studio di problemi discreti, mentre qui ci si occupa di problemi continui + +# Relazione Con Il Mondo Reale + +I problemi del mondo reale possono essere __modellizzati__ in teorie matematiche. + +Le soluzioni matematiche vengono __applicate__ nel mondo reale. + +![[Modellizzazione.png]] diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/2. Error Analysis.md b/triennale/Anno 2/Calcolo Numerico/argomenti/2. Error Analysis.md new file mode 100644 index 000000000..78049a157 --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/2. Error Analysis.md @@ -0,0 +1,142 @@ + +I numeri reali per la maggior parte contengono infinite informazioni, a parte i pochi numeri razionali che ne contengono di limitate. + +# Numeri Razionali + +Sono dei numeri che rispettano la descrizione di un determinato teorema, e che non rispettano altri teoremi come, ad esempio, il teorema degli zeri o quello di Lagrange. + +I numeri razionali sono infiniti, ma noi possiamo lavorare solo con insiemi finiti di numeri. + +```ad-question +title: Problema + +Selezionare un numero finito di valori che possano rappresentare i numeri reali. + +## Distribuzione uniforme + +Scegliere $\varepsilon > 0$, N pari: +$$\varepsilon = \{\varepsilon k: - \frac N2 < k \leq \frac N2, k \in Z\}$$ + +Possiamo rappresentare i numeri reali con l'insieme: +$$ S = \{-\varepsilon\frac N20$ di cifre della mantissa, $m,M$ numeri positivi, definiamo un insieme di __floating point numbers__ + +$$F(\beta, t, m, M) = \{0\} \cup \{\pm \beta^p\sum_{i=1}^t {\beta^{-i}d_i}, \; -m \leq p \leq M, \; 0 \leq d_i < \beta \text{ intero per } i=1,...,t, \; d_1 \neq 0\}$$ + +```ad-example +title: Esempio +collapse: open + +$F := F(10, 2, 2, 3)$ + +Cardinalità: $1 + 2(m+M+1)(\beta - 1)\beta^{t-1}$ + +Max(F): $\Omega=\beta^M\sum_{i=1}^t{\beta^{-i}(\beta - 1)} = \beta^M(1- \beta^{-t})$ + +MinPositivo(F): $\omega = \beta^{-m}\beta^{-1} = \beta^{-m-1}$ + +Per $x \geq \Omega$ i numeri non possono essere rappresentati, mentre per $0 < x < \omega$ c'è un grande errore relativo + +``` + +```ad-example +title: Esempio 2 (Utile per dopo) +collapse: open + +$F(2,2,1,1)$ è composto da 13 numeri + +Questi numeri __non sono uniformi__. Tra $\frac 12$ e $\frac 14$ e tra $\frac 12$ e 1 c'è lo stesso numero di elementi di $F$. + +![[Pasted image 20230125111411.png]] + +Data $S = \{x \in R : \omega \leq x \leq \Omega\}$ costruiamo una __funzione di rappresentazione__ +$$fl : R \to F \cup\{\pm \infty\}$$ + +Con una delle due regole, dato $x=\beta^p \sum_{i=1}^\infty {\beta^{-i}d_i} \in S$: + +- Troncamento: $\tilde x = fl(x) = \beta^p \sum_{i=1}^t {\beta^{-i}d_i}$ +- Arrotondamento: $\tilde x = fl(x)$ il troncamento di $x + \frac{\beta^{p-t-1}}2$ + +Se $x > \Omega$ impostiamo $fl(x) = \infty$ (__overflow__), se $0\leq x \leq \omega$ impostiamo $fl(x) = 0$ (__underflow__), stesso ragionamento per i numeri negativi. + +``` + +^59010a + +Definiamo la __precisione macchina__ come $u=\beta^{-t+1}$ per il troncamento e $u=\frac{\beta^{-t+1}}2$ per l'arrotondamento. + +```ad-important +title: Teorema + +Dato $x \in S$ abbiamo il seguente limite per l'errore relativo: +$$|\frac{fl(x) - x}x| < u$$ + +``` + +__$F$ deve seguire determinate proprietà algebriche__: + +$$x,y \in F \centernot \implies x + y \in F$$ + +Dobbiamo definire delle __operazioni__ con i numeri a virgola mobile. + +Assumendo che esiste una somma a virgola mobile $\oplus$ tale che se $x,y \in F$ (se non avviene overflow): + +$$x\oplus y \in F, \quad x \oplus y = (x+y)(1 + \varepsilon), \quad |\varepsilon| < u$$ + +Similmente definiamo $\otimes, \ominus, \oslash$. + +Un'idea è quella di definire $x \oplus y = fl(x+y)$ ma i dettagli sono più complicati. + +Infatti le varie operazioni seguono solo alcune delle proprietà delle operazioni elementari: + +- commutatività della somma +- commutatività del prodotto +- $x \oslash x = 1$ + +Non seguono infatti: + +- associatività di somma e prodotto +- proprietà distributiva +- semplificazione +- potrebbe succedere che $x \otimes y = z \otimes y, \quad y \neq 0, \quad x \neq z$ + +[[3. Errore Per Le Funzioni Razionali]] \ No newline at end of file diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md b/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md new file mode 100644 index 000000000..8c9fd1285 --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md @@ -0,0 +1,92 @@ + + +Data una funzione razionale $f: R^2 \to R$ che sia $f = \frac pq$ con p e q polinomi. + +Dall'analisi matematica sappiamo che $f$ è definita e differenziabile per $q \neq 0$ (assumendo che p e q siano primi). + +Ci sono due tipi di errori nella valutazione di f con i valori in virgola mobile + +## Errore Inerente + +Non valutiamo f(x) ma valutiamo $f(\tilde x)$ dove $\tilde x = fl(x)$ [[2. Error Analysis#^59010a|funzione approssimata]] + +$$\varepsilon_{IN} = \frac{f(\tilde x) -f(x)}{f(x)}, \quad f(x) \neq 0$$ + +Se l'errore inerente è relativamente piccolo possiamo dire che il problema è __ben condizionato__, altrimenti che è __mal condizionato__. + +L'errore inerente può essere definito anche per funzioni __non razionali__ + +## Errore Algoritmico + +Non si valuta $f(\tilde x)$ ma si valuta $\tilde f (\tilde x)$ + +$$\varepsilon_{ALG} = \frac{\tilde f (\tilde x ) - f (\tilde x )}{f(\tilde x )}, \quad f(\tilde x ) \neq 0$$ + +Se l'errore algoritmico è relativamente piccolo possiamo dire che l'algoritmo $f$ è __numericamente stabile__, altrimenti che è __numericamente instabile__. + +L'errore algoritmico può essere definito anche per le funzioni elementari che vengono trattate come operazioni. + +## Errore Totale + +$$\varepsilon_{TOT} = \frac {\tilde f(\tilde x ) - f(x)}{f(x)}, \quad f(x) \neq 0$$ +Dà una misura genuina dell'errore nella valutazione. + +La situazione ideale è $|\varepsilon_{tot}| < u$, ma in pratica è sufficiente $|\varepsilon_{tot}| < Mu$ con M costante. + +```ad-important +title: Teorema + +Dato $x \in R^n \backslash \{0\}$ e $f : R^n \to R$ razionale con $f(x) \neq 0$, dove $\tilde x = fl(x)$, allora + +$$\varepsilon_{TOT} = \varepsilon_{IN} + \varepsilon_{ALG} + \varepsilon_{IN}\varepsilon_{ALG}$$ + +``` + +Se $\varepsilon_{IN}$ e $\varepsilon_{ALG}$ tendono a 0 con $u \to 0$, abbiamo +$$\varepsilon_{TOT} = \varepsilon_{IN} + \varepsilon_{ALG} + o(u) = \varepsilon_{IN} + \varepsilon_{ALG}$$ + +``````ad-example +collapse: open + +Sia $\tilde x = fl(x) = x(1 + \varepsilon_1), \quad |\varepsilon_1| < u$ ottenuto come +$$\frac {\tilde x - x}x = \varepsilon_1 \iff \tilde x - x = x\varepsilon_1 \iff \tilde x = x(1 + \varepsilon_1)$$ + +Abbiamo per $x \neq 0$: + +$$\varepsilon_{IN} = \frac {\tilde x^2 - x^2}{x^2} = \frac{[x(1+\varepsilon_1)]^2 - x ^2}{x^2} = \frac{x^2(1 + \varepsilon_1)^2 - x^2}{x^2} = \frac{x^2[(1 + \varepsilon_1)^2 - 1]}{x^2} = $$ +$$ = 2 \varepsilon + \varepsilon^2$$ + +Dato che a noi interessa quello che succede con $u \to 0$, possiamo considerare solo i termini più lenti. + +$$ |\varepsilon_{IN}| = |2\varepsilon + \varepsilon^2| \leq 2|\varepsilon| + |\varepsilon^2| < 2u + u^2 = 2u, \quad u \to 0$$ + +è [[4. Problemi Ben Posti#^3243f8|ben condizionato]]. + +__Esiste un'altra formula__ per calcolare l'errore inerente: + +$$\varepsilon_{IN} = \frac x{f(x)}f'(\xi)\varepsilon_x$$ + +Dove $\varepsilon_x = \varepsilon_1$ è la rappresentazione dell'errore in x se $f \in C^2(conv(x, \tilde x))$ allora: +$$\varepsilon_{IN} = \frac x{f(x)} f'(x)\varepsilon_x + o(u) = \frac {x \cdot 2x}{x^2}\varepsilon_x = 2\varepsilon_x$$ + +```ad-done +title: Dimostrazione + +Uso il teorema di Lagrange applicato alla funzione in un intorno di 0 + +$$f(x) = f(0) + f'(0)x + o(x) \doteq f(0) + f'(0)x$$ + + +``` + +```ad-important + +Il termine $\varepsilon_{IN} = \frac{f(\tilde x) - f(x)}{f(x)} \doteq \frac x{f(x)}f'(x)\varepsilon_x$ si chiama __fattore di amplificazione__ e misura l'amplificazione dell'errore + +``` + + +`````` + +^f13037 + diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/4. Problemi Ben Posti.md b/triennale/Anno 2/Calcolo Numerico/argomenti/4. Problemi Ben Posti.md new file mode 100644 index 000000000..23f73e934 --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/4. Problemi Ben Posti.md @@ -0,0 +1,32 @@ +Un problema è composto di tre parti: + +- Dati +- Incognite +- Condizioni +```ad-example + +Sistemi lineari + +Data una matrice A (coefficienti) e un vettore b (lato destro), trovare tutti i vettori x (incognite) tali che $Ax = b$ + +``` + +Le incognite sono funzioni dei dati e nella pratica, possono essere presentate solo delle approssimazioni. + +```ad-important +title: Problemi ben posti + +- Hanno una soluzione +- La soluzione è unica +- La soluzione dipende continuamente dai dati + +La dipendenza continua dai dati è meno ovviamente importante: +- I dati nei problemi reali sono affetti da errori +- I calcoli vengono eseguiti da un'aritmetica finita e ci sono errori di arrotondamento + +``` + +^3243f8 + +# Dipendenza Continua Dai Dati + diff --git "a/triennale/Anno 2/Calcolo Numerico/argomenti/5. Derivata di pi\303\271 variabili.md" "b/triennale/Anno 2/Calcolo Numerico/argomenti/5. Derivata di pi\303\271 variabili.md" new file mode 100644 index 000000000..a25b2bc98 --- /dev/null +++ "b/triennale/Anno 2/Calcolo Numerico/argomenti/5. Derivata di pi\303\271 variabili.md" @@ -0,0 +1,42 @@ +# Derivata Di Fréchet + +$$f:R^n \to R$$ +$$Df(x_0) : R^2 \to R$$ +Se $x = (x_1,...,x_n), \quad x_i \neq 0, \quad f(x) \neq 0$ l'[[3. Errore Per Le Funzioni Razionali#Errore Inerente|errore inerente]] è: +$$\varepsilon_{IN} \doteq c_1\varepsilon_1 + ... + c_n\varepsilon_n, \quad c_i = \frac {x_i}{f(x)}\frac {\partial f} +{\partial x_i}(x), \quad \varepsilon_i = \frac {\tilde x_i - x_i}{x_i}$$ + +dove $\varepsilon_i$ è l'[[3. Errore Per Le Funzioni Razionali|errore di rappresentazione]] e $c_i$ il [[3. Errore Per Le Funzioni Razionali#^f13037|coefficiente di amplificazione]] + +$$\varepsilon_{IN} \doteq \frac {x_1}{f(x)}\frac {\partial f} +{\partial x_1}(x)\varepsilon_1 + ... + \frac {x_n}{f(x)}\frac {\partial f} +{\partial x_n}(x)\varepsilon_n$$ + +```ad-example +collapse: open + +Data $f(x,y) = xy$, la moltiplicazione. Allora $\varepsilon_{IN} \doteq c_x\varepsilon_x + c_y\varepsilon_y$, dove + +$$c_x = \frac x{f(x,y)} \frac {\partial f}{\partial x}(x,y) = \frac x{xy} y = 1,\quad c_y = \frac y{f(x,y)}\frac{\partial f}{\partial y}(x,y) = \frac y{xy}x = 1$$ + +Abbiamo quindi +$$c_x = 1, \quad c_y = 1$$ +E il problema è [[4. Problemi Ben Posti#^3243f8|ben posto]]. +``` + +```ad-example +collapse: open + +Data $f(x,y) = x + y$ la somma. Allora $\varepsilon_{IN} \doteq c_x\varepsilon_x + c_y\varepsilon_y$, dove +$$c_x = \frac x{f(x,y)} \frac {\partial f}{\partial x}(x,y) = \frac x{x + y},\quad c_y = \frac y{f(x,y)}\frac{\partial f}{\partial y}(x,y) = \frac y{x + y}$$ + +Abbiamo quindi +$$c_x = \frac x{x+y}, \quad c_y = \frac y{x + y}$$ +E il problema diventa mal posto quando $x \approx -y$ + +``` + +^27d1db + +[[#^27d1db|Questo fenomeno]] si chiama __cancellazione numerica__ +$$|\varepsilon_{IN}| = |\frac{x\varepsilon_x + y\varepsilon_y}{x + y}| \leq \frac{|x\varepsilon_x| + |y\varepsilon_y|}{|x+y|}$$ diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/6. Valutazione di un polinomio.md b/triennale/Anno 2/Calcolo Numerico/argomenti/6. Valutazione di un polinomio.md new file mode 100644 index 000000000..0e369ba67 --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/6. Valutazione di un polinomio.md @@ -0,0 +1,71 @@ + +Dati $p$ e $x$, calcolare $p(x)$ + +- $R^n(K^n)$ vettori con $n$ componenti in $R(K)$ +- $R_n(x)(K_n(x))$ polinomi di grado al più n con coefficienti reali +- $R^{nxm}(K^{nxm})$ matrici n x m con elementi in R +- $R^{n_1 x n_2 x ... x n_l}$ tensore di $l$ dimensioni +- $p(x) = a_0 + a_1 x + ... + a_n x^n$ + +In computer grafica quasi tutto dipende da delle curve, le superfici possono essere curve, ecc... + +Il calcolatore per disegnare il grafico di una funzione conta numerosi puntini all'interno del grafico della funzione e ne valuta il valore. + +$$\sum_{i=0}^n{a_ix^i}$$ + + s = 0 + for i = 0:n + s = s + a(i) * x^i + end + +ho $\frac{(n+1)(n+2)}2$ operazioni + + s = 0 + p = 1 + for i = 0:n + s = s + a(i) * p + p = p * x + end + +il numero totale è $3(n+1)$ operazioni + +# Algoritmo Di Ruffini-Horner + +$$a_0 + a_1 x + a_2 x^2 + a_3 x^3$$ + +$$((a_3 x + a_2)x + a_1) x + a_0$$ + +$$((...((a_n x + a_{n - 1})x + a_{n-2})x + ...)x + a_1)x + a_0$$ + +$$s_n = a_ns_i =s_{i + 1}x + a_i$$ + +$$\begin{cases} s_n = a_n \\ s_i = s_{i+1}x+a_i\end{cases}\quad \text{metodo di Ruffini-Horner}$$ + + s = a(n) + for i = n-1:0 + s = s * x + a(i) + end + +$2n$ operazioni + +```ad-check +title: Dimostrazione per induzione + +Passo base: + +Per $n=0$ il polinomio è $p(x) = a_0$ per ogni $x$, il passo base quindi è verificato. + +Passo induttivo: + +La proposizione è vera per polinomi di grado $n-1 \implies$ vera per polinomi di grado $n$ + +$$p(x) = a_0 + x(a_1 + a_2 x + ... + a_{n-1} x^{n-2} + a_n x^{n-1}) = a_0 + xq(x)$$ +Dove il polinomio $q$ ha grado $n-1$ e può essere scomposto con il teorema di Ruffini-Horner + +``` + +# Funzioni Razionali con Ruffini-Horner + +Il teorema di Ruffini-Horner può essere usato anche per una funzione razionale del tipo:$$r(x)=\frac{p(x)}{q(x)}$$ +Dove $p$ ha grado $m$ e $q$ ha grado $n$, è sufficiente applicare l'algoritmo a $p(x)$ e $q(x)$ separatamente e poi dividere i risultati, richiedendo $2(m+n) + 1$ operazioni aritmetiche. + diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/7. Vettori.md b/triennale/Anno 2/Calcolo Numerico/argomenti/7. Vettori.md new file mode 100644 index 000000000..a7a088cd5 --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/7. Vettori.md @@ -0,0 +1,46 @@ +$$w,v \in K^n$$ + +$$v=$$ + +$$w = $$ + +$$v + w = $$ + +# Prodotto Riga per Colonna Tra Due Vettori + +$$v^t \cdot w = [v_1,...,v_n]\begin{bmatrix} +w_1 \\ \vdots\\w_n +\end{bmatrix} = v_1w_1 + v_2 w_2 +... + v_nw_n$$ + +# Prodotto Matrice per Vettore + +$$A \in K^{mxn}$$ +$$b \in K^n$$ +$$Ab = c \in K^m$$ +$$A = $$ + +Faccio il prodotto riga per colonna tra le righe della matrice e il vettore. + +Ci sono anche altre interpretazioni, come quello impiegato nel deep learning + +# Prodotto Tra Matrici Di Strassen + +Algoritmo per semplificare il prodotto tra due matrici 2x2. + +La moltiplicazione riga per colonna impiega 8 moltiplicazioni e 4 addizioni. + +L'algoritmo di Strassen impiega 7 moltiplicazioni e 18 addizioni. + +Può sembrare che impieghi un tempo peggiore rispetto alla riga per colonna, ma la moltiplicazione tra due vettori impiega molto di più che l'addizione, rendendo quindi l'algoritmo più veloce. + +L'algoritmo viene poi trasformato in moltiplicazioni in blocchi. Rendendo qualsiasi moltiplicazione tra matrici $2^nx2^n$ come 7 moltiplicazioni e alcune addizioni tra 4 matrici $2^{(n-1)}x2^{(n-1)}$ ricorsivamente. + +Questo ha fatto scaturire una corsa all'algoritmo più veloce per risoluzione di prodotti tra matrici. + +Ma questi algoritmi non vengono usati. + +# Somme Di Elementi in Parallelo + +Ho un vettore di $2^n$ elementi che devono essere sommati tra loro. + +Potrei scorrere tutto il vettore e sommare il primo elemento con il secondo poi con il terzo e così via, però sarebbe troppo lento. Posso quindi sommare in parallelo gli elementi a due a due, creando così un secondo vettore di $2^{n-1}$ elementi e ripeto l'operazione finché non arrivo a un vettore di 1 elemento. Il risultato è lo stesso ma eseguendo le operazioni parallelamente con vettori molto grandi l'algoritmo è molto più veloce. \ No newline at end of file diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/8. Matrici.md b/triennale/Anno 2/Calcolo Numerico/argomenti/8. Matrici.md new file mode 100644 index 000000000..966036dd1 --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/8. Matrici.md @@ -0,0 +1,21 @@ +Il [[6. Valutazione di un polinomio#Algoritmo Di Ruffini-Horner|metodo di Ruffini-Horner]] può essere interpretato come moltiplicazioni di matrici, che quindi può essere eseguito in parallelo impiegando $2 \log_2(n)$ cicli macchina. +$$\begin{bmatrix} +x & a_0\\0&1 +\end{bmatrix}\begin{bmatrix} +x & a_1\\0&1 +\end{bmatrix} ... \begin{bmatrix} +x & a_n\\0&1 +\end{bmatrix}\begin{bmatrix} +0\\1 +\end{bmatrix}$$ +Queste matrici hanno una struttura particolare per cui si possono effettuare meno operazioni, aggiungendo che queste operazioni possono essere eseguite in parallelo non si ottiene un considerevole vantaggio. +# Matrice Sparsa + +Una matrice che ha molti 0: + +$$A \in K^{nxn}$$ + +$$numZeri(A) << n^2$$ + +si riesce a implementare il [[7. Vettori#Prodotto Matrice per Vettore|prodotto matrice-vettore]] con $2l$ operazione anziché con $2n^2$ + diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md b/triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md new file mode 100644 index 000000000..9e369db72 --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md @@ -0,0 +1,110 @@ + +- Sottospazi vettoriali +- Matrici diagonali (prodotto matrice-vettore, prodotto righe per colonne) + +# Matrici Diagonali + +$$\begin{bmatrix} +1 & 0 & 0\\ +0 & 1 & 0\\ +0 & 0 & 1 +\end{bmatrix}$$ + +$$D_n = \{A \in K^{nxn},, a_{ij} = 0, i \neq j\}$$ + +``````ad-question +title: Le matrici diagonali sono sparse? + +$$ numZeri(A) \leq n$$ + +$$\lim\limits_{n \to \infty}{\frac {numZeri(A)}{n^2}} = 0$$ + +```ad-check + +Sono strutture sparse, sono sufficienti $2p + +``` +`````` + +``````ad-question +title: è un sottospazio vettoriale? + +$$A,B \in D_n, \quad \alpha \in K \implies A + B \in D_n, \quad \alpha A \in D_n$$ + +```ad-check +title: Somma tra matrici + +Per dimostrare la somma basta dimostrare che $c_{ij} = 0, i \neq j$: +$$c_{ij} = a_{ij} + b_{ij} = 0 + 0 = 0, i \neq j$$ + + +``` +```ad-check +title: Moltiplicazione per uno scalare + +Stessa cosa della somma: +$$c_{ij} = \alpha \cdot a_{ij} = \alpha \cdot 0 = 0, i \neq j$$ + +``` + +`````` + +``````ad-question +title: è una sottoalgebra di $K^{nxn}$? + +Un'algebra è un sottospazio vettoriale in cui è definito un prodotto compatibile con le operazioni dello spazio vettoriale. + +V è un gruppo se dati $A,B,C \in V$ e $\alpha \in K$: +- $A(B + C) = AB + AC$ +- $(A + B)C = AC + BC$ +- $\alpha(AB) = (\alpha A)B = A(\alpha B)$ + +Le matrici nxn sono un'algebra con il prodotto riga per colonna + +```ad-check +title: Sottoalgebra + +$$A,B \in D_n \implies AB \in D_n$$ +$$C := AB$$ +$$c_{ij} = \sum_{l = 1}^n{a_{il}b_{lj}} = a_{ii}b_{ij}$$ + + + +``` + + +`````` + +## Algoritmi per Il Prodotto Tra Matrici Diagonali + +### Algoritmo Non Strutturato + + for i = 1:n + for j = 1:n + c(i,j) = 0 + for l = 1:n + c(i,j) = c(i,j) + a(i,l) * b(l,j) + + +L'algoritmo strutturato si può ottenere spesso partendo dal codice di quello non strutturato + +## Algoritmo Strutturato + + c = 0 + for i = 1:n + c(i,i) = a(i,i) * b(i,i) + +# Strutture E Algoritmi Strutturati + +Strutture: sottospazi vettoriali, sottoalgebre + +Algoritmo strutturato: algoritmo che sfrutta la struttura + +```ad-example +collapse: open + +- Matrici diagonali: sottoalgebra +- Matrici triangolari: sottoalgebra +- Matrici tridiagonali: sottospazio vettoriale + +``` diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md new file mode 100644 index 000000000..a27b7eb92 --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md @@ -0,0 +1,130 @@ +$$Ax=b$$ + +$$A \in K^{nxn}, x \in K^n, b \in K^n$$ + +$$\begin{cases}a_{1,1}x_1 +. .. a_{1,n}x_n=b_1 \\ a_{2,1}x_1 +. .. a_{2,n}x_n=b_2 \\\vdots\\ a_{n,1}x_1 +. .. a_{n,n}x_n=b_n \end{cases}$$ + +# Teorema Di Rouché-Capelli + +Il sistema lineare Ax = b ammette soluzione se e solo se rango(A) = rango(A|b). + +Se esiste almeno una soluzione allora l'insieme delle soluzioni è un sottospazio affine di dimensione n-rango(A). + +$K^n$ spazio vettoriale + +v sottospazio vettoriale di dimensione l $o \leq l \leq n$ + +$v_0 \in K^n$ la traslazione di $v_0, t_0$ è l'applicazione + +$$t_{v_0}:K^n \to K^n$$ + +$$v \to v+v_0$$ + +L'insieme $W=\{v+v_0, v \in V, v_0 \in K^n\}$ è un sottospazio affine di $K^n$ di dimensione dim(V). + +```ad-info + +- I sottospazi affini di dimensione 1 sono detti rette +- I sottospazi affini di dimensione 2 sono detti piani +- I sottospazi affini di dimensione n-1 sono detti iperpiani +- I sottospazi affini di dimensione 0 sono detti punti (non sono sottospazi) +``` + +$Ax=b \iff c_1x_1+c_2x_2+...+c_nx_n=b \iff$ b è combinazione lineare delle colonne di A $\iff$ rango(A)=rango(A|b) + +$$|c_1|...|c_n|=\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}$$ + +# Problema Ben Posto + +- una soluzione esiste +- la soluzione è unica +- la soluzione dipende in modo continuo dai dati +la soluzione è unica $\iff rk(A)=rk(A|b)=n \iff n-rk(A)=0$ +```ad-important +title: Teorema + +Il sistema lineare Ax=b è ben posto se e solo se A è quadrata e invertibile +``` + +## Caso 1 + +$mn \quad rk(A)=rk(A|b)=n \iff$ la soluzione non è unica, anche se il sistema ammette una soluzione unica per un dato b, esiste $c: \forall \varepsilon >0$ il sistema $Ax=c+\varepsilon c$ non ammette soluzioni $\implies$ la soluzione non può dipendere in modo continuo dai dati +$$Im(A) =\{v \in K^n, v=Ax, x\in R^n\} \subseteq K^n$$ +$$dim(Im(A))=rk(A)\leq n0$$ +per ogni $\varepsilon >0$ questo sistema non ha soluzione se esiste $y: Ay=b+c\varepsilon$ +$$Ay=Ax+c\varepsilon \quad Ay-Ax=c\varepsilon \quad A(\frac{y-x}{\varepsilon})=c$$ +è una contraddizione. + +## Caso 3 + +$m=n\quad det(A)=0 \implies$ la soluzione non esiste o non è unica $\implies$ non è ben posto + +## Caso 4 + +A quadrata, $m=n\quad det(A) \neq 0 \implies$ la soluzione è unica. +Dipende in modo continuo dai dati? +$$x=A^{-1}b\quad (A^{-1})_{ij}=\frac{det(\bar{A}^{ij})}{det(A)}(-1)^{i+j}$$ +$$x_i=\sum_{l=1}^n{(A^{-1})_{il}b_l}=\sum_{l=1}^n{\frac{det(\bar{A}^{il})}{det(A)}(-1)^{i+l}b_l},\quad i=1,..,n$$ +è una funzione continua di $a_{il}, b_l$ poiché è razionale + +## Algoritmi Di Risoluzione + +### Caso Facile (matrici triangolari) + +$$T_n=\{A\in K^{nxn}, a_{ij}=0, i>j\}$$ +$$\begin{bmatrix}a_{1,1}&...&a_{1,n}\\\vdots &\ddots &\vdots\\ 0 &...& a_{n,n} \end{bmatrix}$$ +Sistema lineare Tx=b dove T è triangolare superiore +$$\begin{cases}t_{1,1}x_1 + t_{1,2}x_2 + t_{1,3}x_3=b1 \\ \quad\quad\quad\quad t_{2,2}x_2+t_{2,3}x_3=b_2 \\ \quad\quad\quad\quad\quad\quad\quad\quad t_{3,3}x_3=b_3 \end{cases}$$ +Calcolo il termine i-esimo +$$t_{i,i}x_i+t_{i,i+1}x_{i+1}+...+t_{i,n}x_n=b_i$$ +$$x_i=\frac{b_i-t_{i,i+1}x_{i+1}-...-t_{i,n}x_n}{t_{i,i}}=\frac{b_i - \sum_{j=i+1}^n{t_{i,j}x_j}}{t_{i,j}}$$ +$$Tx=b \implies t_{i,1}x_1+t_{i,2}x_2+...+t_{i,i}x_{i}+...=b_i$$ +$$\sum_{j=i}^n{t_{i,j}x_j}=b_i$$ + +### Algoritmo Di Sostituzione All'indietro + + for i=n:-1:1 + s=b(i) + for j= i+1:n + s = s - t(i,j) * x(j) + x(i) = s / t(i,i) + +costo di un passo del ciclo esterno: 2(n-i)+1 ops +$$\sum_{i=1}^n{2(n-i)+1} = 2\sum_{i=1}^n{2n-2i+1}=2\sum_{i=1}^n{n}-2\sum_{i=1}^n{i}+\sum_{i=1}^n{1}=2n^2-n(n+1)+n=n^2 ops$$ + +```ad-note + +L'algoritmo fallisce se $t_{i,i}=0$ + +Si può dimostrare che $det(T)=t_{1,1}t_{2,2}...t_{n,n}$ + +$det \neq 0 \iff t_{i,i} \neq 0 \iff$ l'algoritmo è applicabile +``` + +### Algoritmo Di Sostituzione in Avanti + +Per sistemi triangolari inferiori +$$\begin{cases} +t_{1,1}x_1&&&&=b_1\\ +t_{2,1}x_1&+t_{2,2}x_2&&&=b_2\\ +t_{3,1}x_1&+t_{3,2}x_2&+t_{3,3}x_3&&=b_2\\ +t_{4,1}x_1&+t_{4,2}x_2&+t_{4,3}x_3&+t_{4,4}x_4&=b_2\\ +\end{cases}$$ +$$\begin{bmatrix} +*&0&0&0\\ +*&*&0&0\\ +*&*&*&0\\ +*&*&*&* +\end{bmatrix}$$ + + for i = 1:n + s = b(i) + for j = 1:i-1 + s = s - t(i,j) * x(j) + x(i) = s / t(i,i) \ No newline at end of file diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a1. Algoritmo di Gauss.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a1. Algoritmo di Gauss.md new file mode 100644 index 000000000..37c2678d4 --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a1. Algoritmo di Gauss.md @@ -0,0 +1,145 @@ +22-11-2022 + +Ax=b, A quadrata e invertibile + + for h=1:n-1: + for i = h+1:n: + l(i,h)=a(i,h) / a(h,h) + a(i,h) = 0 + for j=h+1:n: + a(i,j) -= l(i,h) * a(h,j) + b(i) -= l(i,h) * b(h) + +L'algoritmo per come è implementato richiede + +$$\sum_{h=1}^{n-1}{2(n-h)^2} = \frac 23n^3$$ + + +$$\begin{bmatrix} +a_{11},&a_{12},&a_{13},&\dots \\ +a_{21},&a_{22},&a_{23},&\dots \\ +\vdots,&\vdots,&\vdots,&\ddots +\end{bmatrix}$$ +Il primo passo l'algoritmo ha come pivot $a_{11}$ +Non ce se la fa a staglie dietro mannaggia + +Dato $A\in K^{nxn}$ il minore principale di testa di ordine $j \leq h \leq n$ di A è la matrice $M_h \in K^{nxn}: (M_n)_{ij} = a_{ij}$ + +```ad-important +title: Teorema + +Sia $A \in K^{nxn}$ e siano $M_1,...M_n$ i suoi minori principali di testa. Il metodo di Gauss è applicabile se e solo se $det(M_h) \neq 0, h=1,...,n-1$ + +``` + +``````ad-attention +title: Dimostrazione + +Supponiamo il metodo di Gauss sia applicabile $\implies$ si può ottenere la matrice $$U=A_n=\begin{bmatrix} +a_{11} & * & \\ +& a_{22} & & \\ +&& a_{33} &\\ +& 0 && a_{44} +\end{bmatrix}$$ +Sia $N_h$ il minore principale di testa di U di ordine h, esso è una matrice triangolare e il suo determinante è dato dagli elementi sulla sua diagonale, quindi $det(N_h) = u_{11} u_{22} ... u_{hh}$ + +Siccome il metodo è applicabile allora $$a_{ii} \neq 0, i=1,...,n-1 \implies a_{11} ... a_{hh} \neq 0\implies det(N_h) \neq 0, h=1,...,n-1$$ + +```ad-done + +Si ha che $det(N_h) = det(M_h) \implies det(M_h) \neq 0, h=1,...,n-1$ poiché le operazioni (le combinazioni di righe) del metodo di Gauss non cambiano il determinante della matrice a cui si applica, ma non cambia neanche quello dei suoi minori principali di testa +``` +```ad-failure + +Se il metodo di Gauss fallisce al passo h allora $a_{hh}=0 \quad(a_{jj} \neq 0, j=1,..,h-1)$ e si può costruire la matrice +$$A_h = \begin{bmatrix} +a_{11} & & & &\\ +& \ddots & & & \\ +& & a_{hh}=0 & \\ +& & & & & +\end{bmatrix}$$ +Sia $N_h$ il minore principale di testa di ordine h di $A_h$, $det(N_h)=0$ + +Ma $det(M_h)=det(N_h)$ per quanto detto prima, quindi il metodo non è applicabile +``` +`````` + +Se A è una matrice definita positiva allora la condizione del teorema è verificata + +$A \in R^{nxn}$ è definita positiva se: +- $A^T=A$ (simmetrica) +- $v^T Av > 0$ se $v \in R^n \setminus \{0\}$ + +Se A è a dominanza diagonale allora la condizione del teorema è verificata + +A è a dominanza diagonale (stretta) se +- $|a_{ii}| > \sum_{j=1, j\neq i}^n{|a_{ij}|}$ + +A è a dominanza diagonale se: +- $|a_{ii}| \geq \sum_{j=1, j\neq i}^n{|a_{ij}|}$ +- vale il maggiore stretto per almeno un indice + +# Sistema lineare +Come si risolve un sistema lineare? +$$A_1=A\to A_2 \to A_3 \to ... \to A_n = U$$ +$$[A|b] = [A_1|b_1] \to [A_2|b_2]$$ + + for h=1:n-1: + for i = h+1:n: + l(i,h)=a(i,h) / a(h,h) + for j=h+1:n: + a(i,j) = a(i,j) - l(i,h) * a(h,j) + a(i,h) = 0 + b(i) = b(i) - l(i,h) * b(h) + +Si può interpretare il metodo di Gauss come operazioni sulle equazioni +$$Ax=b \iff A_1x=b_1 \iff ... \iff A_nx=b_n \iff Ux=\tilde{b}$$ +Applicando il metodo di Gauss si parte da Ax=b e si ottiene un sistema triangolare equivalente $Ux=\tilde{b}$ che si può risolvere con il metodo di [[a0. Sistemi lineari#Algoritmo Di Sostituzione All'indietro|sostituzione all'indietro]] + +Qual è il costo computazionale? +$\frac 23 n^3$ per l'eliminazione +$n^2$ per per la soluzione di $Ux=\tilde{b}$ +$\sum_{h=1}^n{2(n-h) \sim n^2}$ per il calcolo di $\tilde{b}$ + +```ad-note +$det(A_1) = det(A_2) = ... = det(U)$ perché le operazioni che eseguiamo non modificano il determinante. +è sufficiente quindi applicare il metodo di Gauss con $\frac 23 n^3$ operazioni (e n-1 moltiplicazioni che si possono trascurare) + +``` + +# Sistemi Lineari Con Membro Destro Multiplo + +$$Ax=b_1, Ax=b_2,...,Ax=b_l$$ +risolvendo indipendentemente i sistemi lineari sono richieste $\frac 23 n^3 l$ operazioni. + +Applico il metodo di Gauss ad A modificando consistentemente anche le colonne relative ai termini noti +$$[A|b_1b_2...b_l]=[A_1|b_1^{(1)}b_2^{(1)},...b_l^{(1)}] \to [A_2|b_1^{(2)}b_2^{(2)}...b_l^{(2)}] \to ... \to[A_n|b_1^{(n)}b_2^{(n)}...b_l^{(n)}]=[U|\tilde{b_1} \tilde{b_2}...\tilde{b_3}]$$ +$$Ux=\tilde{b_1},Ux=\tilde{b_2},...,Ux=\tilde{b_l}$$ +Risolvendo in "parallelo" $\frac 23 n^3l$ operazioni. + +Quante operazioni in più al primo passo? $2l(n-1)$ +al secondo passo? $2l( n-2)$ +al passo i? $2l(n-i)$ +in totale saranno $\sum_{i=1}^{n-1}{2l(n-i)}$ +$$\sum_{i=1}^{n-1}{2l(n-i)} = 2l \sum_{i=1}^{n-1}{(n-i)}\sim\frac {2ln^2}2 = ln^2$$ +$\frac 23 n^3 + ln^2$ operazioni anziché $\frac 23 n^3 l$ per trovare i sistemi $Ux=\tilde{b_1},...,Ux=\tilde{b_l}$ per risolverli sono necessarie l sostituzioni all'indietro e quindi altre $ln^2$ operazioni. + +In totale sono $\frac 23 n^3 + 2n^2 l$ operazioni per risolvere l sistemi lineari con sistemi coefficienti + + +## Calcolare l'inversa di una matrice +X è l'inversa di A (nxn) +se AX=I +$$A[X_1|X_2|...|X_n] = [F_1|F_2|...|F_n]$$ +$$[AX_1|AX_2|...|AX_n] = [F_1|F_2|...|F_n]$$ +$$AX_1=F_1,AX_2=F_2,...,AX_n=F_n$$ +```ad-note +title: Nota + +Il problema di calcolo dell'inversa si riconduce ad un problema di risolvere un sistema lineare con membro destro multiplo + +``` + +Utilizzando l'algoritmo precedente ci vogliono +$$\frac 23 n^3 + 2n^2n = \frac 83 n^3$$ +In realtà si può calcolare l'inversa con $2n^3$ operazioni \ No newline at end of file diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a2. Matrici tridiagonali.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a2. Matrici tridiagonali.md new file mode 100644 index 000000000..de70f3357 --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a2. Matrici tridiagonali.md @@ -0,0 +1,55 @@ +$$A=\begin{bmatrix} +*&*&&&&\\ +*&*&*&&&\\ +&*&*&*&&\\ +&&*&*&*&\\ +&&&*&*&*\\ +&&&&*&*&*\\ +&&&&&*&* +\end{bmatrix}$$ +$T_n=\{A\in K^{nxn}, a_{ij}=0, |i-j| > 1\}$ è un sottospazio vettoriale poiché è una def. dalla posizione degli elementi nulli se sommo o moltiplico per uno scalare elementi nulli rimangono nulli. + +```ad-fail +title: sono una sottoalgebra? + +è vero che $A,B \in T_n \implies AB \in T_n$? +$$ + +\begin{bmatrix} + +1&1&0\\ + +1&1&1\\ + +0&1&1 + +\end{bmatrix} + += \begin{bmatrix} + +\end{bmatrix} + +$$ +``` + +```ad-question +title: Qual è la dimensione vettoriale? + + +$$3n-2$$ +``` + +Trovare un algoritmo che calcola il prodotto di una matrice diagonale $A \in T_n$ per un vettore $b \in K^n$ e che non richieda più di 6n operazioni e darne un'implementazione in pseudocodice (non usare il prodotto matrice sparsa-vettore) + +Primo approccio: scrivo la matrice e il vettore e vedo cosa succede + +Secondo approccio: scrivo il componente $c_i$ + +algoritmo mio + + prodotto(A,b): + c[1] = A[1,1] * b[1] + for i=:2:n: + c[i-1] += A[i-1,i] * b[i] + c[i] = A[i,i-1] * b[i-1] + A[i, i] * b[i] + \ No newline at end of file diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a3. Fattorizzazione LU.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a3. Fattorizzazione LU.md new file mode 100644 index 000000000..a8851eed0 --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a3. Fattorizzazione LU.md @@ -0,0 +1,94 @@ + +Algoritmo di Gauss $\iff$ fatt. LU + +$$A_1\to A_2 \to ...\to U=A_n = \begin{bmatrix} +* & * & * & * \\ +0&* &*&*\\ +0&0&*&*\\ +0&0&0&* +\end{bmatrix}$$ +costruiamo una matrice L triangolare inferiore con 1 sulle diagonali tale che $L_{ij}$ è definito dal metodo di Gauss $i>j$ +$$L=\begin{bmatrix} +1&0&0\\ +l_{ij}&1&0\\ +l_{ij}&l_{ij}&1 +\end{bmatrix}$$ +$$l_{ih}=\frac{a_{ih}}{a_{hh}} \quad l_{ij}=\frac{a_{ij}}{a_{jj}}$$ +$$a_{ij}^{(h+1)}=\begin{cases} +a_{ij}^{(h)}-l_{ih}a_{hj}^{(h)},& i>h,j\geq h\\ +a_{ij}^{(h)}&alternative +\end{cases}$$ + +``````ad-important +title: Teorema + +$A=LU$ (con le valutazioni viste sopra) + +```ad-check +title: Dimostrazione + +$$A=LU \iff a_{ij}=(LU)_{ij}\quad i,j=1,..,n$$ +$$(LU)_{ij} = \sum_{h=1}^n{l_{ih}u_{hj}}=\sum_{h=1}^{min\{i,j\}}{l_{ih}u_{hj}} = a_{ij}$$ + +$$i>j \quad a_{ij}=\sum_{h=1}^j{l_{ih}u_{hj}}=\sum_{h=1}^j{l_{ih}a_{hj}^{(h)}} = \sum_{h=1}^j{a_{ij}^{(h)} - a_{ij}^{(h+1)}} = a_{ij}^{(1)} - a_{ij}^{(2)} + $$ +$$u_{hj}=a_{hj}^{(h)}\quad a_{ij}^{(h+1)} = a_{ij}^{(h)} - l_{ih}a_{ij}^{(h)}$$ + +``` + + +`````` + +# Implementazione Del Metodo Di Gauss + +[[a1. Algoritmo di Gauss]] + +Si parte dalla matrice completa $[A_1|b_1]$ e si eliminano tutti gli elementi della prima colonna a parte il primo, per fare questo si usa la somma di una riga per il multiplo della prima. Il valore per cui si moltiplica è dato dal rapporto dell'elemento nella posizione da eliminare per l'elemento sulla diagonale. + +```ad-example +collapse: open + +Al passo 1, se voglio eliminare l'elemento nella riga 2 ($a_{21}$) devo eseguire l'operazione $r2 = r2 - \frac {a_{21}}{a_{11}}r1$ + +``` + +Bisogna ripetere questi passaggi per ogni colonna usando come pivot la diagonale della matrice + +Il rapporto è poi utilizzato per formare la matrice L, infatti ogni elemento di L: +$$l_{ij} = \frac{a_{ij}}{a_{jj}}$$ + +Il risultato saranno poi due matrici triangolari: +- U triangolare superiore +- L triangolare inferiore + + + + +# Come Si Risolve Un Sistema Lineare? + +$A=LU$ calcoliamo la fattorizzazione $LU$ di $A$ con il metodo di Gauss + +$$Ax=b \iff LUx=b \iff L(Ux) = b\implies \begin{cases} +Ly=b\\ +Ux=y +\end{cases}$$ +Risolviamo il primo sistema $Ly=b$ con l'[[a0. Sistemi lineari#Algoritmo Di Sostituzione in Avanti|algoritmo di sostituzione in avanti]] e poi $Ux=y$ con l'[[a0. Sistemi lineari#Algoritmo Di Sostituzione All'indietro|algoritmo di sostituzione all'indietro]] ottenendo x. + +## Quante Operazioni Richiede? +Il costo totale delle operazioni sulla matrice $A$ è di$$\sum_{k=1}^{n-1}{[2(n-k)^2+(n-k)]}$$ +Di cui: + +- $\frac 23 n^3$ per la fattorizzazione +- $n^2+n^2$ per le due sostituzioni di $\tilde b$ e la soluzione di $Ux = \tilde b$ (trascurabili) + +il costo computazionale è lo stesso ($y= \tilde b$) +$$\begin{bmatrix} +\varepsilon & 1\\ +1&0 +\end{bmatrix} +\begin{bmatrix} +x\\y +\end{bmatrix}= +\begin{bmatrix} +1+\varepsilon\\1 +\end{bmatrix}$$ +il pivot per $\varepsilon >0$ non è nullo e la soluzione è unica $x=\begin{bmatrix}1\\1\end{bmatrix}$ diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a3a. Esercizio.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a3a. Esercizio.md new file mode 100644 index 000000000..ceb5b85eb --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a3a. Esercizio.md @@ -0,0 +1,89 @@ +Descrivere una variante strutturata dell'algoritmo di Gauss che richieda non più di 10n operazioni per la soluzione del sistema lineare $Ax=b$ con A matrice tridiagonale. + +$$[A|b]\to ... \to [U|\tilde{b}]$$ + +$$Ux=\tilde{b}$$ + +$\frac 23n^3$ operazioni algoritmo di Gauss applicato ad A + +$n^2$ operazioni per aggiornare b + +$n^3$ operazioni per risolvere $Ux=\tilde b$ + +A è tridiagonale se $a_{ij} = 0, |i-j|>1$ + +Al primo passo del metodo di Gauss è necessario eliminare gli elementi + +$$a_{21},...,a_{n1}$$ + +Alcuni di essi sono già nulli e quindi alcune operazioni si possono evitare + +$$r^T_2 \gets r^T_2-l_{21}r^T_1$$ + +Nel caso tridiagonale la combinazione di righe modifica solo un elemento e quindi sono richieste solo 2 operazioni (e una divisione per calcolare $l_{21}$) il primo passo richiede 3 operazioni. + +Al secondo passo si osserva che la sottomatrice $B_2$ è tridiagonale, è sufficiente quindi la combinazione lineare + +$$r_3^T \gets r_3^T - l_{32} r_2^T$$ + +che richiede 3 operazioni. + +L'algoritmo richiede 3(n-1) passi in totale + +Implementazione: + +Algoritmo originale di gauss + + for h=1:n-1 + for i=h+1:n + l(i,h) = a(i,h) / a(h,h) + for j=h+1:n + a(i,j)=a(i,j)-l(i,h)*a(h,j) + +Algoritmo modificato + + for h=1:n-1 + l(h+1, h) = a(h+1, h) / a(h,h) + a(h+1,h+1) = a(h+1,h+1) - l(h+1,h)*a(h,h+1) + a(h+1,h=0)=0 + b(h+1) = b(h+1) - l(h+1, h)*b(h) + +Per calcolare U sono sufficienti 3(n-1) operazioni, per calcolare $\tilde b$ sono sufficienti 2(n-1) operazioni + +Risolvere $Ux = \tilde b$ sono sufficienti 3(n-1) + 1 operazioni + +$8n + +$$\begin{bmatrix} +a_{11}&a_{12}&0&...&...&0\\ +&a_{22}&a_{23}&0&...&0\\ +&&\ddots&\ddots&\ddots&\vdots\\ +&&&&&a_{nn} +\end{bmatrix} + +\begin{bmatrix} +x_1\\x_2\\\vdots\\x_n +\end{bmatrix}= +\begin{bmatrix} +\tilde{b_1}\\\tilde{b_2}\\\vdots\\\tilde{b_n}\\ +\end{bmatrix}$$ + +Si può risolvere con O(n) operazioni? + +Modo 1 + +$$i-esima\to\begin{cases} +U_{11}x_1+U_{12}x_2=\tilde{b_1}\\ +U_{22}x_2+U_{23}x_3=\tilde{b_2}\\ +\vdots\\ +U_{ii}x_i+U_{i i+1}x_{i+1}=\tilde{b_i}\\ +\vdots\\ +U_{nn}x_n=\tilde{b_n}\\ +\end{cases}$$ + x(n)=bt(n)/u(n,n) + for i=n-1:-1:1 + x(i) = (bt(i)- u(i,i+1) *x(i+1)) / u(i,i) + +3(n-1) + 1 operazioni $\sim$ 3n operazioni + +[[a3. Fattorizzazione LU]] \ No newline at end of file diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a4. Variante del Massimo Pivot Parziale (GEPP).md b/triennale/Anno 2/Calcolo Numerico/argomenti/a4. Variante del Massimo Pivot Parziale (GEPP).md new file mode 100644 index 000000000..0107c3ac8 --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a4. Variante del Massimo Pivot Parziale (GEPP).md @@ -0,0 +1,181 @@ +```ad-summary +title: Riassunto [[a1. Algoritmo di Gauss]] +- Implementazione +- Costo computazionale $\frac 23 n^3$ +- Risolve molti problemi + - sistemi lineari + - inversa + - determinante +- Equivalente alla fattorizzazione LU +- non è sempre applicabile ma ci sono dei criteri di applicabilità (veri per alcune matrici) + +``` + +$A=LU$ in aritmetica esatta + +$\tilde A = \tilde L \tilde U$ in aritmetica finita, ed esiste $|\tilde {A_{ij}} - A_{ij}| \leq \gamma(n)(|\tilde {A_{ij}} | + |\tilde{L_{ij}}||\tilde{U_{ij}}|)$ + +Quindi il metodo di Gauss è un __algoritmo instabile__, non è adeguato. + +# Implementazione + +Ad ogni passo prima dell'eliminazione si scambiano due righe per ottenere il pivot di massimo modulo. + +Al primo passo si cerca sulla riga/colonna il primo elemento di massimo modulo e se questo elemento non è $a_{11}$ allora si scambia la prima riga con quella dell'elemento di massimo modulo. + +$$A_1=\begin{bmatrix} +a_{11} &...\\ +a_{22} &...\\ +a_{33} &...\\ +\vdots & \vdots\\ +a_{nn} &...\\ +\end{bmatrix} \to |a_{p1}| = max\{|a_{11}|,...,|a_{nn}|\}\to B_1= \begin{bmatrix} +a_{p1} &...\\ +a_{22} &...\\ +a_{33} &...\\ +\vdots & \vdots\\ +a_{nn} &...\\ +\end{bmatrix} \to eliminazione \to $$ +$$\to A_2 = \begin{bmatrix} +a_{11} & a_{12}&...\\ +0 & a_{22}&...\\ +0 & a_{32}&...\\ +\vdots & \vdots & \vdots\\ +0 & a_{n2}&...\\ +\end{bmatrix}\to scambio \to B_2 = \begin{bmatrix} +a_{11} & a_{12}&...\\ +0 & a_{p2}&...\\ +0 & a_{32}&...\\ +\vdots & \vdots & \vdots\\ +0 & a_{n2}&...\\ +\end{bmatrix} \to A_3 = ...$$ + +$argmax\{|a_{11}|,...,|a_{nn}|\}$ è l'insieme degli indici in cui il massimo viene raggiunto + + for h=1: n-1 + p = 1; M = abs(a(h,h)); + for i = h+1: n + m = abs(a(i,h)); + if(m > M) + M = m; + p = i; + if p != h + for j=h:n + swap = a(h, j) + a(h, j) = a(p, j) + a(p, j) = swap + swap = b(h) + b(h) = b(p) + b(p) = swap + for i = k+1 : n + //metodo di gauss + +# Costo Computazionale + +I confronti al primo h sono +$$n-(h+1) + 1 = n-h$$ +E i valori assoluti da calcolare +$$n-h+1$$ +$2(n-h) + 1$ confronti al passo $h$ + +Il numero totale è +$$\sum_{h=1}^{n-1}{2(n-h)+1} \sim n^2$$ +$n^2$ è trascurabile rispetto al costo necessario per l'eliminazione ($\frac 23 n^3$) + +# Applicabilità + +$$\begin{bmatrix} +0 & \dots\\ +0 & \vdots\\ +\vdots & \vdots\\ +0 & \dots +\end{bmatrix}$$ +Al primo passo il metodo si arresta se la prima colonna è nulla $\implies$ A non è invertibile $\iff$ det(A)=0 + + +```ad-important +title: Teorema + +Sia $A \in K^{nxn}$ invertibile (condizione sufficiente). Il metodo di Gauss con la variante del massimo pivot parziale è applicabile ad A. + +``` +```ad-fail +title: Dimostrazione + +Se il metodo fallisce al passo $\ell$ allora A non è invertibile perché si avrà che $a_{ll}=0$, di conseguenza se l'elemento con il pivot massimo è 0, tutti gli altri elementi al disotto sono 0 (il GEPP funziona con i valori assoluti). + +Calcoliamo il det(A) sviluppandolo rispetto alla prima colonna, procedendo in questo modo anche per la colonna 2 e così via. + +Si arriva ad $A_l$ tramite operazioni che non modificano il determinante (combinazioni di righe) e scambi che non modificano il segno +$$|det(A)| = |det(A_l)| = 0 \implies det(A)=0$$ + +Il rango della matrice è quindi al più $k-1 < n$, e quindi non è invertibile, di conseguenza non si può utilizzare il GEPP + +``` + +# Matrici Di Permutazione + +Interpretiamo gli scambi di righe tramite operazioni tra matrici: + +$T \in \mathfrak{S}_n \quad t:\{1,...,n\} \to \{1,...,n\}$ biiettiva +$$\varrho:t \to \Pi$$ +$$f_1=\begin{bmatrix} +1\\0\\\vdots\\0 +\end{bmatrix},f_2=\begin{bmatrix} +0\\1\\\vdots\\0 +\end{bmatrix},...,f_n=\begin{bmatrix} +0\\0\\\vdots\\1 +\end{bmatrix}$$ +```ad-example + +$$\sigma(1)=2,\quad\sigma(2) = 1, \quad n=2$$ + +$$f_1=\begin{bmatrix} +1\\0 +\end{bmatrix},f_2=\begin{bmatrix} +0\\1 +\end{bmatrix}$$ + +$$p(\sigma)=[f_{\sigma(1)}|f_{\sigma(2)}] = [f_2|f_1] =\begin{bmatrix} +0&1\\1&0 +\end{bmatrix}$$ + +Date la matrice $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ e la matrice di permutazione $p$: +$$\begin{bmatrix} +0&1\\1&0 +\end{bmatrix}\begin{bmatrix} +a&b\\c&d +\end{bmatrix}=\begin{bmatrix} +c&d\\a&b +\end{bmatrix}$$ + + +``` + +``````ad-important +title: Teorema + +Sia data la [[a4. Variante del Massimo Pivot Parziale (GEPP)#Implementazione|fattorizzazione LU con GEPP]] esiste una matrice di permutazione $\Pi$ tale che$$\Pi A = LU$$ +La matrice $\Pi$ è ottenuta raccogliendo tutti gli scambi di riga effettuati durante [[a4. Variante del Massimo Pivot Parziale (GEPP)#Implementazione|l'algoritmo]] + +```ad-check +title: Sono invertibili? + +Se vogliamo risolvere il sistema $Ax=b$, le sue soluzioni coincideranno con $\Pi Ax = \Pi b \iff LU=\Pi b$ e quindi il problema si risolve permutando il vettore $b$ e risolvendo la [[a3. Fattorizzazione LU#Come Si Risolve Un Sistema Lineare?|fattorizzazione LU]]. + +``` + + +`````` + +# Calcolo Del Determinante +Oltre che a [[a1. Algoritmo di Gauss#Sistemi Lineari Con Membro Destro Multiplo|risolvere sistemi lineari]] l'Algoritmo di Gauss può essere usato per calcolare il determinante e l'inversa di una matrice . + +L'osservazione sta nel fatto che il metodo di gauss non cambia il determinante della matrice, ma solo il suo segno (Se avvengono degli scambi di righe). + +Quindi, con la variante del massimo pivot parziale, si hanno molti scambi di righe che devono essere contati. Si ottiene quindi che $det(A)=det(U)$ se gli scambi sono pari, e $det(A)=-det(U)$ se gli scambi sono dispari. + +Si può scrivere quindi $det(A)=(-1)^sdet(U)$, con $s$ il numero di scambi di righe. + +Ottenendo la triangolare superiore $U$, il determinante si trova solo sulla diagonale, e quindi $$det(U) = (-1)^s \prod_{i=1}^n{u_{ii}}$$ +Il costo del prodotto è di $n-1$ operazioni, quindi è trascurabile rispetto al metodo di gauss. diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a5. Curva di Bezier.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a5. Curva di Bezier.md new file mode 100644 index 000000000..92417b3e2 --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a5. Curva di Bezier.md @@ -0,0 +1,30 @@ +Dati $P_0,P_1,...,P_n$ punti di controllo, la curva + +$$B(t)=\sum_{i=0}^n{(\substack{n\\i})t^i(1-t)^{n-i}P_i}$$ + +è detta curva di Bezier + +```ad-example +title: Esempi + +## 1 + +N=1 $$B_{0,1}(t)=\substack{\sum_i^1{(\substack{1\\i}) t^i( 1-t)^{1-i}P_i} =\\ +(\substack{1\\0}) t^0( 1-t)^{1} P_0 + (\substack{1\\1}) t^1( 1-t)^{0}P_1 = \\ +1 \cdot 1 (1-t)^1P_0 + 1t(1-t)^0P_1}= +(1-t)P_0 + tP_1$$ + + + +``` + + +# Inviluppo convesso di n+1 punti + +L'inviluppo convesso in n+1 punti in $R^n$ è il più piccolo insieme convesso che li contiene + +# Proprietà delle curve di bezier + +Il vettore tangente (derivata) su B in 0 è $n(v_1 - v_0)$ e ci permette di determinare la retta tangente di $\gamma$ in $\gamma (t_n)$ è $\gamma(t_0) + \gamma' (t_0) t$ + + diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a6. Modello della corda elastica.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a6. Modello della corda elastica.md new file mode 100644 index 000000000..ee8af11ee --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a6. Modello della corda elastica.md @@ -0,0 +1,37 @@ +1. Trovare la posizione di una corda elastica soggetta a forza esterna. +2. Trovare una funzione che ne deriva la forma. +3. Trovare il valore di questa funzione in alcuni punti. + +n masse uguali e equidistanti collegate tra loro da molle. + +4. Scriviamo l'equazione dell'equilibrio per singola massa + - Ogni massa si può muovere solo in verticale + - L'equazione dell'equilibrio è $F_y=0$ + - Le forze con componenti verticali su $m_i$ sono + - La forza peso + - La forza elastica della massa $m_{i+1}$ + - La forza elastica della massa $m_{i-1}$ + +$$|F|=k(\ell - \ell_0)=k\ell,\quad \ell_0=0$$ + +Se $y_i$ è la posizione verticale di $m_i$ allora la forza che $m_{i+1}$ esercita su $m_i$ ha componente verticale $k(y_{i+1} - y_{i})$ + +La massa $m_{i-1}$ esercita una forza $k(y_{i-1} - y_i)$ + +L'equazione diventa + +$$\begin{cases} +-mg +k(y_{i+1} - y_i) + k(y_{i-1} - y_i) = 0 \\ +-mg + k(y_2 - y_1) + k(-y_1)= 0\\ +-mg+k(-y_n) + k(y_{n-1} - y_n) = 0 +\end{cases}$$ + +Le incognite sono $y_1,..,y_n$ le incognite delle molle +$$\frac {-mg}k + y_{i+1} - y_i + y_{i-1} - y_i = 0$$ +$$\begin{cases} +-y_{i-1} + 2y_i - y_{i+1} = -\frac{mg}k\\ +2y_1 - y_2 = - \frac{mg}k\\ +2y_n - y_{n-1} = - \frac{mg}k +\end{cases}$$ + +Il sistema ha n equazioni e n incognite, è un sistema lineare del tipo $Ax=b$, con A tridiagonale \ No newline at end of file diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md new file mode 100644 index 000000000..6af9d0947 --- /dev/null +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md @@ -0,0 +1,152 @@ +```ad-hint +title: Problema dell'interpolazione polinomiale + +Dati $x_0,..,x_n \in [a,b]$ e dati $y_0,...,y_n \in R$ trovare un polinomio p(x) di grado al più n tale che: +$$p(x_0) = y_0,\quad p(x_1) = y_1,\quad p(x_n) = y_n$$ + +``` + +$$\begin{cases} +a_0+a_1x_0+a_2x_0^2+...+a_nx_0^n=y_0\\ +\vdots\\ +a_0+a_1x_n+a_2x_n^2+...+a_nx_n^n=y_n +\end{cases} \text{è un sistema lineare}$$ +$$V= +\begin{bmatrix} +1 &x_0&x_0^2&...&x_0^n\\ +1&x_1&x_1^2&...&x_1^n\\ +\vdots&\vdots&\vdots&\ddots&\vdots\\ +1&x_n&x_n^2&...&x_n^n +\end{bmatrix} \text{matrice di Vandermorde} +$$ +```ad-note + +Il problema è ben posto se e solo se V è invertibile:: +$$det(V) = \prod_{\substack{i>j\\i,j=0}}^2{(x_i - x_j)} = (x_1 - x_0)(x_2 - x_1) (x_2 - x_0)$$ + +``` + +``````ad-info +title: Corollario + +Il problema dell'interpolazione polinomiale è ben posto se e solo se i nodi sono distinti. + +```ad-success +title: Dimostrazione + +Siccome è un sistema lineare, deve essere $det(V) \neq 0$ ma: +$$det(v) \neq 0 \substack{\iff & x_i-x_j \neq 0 & i > j \\ +\iff& x_i \neq x_j & i \neq j\\ +\iff &\text{i nodi sono distinti}}$$ + +``` + + +`````` +```ad-example +title: Esempio + +$x_0,x_1 \in [a,b], \quad y_0,y_1 \in R$ il polinomio di interpolazione ha grado al più 1, quindi abbiamo soluzione unica + +$$ p(x) = a + bx$$ + +$$ +\begin{cases} +a+bx_0=y_0\\ +a+bx_1=y_1 +\end{cases} \implies +\begin{bmatrix} +1 & x_0\\ +1 & x_1 +\end{bmatrix}\begin{bmatrix} +a\\b +\end{bmatrix}=\begin{bmatrix} +y_0\\y_1 +\end{bmatrix} +$$ +$$ +\begin{bmatrix} +a\\b +\end{bmatrix}=\begin{bmatrix} +1 & x_0\\ +1 & x_1 +\end{bmatrix}^{-1}\begin{bmatrix} +y_0\\y_1 +\end{bmatrix} +$$ +$$ +\frac 1{x_1-x_0} +$$ +``` + +L'utilizzo principale del polinomio di interpolazione + +# Polinomi Di LAGRANGE Associati a $x_0,...,x_n$ Distinti + +- $L_0(x)= \frac{(x-x_1)(x-x_2)...(x-x_n)}{(x_0 - x_1) (x_0 - x_2) ... (x_0 - x_n)}$ +- ... +- $L_i(x) = \frac{(x-x_1)(x-x_2)...}{}$ + +$$L_i(x) = \prod_{\substack{j=0 \\ j\neq i}}^n \frac{x - x_j}{x_i - x_j} \quad i = 1,...,n$$ + +```ad-example +title: Esempio due nodi + +$x_0 = -1 \neq x_1 = 3$ + +$$L_0(x)=\frac{x - 3}{-1 -3} = \frac {x - 3}{-4} \quad\quad L_1(x) = \frac{x - (-1)}{3 - ( - 1)} = \frac{x + 1}{4}$$ + + +``` + +``````ad-tip +title: Teorema + +Siano $x_0,..,x_n \in [a,b]$ distinti e siano $L_0(x),...,L_n(x)$ i polinomi di LAGRANGE allora + +1. $L_i(x_i) = \delta_{ij} : \begin{cases} 1 & i=j \\ 0 & i \neq j \end{cases}$ +2. 2 polinomi di Lagrange sono una base di $R_n[x]$ +3. Dati $y_0,..,y_n \in R$, il polinomio di interpolazione rispetto a $\begin{bmatrix}x_0 \\ y_0\end{bmatrix},...,\begin{bmatrix}x_n \\ y_n\end{bmatrix}$ è + +$$p(x) = \sum_{i=1}^n {y_iL_i(x)}$$ + +```ad-done +title: Dimostrazione +collapse: true + +1. $L_i(x_j) = \frac{(x_j-x_0) ... (x_j-x_j)...(x_j - x_{i+1})}{...} = 0, \quad i \neq j$ + +$L_i(x_i) = \frac{(x_i - x_0)...(x_i - x_{i-1})(x_i - x_{i+1})}{(x_i - x_0)...(x_i - x_{i-1})(x_i - x_{i+1})} = 1, \quad i = j$ + +2. Siccome $R_n[x]$ ha dimensione n+1 è sufficiente dimostrare che sono linearmente indipendenti +$$\sum {\alpha_i L_i(x)} = 0 \implies \alpha_i = 0\quad i=0,...,n$$ +valutiamo la combinazione lineare in $x_j \quad j = 0,..,n$ +$$0=\sum_{i=0}^n {\alpha_iL_i(x_j)} = \alpha_jL_j(x_j) = \alpha_j$$ + +3. $p(x) = \sum_{i=0}^n{y_iL_i(x)}$ dobbiamo dimostrare che $p(x_j) = y_j$ poiché p ha grado al più n e i nodi sono distinti +$$ p(x_j) = \sum_{i=0}^n {y_i L_i(x_j)} = y_jL_j(x_j) = y_j$$ +``` + + +`````` + +## Algoritmo per Il Calcolo Di p(x) E La Sua Valutazione + +$$ \sum_{i=0} ^ n {y_i\prod_{j \neq i}{\frac{x-x_j}{x_i - x_j}}} = \sum_{i=0} ^ n {y_i\frac{\prod_{j \neq i}{x-x_j}}{\prod_{j \neq i}{x_i - x_j}}}$$ + +# Interpolazione per Approssimare Funzioni + +Si può utilizzare l'interpolazione polinomiale per l'approssimazione di funzioni continue. + +Data $f \in C[a,b]$ e dati $x_0,...,x_n \in [a,b]$ il polinomio di interpolazione è l'unico polinomio di grado __al più $n$__ tale che $p(x_i) = f(x_i)=y_i$. + +Una volta approssimata una funzione $f(x)$ con un polinomio $p_n(x)$ si ha un errore:$$r(x)=f(x)-p_n(x)$$ +detto __resto dell'interpolazione__ + +```ad-important +title: Teorema + +Sia $f \in C^{k+1}[a,b]$ e sia $p_n(x)$ il polinomio di interpolazione relativo a $x_0,...,x_n \in [a,b]$ distinti. Allora $\forall x \in [a,b], \exists \xi (x) \in [a,b] :$ $$r(x)=f(x)-p_n(x)=\frac{f^{(k+1)}(\xi(x))}{(n+1)!}(x-x_0)...(x-x_n)$$ +In generale non è detto che aumentando il numero dei nodi l'approssimazione migliori (anche se la funzione da approssimare è regolare). + +``` diff --git a/triennale/Anno 2/Calcolo Numerico/dispense/dispense_cn_2021.pdf b/triennale/Anno 2/Calcolo Numerico/dispense/dispense_cn_2021.pdf new file mode 100644 index 0000000000000000000000000000000000000000..54b2b4c9814d1835ee6e3a73fa7e46b0d063d76d GIT binary patch literal 710134 zcmce;1#le8lJ_fSiy24E%*?XTlEuu-OcpaUGqc6a%w&PZ%*lBbX0k2SSiYt{*sg*R=IIA~h zSz!YRkVz?7Ce9>voq1xAN7OWW5VW2bGeq@a4#%3S5~oS39DXiB%v9dpiWcI=7xOv6 z?b7LGL1?yaCP3(8ye%oB9l8QZXv&fvbcKD?!TePTyVM^3%n(aCxNFu18*O)}l1}<8 z%?HQ^f_j-*=Nrqn(NOdb_q<24kHCBFym7JnV${R`3Oii1uTz-l;&mJR!#lA|S_$Mf z;VFZkfpyR}-f(2>#YTbz!Dl)iJp(Olic(DOdvH0;2@T%yAy{`&2i^IgZ_>hXP=cN; z4&MsyE8Nj-3ZLT6UX4u~={(x4T14)RgGb2H2aNhCN>`#EjD9MxAbL3X9n!@G;~0V69N>oyFHH&kb01M3o~;m5shUg 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z0NCCF7#v&}E|~ud?S%N+pjHSu2f@O`d^K=&F@KeCiP19@U}DtYg7Gka5Nd$1*cTVn zA1W^Z!_IiZjYoAcw8QF8iScm!AXkLZ3Rr~574*gINwBztXnY7@A~e}-HR#*v91$fOXH zU9cihzYIbdn7#r-6tiQY9ad{Vz|hzngh9lp9Sbn5-j)cVU~~p1VKNRdR7ZgOfy*Qe zh~{6eiJE`}3+m-Ls9i_EP#+dxLewq> z7*;n2o(31k4{UF2UeE`KeGh$wX#5@u8zI=jlzV`YFCAvmRD{*xfdFCg1Avh)ZejKX zn-$f&U;(2sMzD&-s6PW%9aypmeo$(F>2}zxn4CgIDtfL0E3vw3BGiZal903~-yp$Q z2i%-M{W8ex;5>&Y1Ewb-YK!Sz$UvgL6$Jl40-5Ut7#L>=464IW+Xo^`Sd13JQdnIs zz(go6Jn&%Acr3h+8MBe#%}#KG0l@{bwP@WsRM5j=hO~n%i0T(`Qm_~(z_1(;R1#yk zPk;$9T0vR}_2YRWsG&mk2UygYYyeD*?iq;qk~g%a<$#C+qjJRq5l49rFf0xN*Q&7? z6o{?}&A$N*ie{Pg0|_GERwIoIz{r=wBQPxf2kkI90GJTfO#l<4F%NjbG?sS(7(~NS z{Gdb$!x`#8aTr!-3CLml4PaOuBfzkFLV#iV8LBt17z)6!SUAAQSDP{E2X!}y?u2$& zeJ6y{u)9G-cn#fK00SnWw1Vx1(GPSiMk@$~V=(ZKF#NH+p8-a`n}@MBh&G}6 z8QMXR8(AB1Cj`S8ubU&hC^W7IFbMEs<6-r65blI>VzeE^N-0W-R%YG*9|FBQ AF8}}l literal 0 HcmV?d00001 From 2eabf208ee4c4149bb71571956ddc08025a9889c Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 16:56:02 +0200 Subject: [PATCH 02/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/0. Introduzione.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- triennale/Anno 2/Calcolo Numerico/argomenti/0. Introduzione.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/0. Introduzione.md b/triennale/Anno 2/Calcolo Numerico/argomenti/0. Introduzione.md index 95980c1aa..52364afe0 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/0. Introduzione.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/0. Introduzione.md @@ -47,7 +47,7 @@ Non esiste una formula per le equazioni di 5° grado La materia si occupa di dare strumenti teorici (teorie matematiche) per il calcolo numerico di quantità continue. -Si cerca di trovare una soluzione ai problemi continui che sia utilizzabile. Non basta una formula, ma serve un algoritmo ce fornisca una soluzione in un tempo ragionevole tenendo conto che il calcolatore lavora in quantità finite. Vogliamo quindi trattare __quantità infinite__ con __strumenti finiti__. +Si cerca di trovare una soluzione ai problemi continui che sia utilizzabile. Non basta una formula, ma serve un algoritmo che fornisca una soluzione in un tempo ragionevole tenendo conto che il calcolatore lavora in quantità finite. Vogliamo quindi trattare __quantità infinite__ con __strumenti finiti__. # Quantità Continue From ed5bb87730010235f1f6de53bc55de4051c286de Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 16:56:15 +0200 Subject: [PATCH 03/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/2. Error Analysis.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../Anno 2/Calcolo Numerico/argomenti/2. Error Analysis.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/2. Error Analysis.md b/triennale/Anno 2/Calcolo Numerico/argomenti/2. Error Analysis.md index 78049a157..4ea6bd7a5 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/2. Error Analysis.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/2. Error Analysis.md @@ -58,7 +58,7 @@ Il numero $\sum_{i=1}^\infty {\beta^{-i}d_i}$ si dice __mantissa__ ## Floating Point -Data una base di numerazione $\beta \geq 2$, il numero $t>0$ di cifre della mantissa, $m,M$ numeri positivi, definiamo un insieme di __floating point numbers__ +Data una base di numerazione $\beta \geq 2$, il numero $t>0$ di cifre della mantissa $m$, $M$ numeri positivi, definiamo un insieme di __floating point numbers__ $$F(\beta, t, m, M) = \{0\} \cup \{\pm \beta^p\sum_{i=1}^t {\beta^{-i}d_i}, \; -m \leq p \leq M, \; 0 \leq d_i < \beta \text{ intero per } i=1,...,t, \; d_1 \neq 0\}$$ From 9a5ccfcf79053ac4f566f5268acd3651094ef814 Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 16:58:21 +0200 Subject: [PATCH 04/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../argomenti/3. Errore Per Le Funzioni Razionali.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md b/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md index 8c9fd1285..f9df1646a 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md @@ -1,6 +1,6 @@ -Data una funzione razionale $f: R^2 \to R$ che sia $f = \frac pq$ con p e q polinomi. +Data una funzione razionale $f: R^2 \to R$ che sia $f = \frac pq$ con $p$ e $q$ polinomi. Dall'analisi matematica sappiamo che $f$ è definita e differenziabile per $q \neq 0$ (assumendo che p e q siano primi). From bddeca65dd1206b2532f0c29738f4355eaba1b8e Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 16:58:53 +0200 Subject: [PATCH 05/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../argomenti/3. Errore Per Le Funzioni Razionali.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md b/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md index f9df1646a..6ae455ce0 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md @@ -2,7 +2,7 @@ Data una funzione razionale $f: R^2 \to R$ che sia $f = \frac pq$ con $p$ e $q$ polinomi. -Dall'analisi matematica sappiamo che $f$ è definita e differenziabile per $q \neq 0$ (assumendo che p e q siano primi). +Dall'analisi matematica sappiamo che $f$ è definita e differenziabile per $q \neq 0$ (assumendo che $p$ e $q$ siano primi). Ci sono due tipi di errori nella valutazione di f con i valori in virgola mobile From f39886e64d0a2981822fe5319309cb65029a235e Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 16:59:11 +0200 Subject: [PATCH 06/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../argomenti/3. Errore Per Le Funzioni Razionali.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md b/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md index 6ae455ce0..9a0442805 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md @@ -4,7 +4,7 @@ Data una funzione razionale $f: R^2 \to R$ che sia $f = \frac pq$ con $p$ e $q$ Dall'analisi matematica sappiamo che $f$ è definita e differenziabile per $q \neq 0$ (assumendo che $p$ e $q$ siano primi). -Ci sono due tipi di errori nella valutazione di f con i valori in virgola mobile +Ci sono due tipi di errori nella valutazione di $f$ con i valori in virgola mobile ## Errore Inerente From 392281925cf6b147fc7ef91edda82d984ef05c53 Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 16:59:40 +0200 Subject: [PATCH 07/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../argomenti/3. Errore Per Le Funzioni Razionali.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md b/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md index 9a0442805..a18a26eca 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md @@ -8,7 +8,7 @@ Ci sono due tipi di errori nella valutazione di $f$ con i valori in virgola mobi ## Errore Inerente -Non valutiamo f(x) ma valutiamo $f(\tilde x)$ dove $\tilde x = fl(x)$ [[2. Error Analysis#^59010a|funzione approssimata]] +Non valutiamo $f(x)$ ma valutiamo $f(\tilde x)$ dove $\tilde x = fl(x)$ [[2. Error Analysis#^59010a|funzione approssimata]] $$\varepsilon_{IN} = \frac{f(\tilde x) -f(x)}{f(x)}, \quad f(x) \neq 0$$ From d0899a145eb572b6f42a6f1bd5c93695fd872d12 Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 17:02:31 +0200 Subject: [PATCH 08/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../argomenti/3. Errore Per Le Funzioni Razionali.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md b/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md index a18a26eca..5b5f49468 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md @@ -31,7 +31,7 @@ L'errore algoritmico può essere definito anche per le funzioni elementari che v $$\varepsilon_{TOT} = \frac {\tilde f(\tilde x ) - f(x)}{f(x)}, \quad f(x) \neq 0$$ Dà una misura genuina dell'errore nella valutazione. -La situazione ideale è $|\varepsilon_{tot}| < u$, ma in pratica è sufficiente $|\varepsilon_{tot}| < Mu$ con M costante. +La situazione ideale è $|\varepsilon_{tot}| < u$, ma in pratica è sufficiente $|\varepsilon_{tot}| < Mu$ con $M$ costante. ```ad-important title: Teorema From a62fa2f5cb17eae1d15164af4da6f05f788bb38c Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 17:02:44 +0200 Subject: [PATCH 09/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../argomenti/3. Errore Per Le Funzioni Razionali.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md b/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md index 5b5f49468..4a8b463b2 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/3. Errore Per Le Funzioni Razionali.md @@ -66,7 +66,7 @@ __Esiste un'altra formula__ per calcolare l'errore inerente: $$\varepsilon_{IN} = \frac x{f(x)}f'(\xi)\varepsilon_x$$ -Dove $\varepsilon_x = \varepsilon_1$ è la rappresentazione dell'errore in x se $f \in C^2(conv(x, \tilde x))$ allora: +Dove $\varepsilon_x = \varepsilon_1$ è la rappresentazione dell'errore in $x$ se $f \in C^2(conv(x, \tilde x))$ allora: $$\varepsilon_{IN} = \frac x{f(x)} f'(x)\varepsilon_x + o(u) = \frac {x \cdot 2x}{x^2}\varepsilon_x = 2\varepsilon_x$$ ```ad-done From a7d6e6ac48269eac3baf632ec1f40f44a8528cda Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 17:06:09 +0200 Subject: [PATCH 10/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/4. Problemi Ben Posti.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../Anno 2/Calcolo Numerico/argomenti/4. Problemi Ben Posti.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/4. Problemi Ben Posti.md b/triennale/Anno 2/Calcolo Numerico/argomenti/4. Problemi Ben Posti.md index 23f73e934..302c3a1c6 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/4. Problemi Ben Posti.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/4. Problemi Ben Posti.md @@ -7,7 +7,7 @@ Un problema è composto di tre parti: Sistemi lineari -Data una matrice A (coefficienti) e un vettore b (lato destro), trovare tutti i vettori x (incognite) tali che $Ax = b$ +Data una matrice $A$ (coefficienti) e un vettore $b$ (lato destro), trovare tutti i vettori $x$ (incognite) tali che $Ax = b$ ``` From db5b27948aa535a096680d3dee1ff0ee4f1fd9b1 Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 17:06:54 +0200 Subject: [PATCH 11/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/6. Valutazione di un polinomio.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../argomenti/6. Valutazione di un polinomio.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/6. Valutazione di un polinomio.md b/triennale/Anno 2/Calcolo Numerico/argomenti/6. Valutazione di un polinomio.md index 0e369ba67..ba57405fc 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/6. Valutazione di un polinomio.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/6. Valutazione di un polinomio.md @@ -2,7 +2,7 @@ Dati $p$ e $x$, calcolare $p(x)$ - $R^n(K^n)$ vettori con $n$ componenti in $R(K)$ -- $R_n(x)(K_n(x))$ polinomi di grado al più n con coefficienti reali +- $R_n(x)(K_n(x))$ polinomi di grado al più $n$ con coefficienti reali - $R^{nxm}(K^{nxm})$ matrici n x m con elementi in R - $R^{n_1 x n_2 x ... x n_l}$ tensore di $l$ dimensioni - $p(x) = a_0 + a_1 x + ... + a_n x^n$ From 8c7456b80ad1d769c822ced15512249a86556785 Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 17:07:20 +0200 Subject: [PATCH 12/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/6. Valutazione di un polinomio.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../argomenti/6. Valutazione di un polinomio.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/6. Valutazione di un polinomio.md b/triennale/Anno 2/Calcolo Numerico/argomenti/6. Valutazione di un polinomio.md index ba57405fc..188cafb91 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/6. Valutazione di un polinomio.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/6. Valutazione di un polinomio.md @@ -3,7 +3,7 @@ Dati $p$ e $x$, calcolare $p(x)$ - $R^n(K^n)$ vettori con $n$ componenti in $R(K)$ - $R_n(x)(K_n(x))$ polinomi di grado al più $n$ con coefficienti reali -- $R^{nxm}(K^{nxm})$ matrici n x m con elementi in R +- $R^{nxm}(K^{nxm})$ matrici $n \times m$ con elementi in $R$ - $R^{n_1 x n_2 x ... x n_l}$ tensore di $l$ dimensioni - $p(x) = a_0 + a_1 x + ... + a_n x^n$ From 54bcedc47e836f187f0cf8fc14a8136c52f7b242 Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 17:07:43 +0200 Subject: [PATCH 13/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/7. Vettori.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- triennale/Anno 2/Calcolo Numerico/argomenti/7. Vettori.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/7. Vettori.md b/triennale/Anno 2/Calcolo Numerico/argomenti/7. Vettori.md index a7a088cd5..937d95efc 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/7. Vettori.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/7. Vettori.md @@ -25,7 +25,7 @@ Ci sono anche altre interpretazioni, come quello impiegato nel deep learning # Prodotto Tra Matrici Di Strassen -Algoritmo per semplificare il prodotto tra due matrici 2x2. +Algoritmo per semplificare il prodotto tra due matrici $2 \times 2$. La moltiplicazione riga per colonna impiega 8 moltiplicazioni e 4 addizioni. From ced48ac35abe71782b399a16ba39dd69994484ee Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 17:08:02 +0200 Subject: [PATCH 14/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/7. Vettori.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- triennale/Anno 2/Calcolo Numerico/argomenti/7. Vettori.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/7. Vettori.md b/triennale/Anno 2/Calcolo Numerico/argomenti/7. Vettori.md index 937d95efc..793028001 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/7. Vettori.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/7. Vettori.md @@ -33,7 +33,7 @@ L'algoritmo di Strassen impiega 7 moltiplicazioni e 18 addizioni. Può sembrare che impieghi un tempo peggiore rispetto alla riga per colonna, ma la moltiplicazione tra due vettori impiega molto di più che l'addizione, rendendo quindi l'algoritmo più veloce. -L'algoritmo viene poi trasformato in moltiplicazioni in blocchi. Rendendo qualsiasi moltiplicazione tra matrici $2^nx2^n$ come 7 moltiplicazioni e alcune addizioni tra 4 matrici $2^{(n-1)}x2^{(n-1)}$ ricorsivamente. +L'algoritmo viene poi trasformato in moltiplicazioni in blocchi. Rendendo qualsiasi moltiplicazione tra matrici $2^n \times 2^n$ come 7 moltiplicazioni e alcune addizioni tra 4 matrici $2^{(n-1)} \times 2^{(n-1)}$ ricorsivamente. Questo ha fatto scaturire una corsa all'algoritmo più veloce per risoluzione di prodotti tra matrici. From 367b883b101e8318af60fad16188f4ef56c9fe73 Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 17:08:25 +0200 Subject: [PATCH 15/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/8. Matrici.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- triennale/Anno 2/Calcolo Numerico/argomenti/8. Matrici.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/8. Matrici.md b/triennale/Anno 2/Calcolo Numerico/argomenti/8. Matrici.md index 966036dd1..ffc2233d4 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/8. Matrici.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/8. Matrici.md @@ -17,5 +17,5 @@ $$A \in K^{nxn}$$ $$numZeri(A) << n^2$$ -si riesce a implementare il [[7. Vettori#Prodotto Matrice per Vettore|prodotto matrice-vettore]] con $2l$ operazione anziché con $2n^2$ +si riesce a implementare il [[7. Vettori#Prodotto Matrice per Vettore|prodotto matrice-vettore]] con $2l$ operazioni anziché con $2n^2$ From ff438d0ec01c5d6675d6857144b4820a1c674fe2 Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 17:08:58 +0200 Subject: [PATCH 16/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md b/triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md index 9e369db72..c28578106 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md @@ -21,7 +21,7 @@ $$\lim\limits_{n \to \infty}{\frac {numZeri(A)}{n^2}} = 0$$ ```ad-check -Sono strutture sparse, sono sufficienti $2p +Sono strutture sparse, sono sufficienti $2p$. ``` `````` From 60c7e0674820578b8f246b37a565789c65acd5a6 Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 17:09:17 +0200 Subject: [PATCH 17/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md b/triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md index c28578106..d475006b4 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md @@ -54,7 +54,7 @@ title: è una sottoalgebra di $K^{nxn}$? Un'algebra è un sottospazio vettoriale in cui è definito un prodotto compatibile con le operazioni dello spazio vettoriale. -V è un gruppo se dati $A,B,C \in V$ e $\alpha \in K$: +$V$ è un gruppo se dati $A,B,C \in V$ e $\alpha \in K$: - $A(B + C) = AB + AC$ - $(A + B)C = AC + BC$ - $\alpha(AB) = (\alpha A)B = A(\alpha B)$ From ecfc8b1c058783885e3523c0d98a668da5f03264 Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 17:09:33 +0200 Subject: [PATCH 18/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md b/triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md index d475006b4..522bba178 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md @@ -59,7 +59,7 @@ $V$ è un gruppo se dati $A,B,C \in V$ e $\alpha \in K$: - $(A + B)C = AC + BC$ - $\alpha(AB) = (\alpha A)B = A(\alpha B)$ -Le matrici nxn sono un'algebra con il prodotto riga per colonna +Le matrici $n \times n$ sono un'algebra con il prodotto riga per colonna ```ad-check title: Sottoalgebra From 87ea94b5333166427f2f20ca79cddc2db263da5e Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 17:09:57 +0200 Subject: [PATCH 19/37] Update triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md index a27b7eb92..5ed663412 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md @@ -6,7 +6,7 @@ $$\begin{cases}a_{1,1}x_1 +. .. a_{1,n}x_n=b_1 \\ a_{2,1}x_1 +. .. a_{2,n}x_n=b_ # Teorema Di Rouché-Capelli -Il sistema lineare Ax = b ammette soluzione se e solo se rango(A) = rango(A|b). +Il sistema lineare $Ax = b$ ammette soluzione se e solo se rango(A) = rango(A|b). Se esiste almeno una soluzione allora l'insieme delle soluzioni è un sottospazio affine di dimensione n-rango(A). From 8853d4fa13fcc2ca14261b05100194320bb1b3ee Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Wed, 16 Aug 2023 17:15:21 +0200 Subject: [PATCH 20/37] Apply suggestions from code review MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../argomenti/a0. Sistemi lineari.md | 20 ++++++------ .../argomenti/a1. Algoritmo di Gauss.md | 31 +++++++++---------- .../argomenti/a2. Matrici tridiagonali.md | 2 +- 3 files changed, 26 insertions(+), 27 deletions(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md index 5ed663412..3c9dcc78e 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md @@ -12,7 +12,7 @@ Se esiste almeno una soluzione allora l'insieme delle soluzioni è un sottospazi $K^n$ spazio vettoriale -v sottospazio vettoriale di dimensione l $o \leq l \leq n$ +$v$ sottospazio vettoriale di dimensione $l$ con $o \leq l \leq n$ $v_0 \in K^n$ la traslazione di $v_0, t_0$ è l'applicazione @@ -20,17 +20,17 @@ $$t_{v_0}:K^n \to K^n$$ $$v \to v+v_0$$ -L'insieme $W=\{v+v_0, v \in V, v_0 \in K^n\}$ è un sottospazio affine di $K^n$ di dimensione dim(V). +L'insieme $W=\{v+v_0, v \in V, v_0 \in K^n\}$ è un sottospazio affine di $K^n$ di dimensione $dim(V)$. ```ad-info - I sottospazi affini di dimensione 1 sono detti rette - I sottospazi affini di dimensione 2 sono detti piani -- I sottospazi affini di dimensione n-1 sono detti iperpiani +- I sottospazi affini di dimensione $n-1$ sono detti iperpiani - I sottospazi affini di dimensione 0 sono detti punti (non sono sottospazi) ``` -$Ax=b \iff c_1x_1+c_2x_2+...+c_nx_n=b \iff$ b è combinazione lineare delle colonne di A $\iff$ rango(A)=rango(A|b) +$Ax=b \iff c_1x_1+c_2x_2+...+c_nx_n=b \iff$ $b$ è combinazione lineare delle colonne di $A$ $\iff$ rango(A)=rango(A|b) $$|c_1|...|c_n|=\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}$$ @@ -43,7 +43,7 @@ la soluzione è unica $\iff rk(A)=rk(A|b)=n \iff n-rk(A)=0$ ```ad-important title: Teorema -Il sistema lineare Ax=b è ben posto se e solo se A è quadrata e invertibile +Il sistema lineare $Ax=b$ è ben posto se e solo se $A$ è quadrata e invertibile ``` ## Caso 1 @@ -52,7 +52,7 @@ $mn \quad rk(A)=rk(A|b)=n \iff$ la soluzione non è unica, anche se il sistema ammette una soluzione unica per un dato b, esiste $c: \forall \varepsilon >0$ il sistema $Ax=c+\varepsilon c$ non ammette soluzioni $\implies$ la soluzione non può dipendere in modo continuo dai dati +$m>n \quad rk(A)=rk(A|b)=n \iff$ la soluzione non è unica, anche se il sistema ammette una soluzione unica per un dato $b$, esiste $c: \forall \varepsilon >0$ il sistema $Ax=c+\varepsilon c$ non ammette soluzioni $\implies$ la soluzione non può dipendere in modo continuo dai dati $$Im(A) =\{v \in K^n, v=Ax, x\in R^n\} \subseteq K^n$$ $$dim(Im(A))=rk(A)\leq nj\}$$ $$\begin{bmatrix}a_{1,1}&...&a_{1,n}\\\vdots &\ddots &\vdots\\ 0 &...& a_{n,n} \end{bmatrix}$$ -Sistema lineare Tx=b dove T è triangolare superiore +Sistema lineare $Tx=b$ dove $T$ è triangolare superiore $$\begin{cases}t_{1,1}x_1 + t_{1,2}x_2 + t_{1,3}x_3=b1 \\ \quad\quad\quad\quad t_{2,2}x_2+t_{2,3}x_3=b_2 \\ \quad\quad\quad\quad\quad\quad\quad\quad t_{3,3}x_3=b_3 \end{cases}$$ -Calcolo il termine i-esimo +Calcolo il termine $i$-esimo $$t_{i,i}x_i+t_{i,i+1}x_{i+1}+...+t_{i,n}x_n=b_i$$ $$x_i=\frac{b_i-t_{i,i+1}x_{i+1}-...-t_{i,n}x_n}{t_{i,i}}=\frac{b_i - \sum_{j=i+1}^n{t_{i,j}x_j}}{t_{i,j}}$$ $$Tx=b \implies t_{i,1}x_1+t_{i,2}x_2+...+t_{i,i}x_{i}+...=b_i$$ @@ -95,7 +95,7 @@ $$\sum_{j=i}^n{t_{i,j}x_j}=b_i$$ s = s - t(i,j) * x(j) x(i) = s / t(i,i) -costo di un passo del ciclo esterno: 2(n-i)+1 ops +costo di un passo del ciclo esterno: $2(n-i)+1$ ops $$\sum_{i=1}^n{2(n-i)+1} = 2\sum_{i=1}^n{2n-2i+1}=2\sum_{i=1}^n{n}-2\sum_{i=1}^n{i}+\sum_{i=1}^n{1}=2n^2-n(n+1)+n=n^2 ops$$ ```ad-note diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a1. Algoritmo di Gauss.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a1. Algoritmo di Gauss.md index 37c2678d4..57c6d0be1 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a1. Algoritmo di Gauss.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a1. Algoritmo di Gauss.md @@ -1,6 +1,5 @@ -22-11-2022 -Ax=b, A quadrata e invertibile +$Ax=b$, con $A$ quadrata e invertibile for h=1:n-1: for i = h+1:n: @@ -23,7 +22,7 @@ a_{21},&a_{22},&a_{23},&\dots \\ Il primo passo l'algoritmo ha come pivot $a_{11}$ Non ce se la fa a staglie dietro mannaggia -Dato $A\in K^{nxn}$ il minore principale di testa di ordine $j \leq h \leq n$ di A è la matrice $M_h \in K^{nxn}: (M_n)_{ij} = a_{ij}$ +Dato $A\in K^{nxn}$ il minore principale di testa di ordine $j \leq h \leq n$ di $A$ è la matrice $M_h \in K^{nxn}: (M_n)_{ij} = a_{ij}$ ```ad-important title: Teorema @@ -41,7 +40,7 @@ a_{11} & * & \\ && a_{33} &\\ & 0 && a_{44} \end{bmatrix}$$ -Sia $N_h$ il minore principale di testa di U di ordine h, esso è una matrice triangolare e il suo determinante è dato dagli elementi sulla sua diagonale, quindi $det(N_h) = u_{11} u_{22} ... u_{hh}$ +Sia $N_h$ il minore principale di testa di $U$ di ordine $h$, esso è una matrice triangolare e il suo determinante è dato dagli elementi sulla sua diagonale, quindi $det(N_h) = u_{11} u_{22} ... u_{hh}$ Siccome il metodo è applicabile allora $$a_{ii} \neq 0, i=1,...,n-1 \implies a_{11} ... a_{hh} \neq 0\implies det(N_h) \neq 0, h=1,...,n-1$$ @@ -51,31 +50,31 @@ Si ha che $det(N_h) = det(M_h) \implies det(M_h) \neq 0, h=1,...,n-1$ poiché le ``` ```ad-failure -Se il metodo di Gauss fallisce al passo h allora $a_{hh}=0 \quad(a_{jj} \neq 0, j=1,..,h-1)$ e si può costruire la matrice +Se il metodo di Gauss fallisce al passo $h$ allora $a_{hh}=0 \quad(a_{jj} \neq 0, j=1,..,h-1)$ e si può costruire la matrice $$A_h = \begin{bmatrix} a_{11} & & & &\\ & \ddots & & & \\ & & a_{hh}=0 & \\ & & & & & \end{bmatrix}$$ -Sia $N_h$ il minore principale di testa di ordine h di $A_h$, $det(N_h)=0$ +Sia $N_h$ il minore principale di testa di ordine $h$ di $A_h$, $det(N_h)=0$ Ma $det(M_h)=det(N_h)$ per quanto detto prima, quindi il metodo non è applicabile ``` `````` -Se A è una matrice definita positiva allora la condizione del teorema è verificata +Se $A$ è una matrice definita positiva allora la condizione del teorema è verificata $A \in R^{nxn}$ è definita positiva se: - $A^T=A$ (simmetrica) - $v^T Av > 0$ se $v \in R^n \setminus \{0\}$ -Se A è a dominanza diagonale allora la condizione del teorema è verificata +Se $A$ è a dominanza diagonale allora la condizione del teorema è verificata -A è a dominanza diagonale (stretta) se +$A$ è a dominanza diagonale (stretta) se - $|a_{ii}| > \sum_{j=1, j\neq i}^n{|a_{ij}|}$ -A è a dominanza diagonale se: +$A$ è a dominanza diagonale se: - $|a_{ii}| \geq \sum_{j=1, j\neq i}^n{|a_{ij}|}$ - vale il maggiore stretto per almeno un indice @@ -94,7 +93,7 @@ $$[A|b] = [A_1|b_1] \to [A_2|b_2]$$ Si può interpretare il metodo di Gauss come operazioni sulle equazioni $$Ax=b \iff A_1x=b_1 \iff ... \iff A_nx=b_n \iff Ux=\tilde{b}$$ -Applicando il metodo di Gauss si parte da Ax=b e si ottiene un sistema triangolare equivalente $Ux=\tilde{b}$ che si può risolvere con il metodo di [[a0. Sistemi lineari#Algoritmo Di Sostituzione All'indietro|sostituzione all'indietro]] +Applicando il metodo di Gauss si parte da $Ax=b$ e si ottiene un sistema triangolare equivalente $Ux=\tilde{b}$ che si può risolvere con il metodo di [[a0. Sistemi lineari#Algoritmo Di Sostituzione All'indietro|sostituzione all'indietro]] Qual è il costo computazionale? $\frac 23 n^3$ per l'eliminazione @@ -103,7 +102,7 @@ $\sum_{h=1}^n{2(n-h) \sim n^2}$ per il calcolo di $\tilde{b}$ ```ad-note $det(A_1) = det(A_2) = ... = det(U)$ perché le operazioni che eseguiamo non modificano il determinante. -è sufficiente quindi applicare il metodo di Gauss con $\frac 23 n^3$ operazioni (e n-1 moltiplicazioni che si possono trascurare) +è sufficiente quindi applicare il metodo di Gauss con $\frac 23 n^3$ operazioni (e $n-1$ moltiplicazioni che si possono trascurare) ``` @@ -112,7 +111,7 @@ $det(A_1) = det(A_2) = ... = det(U)$ perché le operazioni che eseguiamo non mod $$Ax=b_1, Ax=b_2,...,Ax=b_l$$ risolvendo indipendentemente i sistemi lineari sono richieste $\frac 23 n^3 l$ operazioni. -Applico il metodo di Gauss ad A modificando consistentemente anche le colonne relative ai termini noti +Applico il metodo di Gauss ad $A$ modificando consistentemente anche le colonne relative ai termini noti $$[A|b_1b_2...b_l]=[A_1|b_1^{(1)}b_2^{(1)},...b_l^{(1)}] \to [A_2|b_1^{(2)}b_2^{(2)}...b_l^{(2)}] \to ... \to[A_n|b_1^{(n)}b_2^{(n)}...b_l^{(n)}]=[U|\tilde{b_1} \tilde{b_2}...\tilde{b_3}]$$ $$Ux=\tilde{b_1},Ux=\tilde{b_2},...,Ux=\tilde{b_l}$$ Risolvendo in "parallelo" $\frac 23 n^3l$ operazioni. @@ -122,14 +121,14 @@ al secondo passo? $2l( n-2)$ al passo i? $2l(n-i)$ in totale saranno $\sum_{i=1}^{n-1}{2l(n-i)}$ $$\sum_{i=1}^{n-1}{2l(n-i)} = 2l \sum_{i=1}^{n-1}{(n-i)}\sim\frac {2ln^2}2 = ln^2$$ -$\frac 23 n^3 + ln^2$ operazioni anziché $\frac 23 n^3 l$ per trovare i sistemi $Ux=\tilde{b_1},...,Ux=\tilde{b_l}$ per risolverli sono necessarie l sostituzioni all'indietro e quindi altre $ln^2$ operazioni. +$\frac 23 n^3 + ln^2$ operazioni anziché $\frac 23 n^3 l$ per trovare i sistemi $Ux=\tilde{b_1},...,Ux=\tilde{b_l}$ per risolverli sono necessarie $l$ sostituzioni all'indietro e quindi altre $ln^2$ operazioni. In totale sono $\frac 23 n^3 + 2n^2 l$ operazioni per risolvere l sistemi lineari con sistemi coefficienti ## Calcolare l'inversa di una matrice -X è l'inversa di A (nxn) -se AX=I +$X$ è l'inversa di $A$ ($n \times n$) +se $AX=I$ $$A[X_1|X_2|...|X_n] = [F_1|F_2|...|F_n]$$ $$[AX_1|AX_2|...|AX_n] = [F_1|F_2|...|F_n]$$ $$AX_1=F_1,AX_2=F_2,...,AX_n=F_n$$ diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a2. Matrici tridiagonali.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a2. Matrici tridiagonali.md index de70f3357..546027500 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a2. Matrici tridiagonali.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a2. Matrici tridiagonali.md @@ -39,7 +39,7 @@ title: Qual è la dimensione vettoriale? $$3n-2$$ ``` -Trovare un algoritmo che calcola il prodotto di una matrice diagonale $A \in T_n$ per un vettore $b \in K^n$ e che non richieda più di 6n operazioni e darne un'implementazione in pseudocodice (non usare il prodotto matrice sparsa-vettore) +Trovare un algoritmo che calcola il prodotto di una matrice diagonale $A \in T_n$ per un vettore $b \in K^n$ e che non richieda più di $6n$ operazioni e darne un'implementazione in pseudocodice (non usare il prodotto matrice sparsa-vettore) Primo approccio: scrivo la matrice e il vettore e vedo cosa succede From cf444d297b6bd821df2a930eeac9277098b7d866 Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Thu, 17 Aug 2023 12:40:23 +0200 Subject: [PATCH 21/37] Apply suggestions from code review MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../argomenti/a3. Fattorizzazione LU.md | 8 ++++---- .../argomenti/a3a. Esercizio.md | 18 +++++++++--------- ...riante del Massimo Pivot Parziale (GEPP).md | 14 +++++++------- .../argomenti/a5. Curva di Bezier.md | 4 ++-- .../a6. Modello della corda elastica.md | 4 ++-- .../argomenti/a7. Interpolazione.md | 12 ++++++------ 6 files changed, 30 insertions(+), 30 deletions(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a3. Fattorizzazione LU.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a3. Fattorizzazione LU.md index a8851eed0..9bf9cf13c 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a3. Fattorizzazione LU.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a3. Fattorizzazione LU.md @@ -7,7 +7,7 @@ $$A_1\to A_2 \to ...\to U=A_n = \begin{bmatrix} 0&0&*&*\\ 0&0&0&* \end{bmatrix}$$ -costruiamo una matrice L triangolare inferiore con 1 sulle diagonali tale che $L_{ij}$ è definito dal metodo di Gauss $i>j$ +costruiamo una matrice $L$ triangolare inferiore con 1 sulle diagonali tale che $L_{ij}$ è definito dal metodo di Gauss $i>j$ $$L=\begin{bmatrix} 1&0&0\\ l_{ij}&1&0\\ @@ -53,11 +53,11 @@ Al passo 1, se voglio eliminare l'elemento nella riga 2 ($a_{21}$) devo eseguire Bisogna ripetere questi passaggi per ogni colonna usando come pivot la diagonale della matrice -Il rapporto è poi utilizzato per formare la matrice L, infatti ogni elemento di L: +Il rapporto è poi utilizzato per formare la matrice $L$, infatti ogni elemento di $L$: $$l_{ij} = \frac{a_{ij}}{a_{jj}}$$ Il risultato saranno poi due matrici triangolari: -- U triangolare superiore +- $U$ triangolare superiore - L triangolare inferiore @@ -71,7 +71,7 @@ $$Ax=b \iff LUx=b \iff L(Ux) = b\implies \begin{cases} Ly=b\\ Ux=y \end{cases}$$ -Risolviamo il primo sistema $Ly=b$ con l'[[a0. Sistemi lineari#Algoritmo Di Sostituzione in Avanti|algoritmo di sostituzione in avanti]] e poi $Ux=y$ con l'[[a0. Sistemi lineari#Algoritmo Di Sostituzione All'indietro|algoritmo di sostituzione all'indietro]] ottenendo x. +Risolviamo il primo sistema $Ly=b$ con l'[[a0. Sistemi lineari#Algoritmo Di Sostituzione in Avanti|algoritmo di sostituzione in avanti]] e poi $Ux=y$ con l'[[a0. Sistemi lineari#Algoritmo Di Sostituzione All'indietro|algoritmo di sostituzione all'indietro]] ottenendo $x$. ## Quante Operazioni Richiede? Il costo totale delle operazioni sulla matrice $A$ è di$$\sum_{k=1}^{n-1}{[2(n-k)^2+(n-k)]}$$ diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a3a. Esercizio.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a3a. Esercizio.md index ceb5b85eb..82932d841 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a3a. Esercizio.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a3a. Esercizio.md @@ -1,16 +1,16 @@ -Descrivere una variante strutturata dell'algoritmo di Gauss che richieda non più di 10n operazioni per la soluzione del sistema lineare $Ax=b$ con A matrice tridiagonale. +Descrivere una variante strutturata dell'algoritmo di Gauss che richieda non più di $10n$ operazioni per la soluzione del sistema lineare $Ax=b$ con $A$ matrice tridiagonale. $$[A|b]\to ... \to [U|\tilde{b}]$$ $$Ux=\tilde{b}$$ -$\frac 23n^3$ operazioni algoritmo di Gauss applicato ad A +$\frac 23n^3$ operazioni algoritmo di Gauss applicato ad $A$ -$n^2$ operazioni per aggiornare b +$n^2$ operazioni per aggiornare $b$ $n^3$ operazioni per risolvere $Ux=\tilde b$ -A è tridiagonale se $a_{ij} = 0, |i-j|>1$ +$A$ è tridiagonale se $a_{ij} = 0, |i-j|>1$ Al primo passo del metodo di Gauss è necessario eliminare gli elementi @@ -28,7 +28,7 @@ $$r_3^T \gets r_3^T - l_{32} r_2^T$$ che richiede 3 operazioni. -L'algoritmo richiede 3(n-1) passi in totale +L'algoritmo richiede $3(n-1)$ passi in totale Implementazione: @@ -48,9 +48,9 @@ Algoritmo modificato a(h+1,h=0)=0 b(h+1) = b(h+1) - l(h+1, h)*b(h) -Per calcolare U sono sufficienti 3(n-1) operazioni, per calcolare $\tilde b$ sono sufficienti 2(n-1) operazioni +Per calcolare $U$ sono sufficienti $3(n-1)$ operazioni, per calcolare $\tilde b$ sono sufficienti $2(n-1)$ operazioni -Risolvere $Ux = \tilde b$ sono sufficienti 3(n-1) + 1 operazioni +Per risolvere $Ux = \tilde b$ sono sufficienti $3(n-1) + 1$ operazioni $8n @@ -68,7 +68,7 @@ x_1\\x_2\\\vdots\\x_n \tilde{b_1}\\\tilde{b_2}\\\vdots\\\tilde{b_n}\\ \end{bmatrix}$$ -Si può risolvere con O(n) operazioni? +Si può risolvere con $O(n)$ operazioni? Modo 1 @@ -84,6 +84,6 @@ U_{nn}x_n=\tilde{b_n}\\ for i=n-1:-1:1 x(i) = (bt(i)- u(i,i+1) *x(i+1)) / u(i,i) -3(n-1) + 1 operazioni $\sim$ 3n operazioni +$3(n-1) + 1$ operazioni $\sim$ $3n$ operazioni [[a3. Fattorizzazione LU]] \ No newline at end of file diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a4. Variante del Massimo Pivot Parziale (GEPP).md b/triennale/Anno 2/Calcolo Numerico/argomenti/a4. Variante del Massimo Pivot Parziale (GEPP).md index 0107c3ac8..1a7344588 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a4. Variante del Massimo Pivot Parziale (GEPP).md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a4. Variante del Massimo Pivot Parziale (GEPP).md @@ -29,7 +29,7 @@ a_{22} &...\\ a_{33} &...\\ \vdots & \vdots\\ a_{nn} &...\\ -\end{bmatrix} \to |a_{p1}| = max\{|a_{11}|,...,|a_{nn}|\}\to B_1= \begin{bmatrix} +\end{bmatrix} \to |a_{p1}| = \max\{|a_{11}|,...,|a_{nn}|\}\to B_1= \begin{bmatrix} a_{p1} &...\\ a_{22} &...\\ a_{33} &...\\ @@ -72,7 +72,7 @@ $argmax\{|a_{11}|,...,|a_{nn}|\}$ è l'insieme degli indici in cui il massimo vi # Costo Computazionale -I confronti al primo h sono +I confronti al primo $h$ sono $$n-(h+1) + 1 = n-h$$ E i valori assoluti da calcolare $$n-h+1$$ @@ -90,21 +90,21 @@ $$\begin{bmatrix} \vdots & \vdots\\ 0 & \dots \end{bmatrix}$$ -Al primo passo il metodo si arresta se la prima colonna è nulla $\implies$ A non è invertibile $\iff$ det(A)=0 +Al primo passo il metodo si arresta se la prima colonna è nulla $\implies$ $A$ non è invertibile $\iff$ $det(A)=0$ ```ad-important title: Teorema -Sia $A \in K^{nxn}$ invertibile (condizione sufficiente). Il metodo di Gauss con la variante del massimo pivot parziale è applicabile ad A. +Sia $A \in K^{nxn}$ invertibile (condizione sufficiente). Il metodo di Gauss con la variante del massimo pivot parziale è applicabile ad $A$. ``` ```ad-fail title: Dimostrazione -Se il metodo fallisce al passo $\ell$ allora A non è invertibile perché si avrà che $a_{ll}=0$, di conseguenza se l'elemento con il pivot massimo è 0, tutti gli altri elementi al disotto sono 0 (il GEPP funziona con i valori assoluti). +Se il metodo fallisce al passo $\ell$ allora $A$ non è invertibile perché si avrà che $a_{ll}=0$, di conseguenza se l'elemento con il pivot massimo è 0, tutti gli altri elementi al disotto sono 0 (il GEPP funziona con i valori assoluti). -Calcoliamo il det(A) sviluppandolo rispetto alla prima colonna, procedendo in questo modo anche per la colonna 2 e così via. +Calcoliamo il $det(A)$ sviluppandolo rispetto alla prima colonna, procedendo in questo modo anche per la colonna 2 e così via. Si arriva ad $A_l$ tramite operazioni che non modificano il determinante (combinazioni di righe) e scambi che non modificano il segno $$|det(A)| = |det(A_l)| = 0 \implies det(A)=0$$ @@ -178,4 +178,4 @@ Quindi, con la variante del massimo pivot parziale, si hanno molti scambi di rig Si può scrivere quindi $det(A)=(-1)^sdet(U)$, con $s$ il numero di scambi di righe. Ottenendo la triangolare superiore $U$, il determinante si trova solo sulla diagonale, e quindi $$det(U) = (-1)^s \prod_{i=1}^n{u_{ii}}$$ -Il costo del prodotto è di $n-1$ operazioni, quindi è trascurabile rispetto al metodo di gauss. +Il costo del prodotto è di $n-1$ operazioni, quindi è trascurabile rispetto al metodo di Gauss. diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a5. Curva di Bezier.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a5. Curva di Bezier.md index 92417b3e2..a79196c8e 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a5. Curva di Bezier.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a5. Curva di Bezier.md @@ -21,10 +21,10 @@ N=1 $$B_{0,1}(t)=\substack{\sum_i^1{(\substack{1\\i}) t^i( 1-t)^{1-i}P_i} =\\ # Inviluppo convesso di n+1 punti -L'inviluppo convesso in n+1 punti in $R^n$ è il più piccolo insieme convesso che li contiene +L'inviluppo convesso in $n+1$ punti in $R^n$ è il più piccolo insieme convesso che li contiene # Proprietà delle curve di bezier -Il vettore tangente (derivata) su B in 0 è $n(v_1 - v_0)$ e ci permette di determinare la retta tangente di $\gamma$ in $\gamma (t_n)$ è $\gamma(t_0) + \gamma' (t_0) t$ +Il vettore tangente (derivata) su $B$ in 0 è $n(v_1 - v_0)$ e ci permette di determinare la retta tangente di $\gamma$ in $\gamma (t_n)$ è $\gamma(t_0) + \gamma' (t_0) t$ diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a6. Modello della corda elastica.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a6. Modello della corda elastica.md index ee8af11ee..a4917cea3 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a6. Modello della corda elastica.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a6. Modello della corda elastica.md @@ -2,7 +2,7 @@ 2. Trovare una funzione che ne deriva la forma. 3. Trovare il valore di questa funzione in alcuni punti. -n masse uguali e equidistanti collegate tra loro da molle. +$n$ masse uguali e equidistanti collegate tra loro da molle. 4. Scriviamo l'equazione dell'equilibrio per singola massa - Ogni massa si può muovere solo in verticale @@ -34,4 +34,4 @@ $$\begin{cases} 2y_n - y_{n-1} = - \frac{mg}k \end{cases}$$ -Il sistema ha n equazioni e n incognite, è un sistema lineare del tipo $Ax=b$, con A tridiagonale \ No newline at end of file +Il sistema ha $n$ equazioni e $n$ incognite, è un sistema lineare del tipo $Ax=b$, con $A$ tridiagonale \ No newline at end of file diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md index 6af9d0947..dd5c1d4c6 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md @@ -1,7 +1,7 @@ ```ad-hint title: Problema dell'interpolazione polinomiale -Dati $x_0,..,x_n \in [a,b]$ e dati $y_0,...,y_n \in R$ trovare un polinomio p(x) di grado al più n tale che: +Dati $x_0,..,x_n \in [a,b]$ e dati $y_0,...,y_n \in R$ trovare un polinomio $p(x)$ di grado al più $n$ tale che: $$p(x_0) = y_0,\quad p(x_1) = y_1,\quad p(x_n) = y_n$$ ``` @@ -21,7 +21,7 @@ $$V= $$ ```ad-note -Il problema è ben posto se e solo se V è invertibile:: +Il problema è ben posto se e solo se $V$ è invertibile: $$det(V) = \prod_{\substack{i>j\\i,j=0}}^2{(x_i - x_j)} = (x_1 - x_0)(x_2 - x_1) (x_2 - x_0)$$ ``` @@ -81,7 +81,7 @@ $$ L'utilizzo principale del polinomio di interpolazione -# Polinomi Di LAGRANGE Associati a $x_0,...,x_n$ Distinti +# Polinomi di Lagrange Associati a $x_0,...,x_n$ Distinti - $L_0(x)= \frac{(x-x_1)(x-x_2)...(x-x_n)}{(x_0 - x_1) (x_0 - x_2) ... (x_0 - x_n)}$ - ... @@ -102,7 +102,7 @@ $$L_0(x)=\frac{x - 3}{-1 -3} = \frac {x - 3}{-4} \quad\quad L_1(x) = \frac{x - ( ``````ad-tip title: Teorema -Siano $x_0,..,x_n \in [a,b]$ distinti e siano $L_0(x),...,L_n(x)$ i polinomi di LAGRANGE allora +Siano $x_0,..,x_n \in [a,b]$ distinti e siano $L_0(x),...,L_n(x)$ i polinomi di Lagrange allora 1. $L_i(x_i) = \delta_{ij} : \begin{cases} 1 & i=j \\ 0 & i \neq j \end{cases}$ 2. 2 polinomi di Lagrange sono una base di $R_n[x]$ @@ -118,12 +118,12 @@ collapse: true $L_i(x_i) = \frac{(x_i - x_0)...(x_i - x_{i-1})(x_i - x_{i+1})}{(x_i - x_0)...(x_i - x_{i-1})(x_i - x_{i+1})} = 1, \quad i = j$ -2. Siccome $R_n[x]$ ha dimensione n+1 è sufficiente dimostrare che sono linearmente indipendenti +2. Siccome $R_n[x]$ ha dimensione $n+1$ è sufficiente dimostrare che sono linearmente indipendenti $$\sum {\alpha_i L_i(x)} = 0 \implies \alpha_i = 0\quad i=0,...,n$$ valutiamo la combinazione lineare in $x_j \quad j = 0,..,n$ $$0=\sum_{i=0}^n {\alpha_iL_i(x_j)} = \alpha_jL_j(x_j) = \alpha_j$$ -3. $p(x) = \sum_{i=0}^n{y_iL_i(x)}$ dobbiamo dimostrare che $p(x_j) = y_j$ poiché p ha grado al più n e i nodi sono distinti +3. $p(x) = \sum_{i=0}^n{y_iL_i(x)}$ dobbiamo dimostrare che $p(x_j) = y_j$ poiché $p$ ha grado al più $n$ e i nodi sono distinti $$ p(x_j) = \sum_{i=0}^n {y_i L_i(x_j)} = y_jL_j(x_j) = y_j$$ ``` From 2e0530a506ed775af7399673a99b0070aac1cb0e Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Thu, 17 Aug 2023 12:45:25 +0200 Subject: [PATCH 22/37] Apply suggestions from code review MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- .../Anno 2/Calcolo Numerico/argomenti/a3. Fattorizzazione LU.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a3. Fattorizzazione LU.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a3. Fattorizzazione LU.md index 9bf9cf13c..90311a412 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a3. Fattorizzazione LU.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a3. Fattorizzazione LU.md @@ -58,7 +58,7 @@ $$l_{ij} = \frac{a_{ij}}{a_{jj}}$$ Il risultato saranno poi due matrici triangolari: - $U$ triangolare superiore -- L triangolare inferiore +- $L$ triangolare inferiore From a4f52d45012e6f964cf8c29dedf92d803efdf68c Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Thu, 17 Aug 2023 12:46:36 +0200 Subject: [PATCH 23/37] Update a3a. Esercizio.md --- triennale/Anno 2/Calcolo Numerico/argomenti/a3a. Esercizio.md | 4 +--- 1 file changed, 1 insertion(+), 3 deletions(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a3a. Esercizio.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a3a. Esercizio.md index 82932d841..0a183843f 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a3a. Esercizio.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a3a. Esercizio.md @@ -52,8 +52,6 @@ Per calcolare $U$ sono sufficienti $3(n-1)$ operazioni, per calcolare $\tilde b$ Per risolvere $Ux = \tilde b$ sono sufficienti $3(n-1) + 1$ operazioni -$8n - $$\begin{bmatrix} a_{11}&a_{12}&0&...&...&0\\ &a_{22}&a_{23}&0&...&0\\ @@ -86,4 +84,4 @@ U_{nn}x_n=\tilde{b_n}\\ $3(n-1) + 1$ operazioni $\sim$ $3n$ operazioni -[[a3. Fattorizzazione LU]] \ No newline at end of file +[[a3. Fattorizzazione LU]] From 55072c6547a62ab4beb8ae5474730bc267a18999 Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Thu, 17 Aug 2023 12:53:14 +0200 Subject: [PATCH 24/37] Update a5. Curva di Bezier.md --- .../argomenti/a5. Curva di Bezier.md | 12 +++++------- 1 file changed, 5 insertions(+), 7 deletions(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a5. Curva di Bezier.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a5. Curva di Bezier.md index a79196c8e..712ed7a23 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a5. Curva di Bezier.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a5. Curva di Bezier.md @@ -5,16 +5,14 @@ $$B(t)=\sum_{i=0}^n{(\substack{n\\i})t^i(1-t)^{n-i}P_i}$$ è detta curva di Bezier ```ad-example -title: Esempi +title: Esempio -## 1 - -N=1 $$B_{0,1}(t)=\substack{\sum_i^1{(\substack{1\\i}) t^i( 1-t)^{1-i}P_i} =\\ -(\substack{1\\0}) t^0( 1-t)^{1} P_0 + (\substack{1\\1}) t^1( 1-t)^{0}P_1 = \\ -1 \cdot 1 (1-t)^1P_0 + 1t(1-t)^0P_1}= -(1-t)P_0 + tP_1$$ +N=1 +$$B_{0,1}(t) = \sum_{i = 0}^1 {(\substack{1 \\ i}) t^i (1-t)^{1-i} P_i } = $$ +$$ = (\substack{1\\0}) t^0( 1-t)^{1} P_0 + (\substack{1\\1}) t^1( 1-t)^{0}P_1 =$$ +$$= 1 \cdot 1 (1-t)^1P_0 + 1t(1-t)^0P_1 = (1-t)P_0 + tP_1$$ ``` From b0f2f442ad350ff693cf3beeacb8828f8398f60f Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Sun, 20 Aug 2023 11:37:59 +0200 Subject: [PATCH 25/37] Update a7. Interpolazione.md --- .../Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md index dd5c1d4c6..5517f89ab 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md @@ -35,9 +35,9 @@ Il problema dell'interpolazione polinomiale è ben posto se e solo se i nodi son title: Dimostrazione Siccome è un sistema lineare, deve essere $det(V) \neq 0$ ma: -$$det(v) \neq 0 \substack{\iff & x_i-x_j \neq 0 & i > j \\ -\iff& x_i \neq x_j & i \neq j\\ -\iff &\text{i nodi sono distinti}}$$ +$$det(v) \neq 0 \iff x_i-x_j \neq 0 +\iff x_i \neq x_j +\iff \text{i nodi sono distinti}$$ ``` From 8977783665966795aa9f5c6bb7a39438d2b88556 Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Sun, 20 Aug 2023 11:44:28 +0200 Subject: [PATCH 26/37] Update a0. Sistemi lineari.md --- .../Calcolo Numerico/argomenti/a0. Sistemi lineari.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md index 3c9dcc78e..1dd00acfd 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md @@ -39,7 +39,7 @@ $$|c_1|...|c_n|=\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}$$ - una soluzione esiste - la soluzione è unica - la soluzione dipende in modo continuo dai dati -la soluzione è unica $\iff rk(A)=rk(A|b)=n \iff n-rk(A)=0$ +la soluzione è unica $\iff rango(A)=rango(A|b)=n \iff n-rango(A)=0$ ```ad-important title: Teorema @@ -48,13 +48,13 @@ Il sistema lineare $Ax=b$ è ben posto se e solo se $A$ è quadrata e invertibil ## Caso 1 -$mn \quad rk(A)=rk(A|b)=n \iff$ la soluzione non è unica, anche se il sistema ammette una soluzione unica per un dato $b$, esiste $c: \forall \varepsilon >0$ il sistema $Ax=c+\varepsilon c$ non ammette soluzioni $\implies$ la soluzione non può dipendere in modo continuo dai dati +$m>n \quad rango(A)=rango(A|b)=n \iff$ la soluzione non è unica, anche se il sistema ammette una soluzione unica per un dato $b$, esiste $c: \forall \varepsilon >0$ il sistema $Ax=c+\varepsilon c$ non ammette soluzioni $\implies$ la soluzione non può dipendere in modo continuo dai dati $$Im(A) =\{v \in K^n, v=Ax, x\in R^n\} \subseteq K^n$$ -$$dim(Im(A))=rk(A)\leq n0$$ per ogni $\varepsilon >0$ questo sistema non ha soluzione se esiste $y: Ay=b+c\varepsilon$ @@ -127,4 +127,4 @@ $$\begin{bmatrix} s = b(i) for j = 1:i-1 s = s - t(i,j) * x(j) - x(i) = s / t(i,i) \ No newline at end of file + x(i) = s / t(i,i) From c517608bfbbb494e773996ead09e24edf564fda2 Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Sun, 20 Aug 2023 11:45:56 +0200 Subject: [PATCH 27/37] Update a0. Sistemi lineari.md --- .../Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md index 1dd00acfd..c26f8df68 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md @@ -12,7 +12,7 @@ Se esiste almeno una soluzione allora l'insieme delle soluzioni è un sottospazi $K^n$ spazio vettoriale -$v$ sottospazio vettoriale di dimensione $l$ con $o \leq l \leq n$ +$v$ sottospazio vettoriale di dimensione $l$ con $0 \leq l \leq n$ $v_0 \in K^n$ la traslazione di $v_0, t_0$ è l'applicazione From c338f11d1f1ea7fbac920daab2efbc60486f1eaf Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Sun, 20 Aug 2023 11:51:02 +0200 Subject: [PATCH 28/37] Update 2. Error Analysis.md --- .../Anno 2/Calcolo Numerico/argomenti/2. Error Analysis.md | 5 +++-- 1 file changed, 3 insertions(+), 2 deletions(-) diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/2. Error Analysis.md b/triennale/Anno 2/Calcolo Numerico/argomenti/2. Error Analysis.md index 4ea6bd7a5..4133c59f6 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/2. Error Analysis.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/2. Error Analysis.md @@ -60,7 +60,8 @@ Il numero $\sum_{i=1}^\infty {\beta^{-i}d_i}$ si dice __mantissa__ Data una base di numerazione $\beta \geq 2$, il numero $t>0$ di cifre della mantissa $m$, $M$ numeri positivi, definiamo un insieme di __floating point numbers__ -$$F(\beta, t, m, M) = \{0\} \cup \{\pm \beta^p\sum_{i=1}^t {\beta^{-i}d_i}, \; -m \leq p \leq M, \; 0 \leq d_i < \beta \text{ intero per } i=1,...,t, \; d_1 \neq 0\}$$ +$$F(\beta, t, m, M) = \{0\} \cup \{\pm \beta^p\sum_{i=1}^t {\beta^{-i}d_i}, \; -m \leq p \leq M\}$$ +$$\text {con} \; 0 \leq d_i < \beta \text{ intero per } i=1,...,t, \; d_1 \neq 0$$ ```ad-example title: Esempio @@ -139,4 +140,4 @@ Non seguono infatti: - semplificazione - potrebbe succedere che $x \otimes y = z \otimes y, \quad y \neq 0, \quad x \neq z$ -[[3. Errore Per Le Funzioni Razionali]] \ No newline at end of file +[[3. Errore Per Le Funzioni Razionali]] From b679b97fde930d5399aebeb3e5d04358c3cdcfbc Mon Sep 17 00:00:00 2001 From: michelevantaggi02 Date: Sun, 20 Aug 2023 12:01:34 +0200 Subject: [PATCH 29/37] Merge dei file modificati --- .../Anno 2/Calcolo Numerico/CalcNum-merged.md | 200 ++++++++---------- .../Calcolo Numerico/CalcNum-merged.pdf | Bin 431444 -> 422804 bytes 2 files changed, 93 insertions(+), 107 deletions(-) diff --git a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md index f60db6a8d..f399d1b10 100644 --- a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md +++ b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md @@ -49,7 +49,7 @@ Non esiste una formula per le equazioni di 5° grado La materia si occupa di dare strumenti teorici (teorie matematiche) per il calcolo numerico di quantità continue. -Si cerca di trovare una soluzione ai problemi continui che sia utilizzabile. Non basta una formula, ma serve un algoritmo ce fornisca una soluzione in un tempo ragionevole tenendo conto che il calcolatore lavora in quantità finite. Vogliamo quindi trattare __quantità infinite__ con __strumenti finiti__. +Si cerca di trovare una soluzione ai problemi continui che sia utilizzabile. Non basta una formula, ma serve un algoritmo che fornisca una soluzione in un tempo ragionevole tenendo conto che il calcolatore lavora in quantità finite. Vogliamo quindi trattare __quantità infinite__ con __strumenti finiti__. # Quantità Continue @@ -168,9 +168,10 @@ Il numero $\sum_{i=1}^\infty {\beta^{-i}d_i}$ si dice __mantissa__ ## Floating Point -Data una base di numerazione $\beta \geq 2$, il numero $t>0$ di cifre della mantissa, $m,M$ numeri positivi, definiamo un insieme di __floating point numbers__ +Data una base di numerazione $\beta \geq 2$, il numero $t>0$ di cifre della mantissa $m$, $M$ numeri positivi, definiamo un insieme di __floating point numbers__ -$$F(\beta, t, m, M) = \{0\} \cup \{\pm \beta^p\sum_{i=1}^t {\beta^{-i}d_i}, \; -m \leq p \leq M, \; 0 \leq d_i < \beta \text{ intero per } i=1,...,t, \; d_1 \neq 0\}$$ +$$F(\beta, t, m, M) = \{0\} \cup \{\pm \beta^p\sum_{i=1}^t {\beta^{-i}d_i}, \; -m \leq p \leq M\}$$ +$$\text {con} \; 0 \leq d_i < \beta \text{ intero per } i=1,...,t, \; d_1 \neq 0$$ ```ad-example title: Esempio @@ -251,20 +252,21 @@ Non seguono infatti: [[3. Errore Per Le Funzioni Razionali]] + # 3. Errore Per Le Funzioni Razionali -Data una funzione razionale $f: R^2 \to R$ che sia $f = \frac pq$ con p e q polinomi. +Data una funzione razionale $f: R^2 \to R$ che sia $f = \frac pq$ con $p$ e $q$ polinomi. -Dall'analisi matematica sappiamo che $f$ è definita e differenziabile per $q \neq 0$ (assumendo che p e q siano primi). +Dall'analisi matematica sappiamo che $f$ è definita e differenziabile per $q \neq 0$ (assumendo che $p$ e $q$ siano primi). -Ci sono due tipi di errori nella valutazione di f con i valori in virgola mobile +Ci sono due tipi di errori nella valutazione di $f$ con i valori in virgola mobile ## Errore Inerente -Non valutiamo f(x) ma valutiamo $f(\tilde x)$ dove $\tilde x = fl(x)$ [[2. Error Analysis#^59010a|funzione approssimata]] +Non valutiamo $f(x)$ ma valutiamo $f(\tilde x)$ dove $\tilde x = fl(x)$ [[2. Error Analysis#^59010a|funzione approssimata]] $$\varepsilon_{IN} = \frac{f(\tilde x) -f(x)}{f(x)}, \quad f(x) \neq 0$$ @@ -287,7 +289,7 @@ L'errore algoritmico può essere definito anche per le funzioni elementari che v $$\varepsilon_{TOT} = \frac {\tilde f(\tilde x ) - f(x)}{f(x)}, \quad f(x) \neq 0$$ Dà una misura genuina dell'errore nella valutazione. -La situazione ideale è $|\varepsilon_{tot}| < u$, ma in pratica è sufficiente $|\varepsilon_{tot}| < Mu$ con M costante. +La situazione ideale è $|\varepsilon_{tot}| < u$, ma in pratica è sufficiente $|\varepsilon_{tot}| < Mu$ con $M$ costante. ```ad-important title: Teorema @@ -322,7 +324,7 @@ __Esiste un'altra formula__ per calcolare l'errore inerente: $$\varepsilon_{IN} = \frac x{f(x)}f'(\xi)\varepsilon_x$$ -Dove $\varepsilon_x = \varepsilon_1$ è la rappresentazione dell'errore in x se $f \in C^2(conv(x, \tilde x))$ allora: +Dove $\varepsilon_x = \varepsilon_1$ è la rappresentazione dell'errore in $x$ se $f \in C^2(conv(x, \tilde x))$ allora: $$\varepsilon_{IN} = \frac x{f(x)} f'(x)\varepsilon_x + o(u) = \frac {x \cdot 2x}{x^2}\varepsilon_x = 2\varepsilon_x$$ ```ad-done @@ -360,7 +362,7 @@ Un problema è composto di tre parti: Sistemi lineari -Data una matrice A (coefficienti) e un vettore b (lato destro), trovare tutti i vettori x (incognite) tali che $Ax = b$ +Data una matrice $A$ (coefficienti) e un vettore $b$ (lato destro), trovare tutti i vettori $x$ (incognite) tali che $Ax = b$ ``` @@ -439,8 +441,8 @@ $$|\varepsilon_{IN}| = |\frac{x\varepsilon_x + y\varepsilon_y}{x + y}| \leq \fra Dati $p$ e $x$, calcolare $p(x)$ - $R^n(K^n)$ vettori con $n$ componenti in $R(K)$ -- $R_n(x)(K_n(x))$ polinomi di grado al più n con coefficienti reali -- $R^{nxm}(K^{nxm})$ matrici n x m con elementi in R +- $R_n(x)(K_n(x))$ polinomi di grado al più $n$ con coefficienti reali +- $R^{nxm}(K^{nxm})$ matrici $n \times m$ con elementi in $R$ - $R^{n_1 x n_2 x ... x n_l}$ tensore di $l$ dimensioni - $p(x) = a_0 + a_1 x + ... + a_n x^n$ @@ -538,7 +540,7 @@ Ci sono anche altre interpretazioni, come quello impiegato nel deep learning # Prodotto Tra Matrici Di Strassen -Algoritmo per semplificare il prodotto tra due matrici 2x2. +Algoritmo per semplificare il prodotto tra due matrici $2 \times 2$. La moltiplicazione riga per colonna impiega 8 moltiplicazioni e 4 addizioni. @@ -546,7 +548,7 @@ L'algoritmo di Strassen impiega 7 moltiplicazioni e 18 addizioni. Può sembrare che impieghi un tempo peggiore rispetto alla riga per colonna, ma la moltiplicazione tra due vettori impiega molto di più che l'addizione, rendendo quindi l'algoritmo più veloce. -L'algoritmo viene poi trasformato in moltiplicazioni in blocchi. Rendendo qualsiasi moltiplicazione tra matrici $2^nx2^n$ come 7 moltiplicazioni e alcune addizioni tra 4 matrici $2^{(n-1)}x2^{(n-1)}$ ricorsivamente. +L'algoritmo viene poi trasformato in moltiplicazioni in blocchi. Rendendo qualsiasi moltiplicazione tra matrici $2^n \times 2^n$ come 7 moltiplicazioni e alcune addizioni tra 4 matrici $2^{(n-1)} \times 2^{(n-1)}$ ricorsivamente. Questo ha fatto scaturire una corsa all'algoritmo più veloce per risoluzione di prodotti tra matrici. @@ -580,7 +582,7 @@ $$A \in K^{nxn}$$ $$numZeri(A) << n^2$$ -si riesce a implementare il [[7. Vettori#Prodotto Matrice per Vettore|prodotto matrice-vettore]] con $2l$ operazione anziché con $2n^2$ +si riesce a implementare il [[7. Vettori#Prodotto Matrice per Vettore|prodotto matrice-vettore]] con $2l$ operazioni anziché con $2n^2$ @@ -610,7 +612,7 @@ $$\lim\limits_{n \to \infty}{\frac {numZeri(A)}{n^2}} = 0$$ ```ad-check -Sono strutture sparse, sono sufficienti $2p +Sono strutture sparse, sono sufficienti $2p$. ``` `````` @@ -643,12 +645,12 @@ title: è una sottoalgebra di $K^{nxn}$? Un'algebra è un sottospazio vettoriale in cui è definito un prodotto compatibile con le operazioni dello spazio vettoriale. -V è un gruppo se dati $A,B,C \in V$ e $\alpha \in K$: +$V$ è un gruppo se dati $A,B,C \in V$ e $\alpha \in K$: - $A(B + C) = AB + AC$ - $(A + B)C = AC + BC$ - $\alpha(AB) = (\alpha A)B = A(\alpha B)$ -Le matrici nxn sono un'algebra con il prodotto riga per colonna +Le matrici $n \times n$ sono un'algebra con il prodotto riga per colonna ```ad-check title: Sottoalgebra @@ -710,13 +712,13 @@ $$\begin{cases}a_{1,1}x_1 +. .. a_{1,n}x_n=b_1 \\ a_{2,1}x_1 +. .. a_{2,n}x_n=b_ # Teorema Di Rouché-Capelli -Il sistema lineare Ax = b ammette soluzione se e solo se rango(A) = rango(A|b). +Il sistema lineare $Ax = b$ ammette soluzione se e solo se rango(A) = rango(A|b). Se esiste almeno una soluzione allora l'insieme delle soluzioni è un sottospazio affine di dimensione n-rango(A). $K^n$ spazio vettoriale -v sottospazio vettoriale di dimensione l $o \leq l \leq n$ +$v$ sottospazio vettoriale di dimensione $l$ con $0 \leq l \leq n$ $v_0 \in K^n$ la traslazione di $v_0, t_0$ è l'applicazione @@ -724,17 +726,17 @@ $$t_{v_0}:K^n \to K^n$$ $$v \to v+v_0$$ -L'insieme $W=\{v+v_0, v \in V, v_0 \in K^n\}$ è un sottospazio affine di $K^n$ di dimensione dim(V). +L'insieme $W=\{v+v_0, v \in V, v_0 \in K^n\}$ è un sottospazio affine di $K^n$ di dimensione $dim(V)$. ```ad-info - I sottospazi affini di dimensione 1 sono detti rette - I sottospazi affini di dimensione 2 sono detti piani -- I sottospazi affini di dimensione n-1 sono detti iperpiani +- I sottospazi affini di dimensione $n-1$ sono detti iperpiani - I sottospazi affini di dimensione 0 sono detti punti (non sono sottospazi) ``` -$Ax=b \iff c_1x_1+c_2x_2+...+c_nx_n=b \iff$ b è combinazione lineare delle colonne di A $\iff$ rango(A)=rango(A|b) +$Ax=b \iff c_1x_1+c_2x_2+...+c_nx_n=b \iff$ $b$ è combinazione lineare delle colonne di $A$ $\iff$ rango(A)=rango(A|b) $$|c_1|...|c_n|=\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}$$ @@ -743,22 +745,22 @@ $$|c_1|...|c_n|=\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}$$ - una soluzione esiste - la soluzione è unica - la soluzione dipende in modo continuo dai dati -la soluzione è unica $\iff rk(A)=rk(A|b)=n \iff n-rk(A)=0$ +la soluzione è unica $\iff rango(A)=rango(A|b)=n \iff n-rango(A)=0$ ```ad-important title: Teorema -Il sistema lineare Ax=b è ben posto se e solo se A è quadrata e invertibile +Il sistema lineare $Ax=b$ è ben posto se e solo se $A$ è quadrata e invertibile ``` ## Caso 1 -$mn \quad rk(A)=rk(A|b)=n \iff$ la soluzione non è unica, anche se il sistema ammette una soluzione unica per un dato b, esiste $c: \forall \varepsilon >0$ il sistema $Ax=c+\varepsilon c$ non ammette soluzioni $\implies$ la soluzione non può dipendere in modo continuo dai dati +$m>n \quad rango(A)=rango(A|b)=n \iff$ la soluzione non è unica, anche se il sistema ammette una soluzione unica per un dato $b$, esiste $c: \forall \varepsilon >0$ il sistema $Ax=c+\varepsilon c$ non ammette soluzioni $\implies$ la soluzione non può dipendere in modo continuo dai dati $$Im(A) =\{v \in K^n, v=Ax, x\in R^n\} \subseteq K^n$$ -$$dim(Im(A))=rk(A)\leq n0$$ per ogni $\varepsilon >0$ questo sistema non ha soluzione se esiste $y: Ay=b+c\varepsilon$ @@ -771,7 +773,7 @@ $m=n\quad det(A)=0 \implies$ la soluzione non esiste o non è unica $\implies$ n ## Caso 4 -A quadrata, $m=n\quad det(A) \neq 0 \implies$ la soluzione è unica. +$A$ quadrata, $m=n\quad det(A) \neq 0 \implies$ la soluzione è unica. Dipende in modo continuo dai dati? $$x=A^{-1}b\quad (A^{-1})_{ij}=\frac{det(\bar{A}^{ij})}{det(A)}(-1)^{i+j}$$ $$x_i=\sum_{l=1}^n{(A^{-1})_{il}b_l}=\sum_{l=1}^n{\frac{det(\bar{A}^{il})}{det(A)}(-1)^{i+l}b_l},\quad i=1,..,n$$ @@ -783,9 +785,9 @@ $$x_i=\sum_{l=1}^n{(A^{-1})_{il}b_l}=\sum_{l=1}^n{\frac{det(\bar{A}^{il})}{det(A $$T_n=\{A\in K^{nxn}, a_{ij}=0, i>j\}$$ $$\begin{bmatrix}a_{1,1}&...&a_{1,n}\\\vdots &\ddots &\vdots\\ 0 &...& a_{n,n} \end{bmatrix}$$ -Sistema lineare Tx=b dove T è triangolare superiore +Sistema lineare $Tx=b$ dove $T$ è triangolare superiore $$\begin{cases}t_{1,1}x_1 + t_{1,2}x_2 + t_{1,3}x_3=b1 \\ \quad\quad\quad\quad t_{2,2}x_2+t_{2,3}x_3=b_2 \\ \quad\quad\quad\quad\quad\quad\quad\quad t_{3,3}x_3=b_3 \end{cases}$$ -Calcolo il termine i-esimo +Calcolo il termine $i$-esimo $$t_{i,i}x_i+t_{i,i+1}x_{i+1}+...+t_{i,n}x_n=b_i$$ $$x_i=\frac{b_i-t_{i,i+1}x_{i+1}-...-t_{i,n}x_n}{t_{i,i}}=\frac{b_i - \sum_{j=i+1}^n{t_{i,j}x_j}}{t_{i,j}}$$ $$Tx=b \implies t_{i,1}x_1+t_{i,2}x_2+...+t_{i,i}x_{i}+...=b_i$$ @@ -799,7 +801,7 @@ $$\sum_{j=i}^n{t_{i,j}x_j}=b_i$$ s = s - t(i,j) * x(j) x(i) = s / t(i,i) -costo di un passo del ciclo esterno: 2(n-i)+1 ops +costo di un passo del ciclo esterno: $2(n-i)+1$ ops $$\sum_{i=1}^n{2(n-i)+1} = 2\sum_{i=1}^n{2n-2i+1}=2\sum_{i=1}^n{n}-2\sum_{i=1}^n{i}+\sum_{i=1}^n{1}=2n^2-n(n+1)+n=n^2 ops$$ ```ad-note @@ -833,12 +835,12 @@ $$\begin{bmatrix} s = s - t(i,j) * x(j) x(i) = s / t(i,i) + # a1. Algoritmo di Gauss -22-11-2022 -Ax=b, A quadrata e invertibile +$Ax=b$, con $A$ quadrata e invertibile for h=1:n-1: for i = h+1:n: @@ -861,7 +863,7 @@ a_{21},&a_{22},&a_{23},&\dots \\ Il primo passo l'algoritmo ha come pivot $a_{11}$ Non ce se la fa a staglie dietro mannaggia -Dato $A\in K^{nxn}$ il minore principale di testa di ordine $j \leq h \leq n$ di A è la matrice $M_h \in K^{nxn}: (M_n)_{ij} = a_{ij}$ +Dato $A\in K^{nxn}$ il minore principale di testa di ordine $j \leq h \leq n$ di $A$ è la matrice $M_h \in K^{nxn}: (M_n)_{ij} = a_{ij}$ ```ad-important title: Teorema @@ -879,7 +881,7 @@ a_{11} & * & \\ && a_{33} &\\ & 0 && a_{44} \end{bmatrix}$$ -Sia $N_h$ il minore principale di testa di U di ordine h, esso è una matrice triangolare e il suo determinante è dato dagli elementi sulla sua diagonale, quindi $det(N_h) = u_{11} u_{22} ... u_{hh}$ +Sia $N_h$ il minore principale di testa di $U$ di ordine $h$, esso è una matrice triangolare e il suo determinante è dato dagli elementi sulla sua diagonale, quindi $det(N_h) = u_{11} u_{22} ... u_{hh}$ Siccome il metodo è applicabile allora $$a_{ii} \neq 0, i=1,...,n-1 \implies a_{11} ... a_{hh} \neq 0\implies det(N_h) \neq 0, h=1,...,n-1$$ @@ -889,31 +891,31 @@ Si ha che $det(N_h) = det(M_h) \implies det(M_h) \neq 0, h=1,...,n-1$ poiché le ``` ```ad-failure -Se il metodo di Gauss fallisce al passo h allora $a_{hh}=0 \quad(a_{jj} \neq 0, j=1,..,h-1)$ e si può costruire la matrice +Se il metodo di Gauss fallisce al passo $h$ allora $a_{hh}=0 \quad(a_{jj} \neq 0, j=1,..,h-1)$ e si può costruire la matrice $$A_h = \begin{bmatrix} a_{11} & & & &\\ & \ddots & & & \\ & & a_{hh}=0 & \\ & & & & & \end{bmatrix}$$ -Sia $N_h$ il minore principale di testa di ordine h di $A_h$, $det(N_h)=0$ +Sia $N_h$ il minore principale di testa di ordine $h$ di $A_h$, $det(N_h)=0$ Ma $det(M_h)=det(N_h)$ per quanto detto prima, quindi il metodo non è applicabile ``` `````` -Se A è una matrice definita positiva allora la condizione del teorema è verificata +Se $A$ è una matrice definita positiva allora la condizione del teorema è verificata $A \in R^{nxn}$ è definita positiva se: - $A^T=A$ (simmetrica) - $v^T Av > 0$ se $v \in R^n \setminus \{0\}$ -Se A è a dominanza diagonale allora la condizione del teorema è verificata +Se $A$ è a dominanza diagonale allora la condizione del teorema è verificata -A è a dominanza diagonale (stretta) se +$A$ è a dominanza diagonale (stretta) se - $|a_{ii}| > \sum_{j=1, j\neq i}^n{|a_{ij}|}$ -A è a dominanza diagonale se: +$A$ è a dominanza diagonale se: - $|a_{ii}| \geq \sum_{j=1, j\neq i}^n{|a_{ij}|}$ - vale il maggiore stretto per almeno un indice @@ -932,7 +934,7 @@ $$[A|b] = [A_1|b_1] \to [A_2|b_2]$$ Si può interpretare il metodo di Gauss come operazioni sulle equazioni $$Ax=b \iff A_1x=b_1 \iff ... \iff A_nx=b_n \iff Ux=\tilde{b}$$ -Applicando il metodo di Gauss si parte da Ax=b e si ottiene un sistema triangolare equivalente $Ux=\tilde{b}$ che si può risolvere con il metodo di [[a0. Sistemi lineari#Algoritmo Di Sostituzione All'indietro|sostituzione all'indietro]] +Applicando il metodo di Gauss si parte da $Ax=b$ e si ottiene un sistema triangolare equivalente $Ux=\tilde{b}$ che si può risolvere con il metodo di [[a0. Sistemi lineari#Algoritmo Di Sostituzione All'indietro|sostituzione all'indietro]] Qual è il costo computazionale? $\frac 23 n^3$ per l'eliminazione @@ -941,7 +943,7 @@ $\sum_{h=1}^n{2(n-h) \sim n^2}$ per il calcolo di $\tilde{b}$ ```ad-note $det(A_1) = det(A_2) = ... = det(U)$ perché le operazioni che eseguiamo non modificano il determinante. -è sufficiente quindi applicare il metodo di Gauss con $\frac 23 n^3$ operazioni (e n-1 moltiplicazioni che si possono trascurare) +è sufficiente quindi applicare il metodo di Gauss con $\frac 23 n^3$ operazioni (e $n-1$ moltiplicazioni che si possono trascurare) ``` @@ -950,7 +952,7 @@ $det(A_1) = det(A_2) = ... = det(U)$ perché le operazioni che eseguiamo non mod $$Ax=b_1, Ax=b_2,...,Ax=b_l$$ risolvendo indipendentemente i sistemi lineari sono richieste $\frac 23 n^3 l$ operazioni. -Applico il metodo di Gauss ad A modificando consistentemente anche le colonne relative ai termini noti +Applico il metodo di Gauss ad $A$ modificando consistentemente anche le colonne relative ai termini noti $$[A|b_1b_2...b_l]=[A_1|b_1^{(1)}b_2^{(1)},...b_l^{(1)}] \to [A_2|b_1^{(2)}b_2^{(2)}...b_l^{(2)}] \to ... \to[A_n|b_1^{(n)}b_2^{(n)}...b_l^{(n)}]=[U|\tilde{b_1} \tilde{b_2}...\tilde{b_3}]$$ $$Ux=\tilde{b_1},Ux=\tilde{b_2},...,Ux=\tilde{b_l}$$ Risolvendo in "parallelo" $\frac 23 n^3l$ operazioni. @@ -960,14 +962,14 @@ al secondo passo? $2l( n-2)$ al passo i? $2l(n-i)$ in totale saranno $\sum_{i=1}^{n-1}{2l(n-i)}$ $$\sum_{i=1}^{n-1}{2l(n-i)} = 2l \sum_{i=1}^{n-1}{(n-i)}\sim\frac {2ln^2}2 = ln^2$$ -$\frac 23 n^3 + ln^2$ operazioni anziché $\frac 23 n^3 l$ per trovare i sistemi $Ux=\tilde{b_1},...,Ux=\tilde{b_l}$ per risolverli sono necessarie l sostituzioni all'indietro e quindi altre $ln^2$ operazioni. +$\frac 23 n^3 + ln^2$ operazioni anziché $\frac 23 n^3 l$ per trovare i sistemi $Ux=\tilde{b_1},...,Ux=\tilde{b_l}$ per risolverli sono necessarie $l$ sostituzioni all'indietro e quindi altre $ln^2$ operazioni. In totale sono $\frac 23 n^3 + 2n^2 l$ operazioni per risolvere l sistemi lineari con sistemi coefficienti ## Calcolare l'inversa di una matrice -X è l'inversa di A (nxn) -se AX=I +$X$ è l'inversa di $A$ ($n \times n$) +se $AX=I$ $$A[X_1|X_2|...|X_n] = [F_1|F_2|...|F_n]$$ $$[AX_1|AX_2|...|AX_n] = [F_1|F_2|...|F_n]$$ $$AX_1=F_1,AX_2=F_2,...,AX_n=F_n$$ @@ -1011,25 +1013,12 @@ $$ 0&1&1 \end{bmatrix} -\begin{bmatrix} - -1&1&0\\ - -1&1&1\\ - -0&1&1 -\end{bmatrix} = \begin{bmatrix} -2&2&1\\ -2&3&2\\ -1&2&2 + \end{bmatrix} $$ - -Come si vede dall'esempio, il prodotto tra due matrici tridiagonali non ha come risultato una matrice tridiagonale, quindi non è una sottoalgebra - ``` ```ad-question @@ -1039,7 +1028,7 @@ title: Qual è la dimensione vettoriale? $$3n-2$$ ``` -Trovare un algoritmo che calcola il prodotto di una matrice diagonale $A \in T_n$ per un vettore $b \in K^n$ e che non richieda più di 6n operazioni e darne un'implementazione in pseudocodice (non usare il prodotto matrice sparsa-vettore) +Trovare un algoritmo che calcola il prodotto di una matrice diagonale $A \in T_n$ per un vettore $b \in K^n$ e che non richieda più di $6n$ operazioni e darne un'implementazione in pseudocodice (non usare il prodotto matrice sparsa-vettore) Primo approccio: scrivo la matrice e il vettore e vedo cosa succede @@ -1066,7 +1055,7 @@ $$A_1\to A_2 \to ...\to U=A_n = \begin{bmatrix} 0&0&*&*\\ 0&0&0&* \end{bmatrix}$$ -costruiamo una matrice L triangolare inferiore con 1 sulle diagonali tale che $L_{ij}$ è definito dal metodo di Gauss $i>j$ +costruiamo una matrice $L$ triangolare inferiore con 1 sulle diagonali tale che $L_{ij}$ è definito dal metodo di Gauss $i>j$ $$L=\begin{bmatrix} 1&0&0\\ l_{ij}&1&0\\ @@ -1112,12 +1101,12 @@ Al passo 1, se voglio eliminare l'elemento nella riga 2 ($a_{21}$) devo eseguire Bisogna ripetere questi passaggi per ogni colonna usando come pivot la diagonale della matrice -Il rapporto è poi utilizzato per formare la matrice L, infatti ogni elemento di L: +Il rapporto è poi utilizzato per formare la matrice $L$, infatti ogni elemento di $L$: $$l_{ij} = \frac{a_{ij}}{a_{jj}}$$ Il risultato saranno poi due matrici triangolari: -- U triangolare superiore -- L triangolare inferiore +- $U$ triangolare superiore +- $L$ triangolare inferiore @@ -1130,7 +1119,7 @@ $$Ax=b \iff LUx=b \iff L(Ux) = b\implies \begin{cases} Ly=b\\ Ux=y \end{cases}$$ -Risolviamo il primo sistema $Ly=b$ con l'[[a0. Sistemi lineari#Algoritmo Di Sostituzione in Avanti|algoritmo di sostituzione in avanti]] e poi $Ux=y$ con l'[[a0. Sistemi lineari#Algoritmo Di Sostituzione All'indietro|algoritmo di sostituzione all'indietro]] ottenendo x. +Risolviamo il primo sistema $Ly=b$ con l'[[a0. Sistemi lineari#Algoritmo Di Sostituzione in Avanti|algoritmo di sostituzione in avanti]] e poi $Ux=y$ con l'[[a0. Sistemi lineari#Algoritmo Di Sostituzione All'indietro|algoritmo di sostituzione all'indietro]] ottenendo $x$. ## Quante Operazioni Richiede? Il costo totale delle operazioni sulla matrice $A$ è di$$\sum_{k=1}^{n-1}{[2(n-k)^2+(n-k)]}$$ @@ -1156,19 +1145,19 @@ il pivot per $\varepsilon >0$ non è nullo e la soluzione è unica $x=\begin{bma # a3a. Esercizio -Descrivere una variante strutturata dell'algoritmo di Gauss che richieda non più di 10n operazioni per la soluzione del sistema lineare $Ax=b$ con A matrice tridiagonale. +Descrivere una variante strutturata dell'algoritmo di Gauss che richieda non più di $10n$ operazioni per la soluzione del sistema lineare $Ax=b$ con $A$ matrice tridiagonale. $$[A|b]\to ... \to [U|\tilde{b}]$$ $$Ux=\tilde{b}$$ -$\frac 23n^3$ operazioni algoritmo di Gauss applicato ad A +$\frac 23n^3$ operazioni algoritmo di Gauss applicato ad $A$ -$n^2$ operazioni per aggiornare b +$n^2$ operazioni per aggiornare $b$ $n^3$ operazioni per risolvere $Ux=\tilde b$ -A è tridiagonale se $a_{ij} = 0, |i-j|>1$ +$A$ è tridiagonale se $a_{ij} = 0, |i-j|>1$ Al primo passo del metodo di Gauss è necessario eliminare gli elementi @@ -1186,7 +1175,7 @@ $$r_3^T \gets r_3^T - l_{32} r_2^T$$ che richiede 3 operazioni. -L'algoritmo richiede 3(n-1) passi in totale +L'algoritmo richiede $3(n-1)$ passi in totale Implementazione: @@ -1206,11 +1195,9 @@ Algoritmo modificato a(h+1,h=0)=0 b(h+1) = b(h+1) - l(h+1, h)*b(h) -Per calcolare U sono sufficienti 3(n-1) operazioni, per calcolare $\tilde b$ sono sufficienti 2(n-1) operazioni - -Risolvere $Ux = \tilde b$ sono sufficienti 3(n-1) + 1 operazioni +Per calcolare $U$ sono sufficienti $3(n-1)$ operazioni, per calcolare $\tilde b$ sono sufficienti $2(n-1)$ operazioni -$8n +Per risolvere $Ux = \tilde b$ sono sufficienti $3(n-1) + 1$ operazioni $$\begin{bmatrix} a_{11}&a_{12}&0&...&...&0\\ @@ -1226,7 +1213,7 @@ x_1\\x_2\\\vdots\\x_n \tilde{b_1}\\\tilde{b_2}\\\vdots\\\tilde{b_n}\\ \end{bmatrix}$$ -Si può risolvere con O(n) operazioni? +Si può risolvere con $O(n)$ operazioni? Modo 1 @@ -1242,10 +1229,11 @@ U_{nn}x_n=\tilde{b_n}\\ for i=n-1:-1:1 x(i) = (bt(i)- u(i,i+1) *x(i+1)) / u(i,i) -3(n-1) + 1 operazioni $\sim$ 3n operazioni +$3(n-1) + 1$ operazioni $\sim$ $3n$ operazioni [[a3. Fattorizzazione LU]] + # a4. Variante del Massimo Pivot Parziale (GEPP) @@ -1280,7 +1268,7 @@ a_{22} &...\\ a_{33} &...\\ \vdots & \vdots\\ a_{nn} &...\\ -\end{bmatrix} \to |a_{p1}| = max\{|a_{11}|,...,|a_{nn}|\}\to B_1= \begin{bmatrix} +\end{bmatrix} \to |a_{p1}| = \max\{|a_{11}|,...,|a_{nn}|\}\to B_1= \begin{bmatrix} a_{p1} &...\\ a_{22} &...\\ a_{33} &...\\ @@ -1323,7 +1311,7 @@ $argmax\{|a_{11}|,...,|a_{nn}|\}$ è l'insieme degli indici in cui il massimo vi # Costo Computazionale -I confronti al primo h sono +I confronti al primo $h$ sono $$n-(h+1) + 1 = n-h$$ E i valori assoluti da calcolare $$n-h+1$$ @@ -1341,21 +1329,21 @@ $$\begin{bmatrix} \vdots & \vdots\\ 0 & \dots \end{bmatrix}$$ -Al primo passo il metodo si arresta se la prima colonna è nulla $\implies$ A non è invertibile $\iff$ det(A)=0 +Al primo passo il metodo si arresta se la prima colonna è nulla $\implies$ $A$ non è invertibile $\iff$ $det(A)=0$ ```ad-important title: Teorema -Sia $A \in K^{nxn}$ invertibile (condizione sufficiente). Il metodo di Gauss con la variante del massimo pivot parziale è applicabile ad A. +Sia $A \in K^{nxn}$ invertibile (condizione sufficiente). Il metodo di Gauss con la variante del massimo pivot parziale è applicabile ad $A$. ``` ```ad-fail title: Dimostrazione -Se il metodo fallisce al passo $\ell$ allora A non è invertibile perché si avrà che $a_{ll}=0$, di conseguenza se l'elemento con il pivot massimo è 0, tutti gli altri elementi al disotto sono 0 (il GEPP funziona con i valori assoluti). +Se il metodo fallisce al passo $\ell$ allora $A$ non è invertibile perché si avrà che $a_{ll}=0$, di conseguenza se l'elemento con il pivot massimo è 0, tutti gli altri elementi al disotto sono 0 (il GEPP funziona con i valori assoluti). -Calcoliamo il det(A) sviluppandolo rispetto alla prima colonna, procedendo in questo modo anche per la colonna 2 e così via. +Calcoliamo il $det(A)$ sviluppandolo rispetto alla prima colonna, procedendo in questo modo anche per la colonna 2 e così via. Si arriva ad $A_l$ tramite operazioni che non modificano il determinante (combinazioni di righe) e scambi che non modificano il segno $$|det(A)| = |det(A_l)| = 0 \implies det(A)=0$$ @@ -1429,7 +1417,7 @@ Quindi, con la variante del massimo pivot parziale, si hanno molti scambi di rig Si può scrivere quindi $det(A)=(-1)^sdet(U)$, con $s$ il numero di scambi di righe. Ottenendo la triangolare superiore $U$, il determinante si trova solo sulla diagonale, e quindi $$det(U) = (-1)^s \prod_{i=1}^n{u_{ii}}$$ -Il costo del prodotto è di $n-1$ operazioni, quindi è trascurabile rispetto al metodo di gauss. +Il costo del prodotto è di $n-1$ operazioni, quindi è trascurabile rispetto al metodo di Gauss. # a5. Curva di Bezier @@ -1442,27 +1430,25 @@ $$B(t)=\sum_{i=0}^n{(\substack{n\\i})t^i(1-t)^{n-i}P_i}$$ è detta curva di Bezier ```ad-example -title: Esempi - -## 1 +title: Esempio -N=1 $$B_{0,1}(t)=\substack{\sum_i^1{(\substack{1\\i}) t^i( 1-t)^{1-i}P_i} =\\ -(\substack{1\\0}) t^0( 1-t)^{1} P_0 + (\substack{1\\1}) t^1( 1-t)^{0}P_1 = \\ -1 \cdot 1 (1-t)^1P_0 + 1t(1-t)^0P_1}= -(1-t)P_0 + tP_1$$ +N=1 +$$B_{0,1}(t) = \sum_{i = 0}^1 {(\substack{1 \\ i}) t^i (1-t)^{1-i} P_i } = $$ +$$ = (\substack{1\\0}) t^0( 1-t)^{1} P_0 + (\substack{1\\1}) t^1( 1-t)^{0}P_1 =$$ +$$= 1 \cdot 1 (1-t)^1P_0 + 1t(1-t)^0P_1 = (1-t)P_0 + tP_1$$ ``` # Inviluppo convesso di n+1 punti -L'inviluppo convesso in n+1 punti in $R^n$ è il più piccolo insieme convesso che li contiene +L'inviluppo convesso in $n+1$ punti in $R^n$ è il più piccolo insieme convesso che li contiene # Proprietà delle curve di bezier -Il vettore tangente (derivata) su B in 0 è $n(v_1 - v_0)$ e ci permette di determinare la retta tangente di $\gamma$ in $\gamma (t_n)$ è $\gamma(t_0) + \gamma' (t_0) t$ +Il vettore tangente (derivata) su $B$ in 0 è $n(v_1 - v_0)$ e ci permette di determinare la retta tangente di $\gamma$ in $\gamma (t_n)$ è $\gamma(t_0) + \gamma' (t_0) t$ @@ -1474,7 +1460,7 @@ Il vettore tangente (derivata) su B in 0 è $n(v_1 - v_0)$ e ci permette di dete 2. Trovare una funzione che ne deriva la forma. 3. Trovare il valore di questa funzione in alcuni punti. -n masse uguali e equidistanti collegate tra loro da molle. +$n$ masse uguali e equidistanti collegate tra loro da molle. 4. Scriviamo l'equazione dell'equilibrio per singola massa - Ogni massa si può muovere solo in verticale @@ -1506,7 +1492,7 @@ $$\begin{cases} 2y_n - y_{n-1} = - \frac{mg}k \end{cases}$$ -Il sistema ha n equazioni e n incognite, è un sistema lineare del tipo $Ax=b$, con A tridiagonale +Il sistema ha $n$ equazioni e $n$ incognite, è un sistema lineare del tipo $Ax=b$, con $A$ tridiagonale # a7. Interpolazione @@ -1514,7 +1500,7 @@ Il sistema ha n equazioni e n incognite, è un sistema lineare del tipo $Ax=b$, ```ad-hint title: Problema dell'interpolazione polinomiale -Dati $x_0,..,x_n \in [a,b]$ e dati $y_0,...,y_n \in R$ trovare un polinomio p(x) di grado al più n tale che: +Dati $x_0,..,x_n \in [a,b]$ e dati $y_0,...,y_n \in R$ trovare un polinomio $p(x)$ di grado al più $n$ tale che: $$p(x_0) = y_0,\quad p(x_1) = y_1,\quad p(x_n) = y_n$$ ``` @@ -1534,7 +1520,7 @@ $$V= $$ ```ad-note -Il problema è ben posto se e solo se V è invertibile:: +Il problema è ben posto se e solo se $V$ è invertibile: $$det(V) = \prod_{\substack{i>j\\i,j=0}}^2{(x_i - x_j)} = (x_1 - x_0)(x_2 - x_1) (x_2 - x_0)$$ ``` @@ -1548,9 +1534,9 @@ Il problema dell'interpolazione polinomiale è ben posto se e solo se i nodi son title: Dimostrazione Siccome è un sistema lineare, deve essere $det(V) \neq 0$ ma: -$$det(v) \neq 0 \substack{\iff & x_i-x_j \neq 0 & i > j \\ -\iff& x_i \neq x_j & i \neq j\\ -\iff &\text{i nodi sono distinti}}$$ +$$det(v) \neq 0 \iff x_i-x_j \neq 0 +\iff x_i \neq x_j +\iff \text{i nodi sono distinti}$$ ``` @@ -1594,7 +1580,7 @@ $$ L'utilizzo principale del polinomio di interpolazione -# Polinomi Di LAGRANGE Associati a $x_0,...,x_n$ Distinti +# Polinomi di Lagrange Associati a $x_0,...,x_n$ Distinti - $L_0(x)= \frac{(x-x_1)(x-x_2)...(x-x_n)}{(x_0 - x_1) (x_0 - x_2) ... (x_0 - x_n)}$ - ... @@ -1615,7 +1601,7 @@ $$L_0(x)=\frac{x - 3}{-1 -3} = \frac {x - 3}{-4} \quad\quad L_1(x) = \frac{x - ( ``````ad-tip title: Teorema -Siano $x_0,..,x_n \in [a,b]$ distinti e siano $L_0(x),...,L_n(x)$ i polinomi di LAGRANGE allora +Siano $x_0,..,x_n \in [a,b]$ distinti e siano $L_0(x),...,L_n(x)$ i polinomi di Lagrange allora 1. $L_i(x_i) = \delta_{ij} : \begin{cases} 1 & i=j \\ 0 & i \neq j \end{cases}$ 2. 2 polinomi di Lagrange sono una base di $R_n[x]$ @@ -1625,18 +1611,18 @@ $$p(x) = \sum_{i=1}^n {y_iL_i(x)}$$ ```ad-done title: Dimostrazione - +collapse: true 1. $L_i(x_j) = \frac{(x_j-x_0) ... (x_j-x_j)...(x_j - x_{i+1})}{...} = 0, \quad i \neq j$ $L_i(x_i) = \frac{(x_i - x_0)...(x_i - x_{i-1})(x_i - x_{i+1})}{(x_i - x_0)...(x_i - x_{i-1})(x_i - x_{i+1})} = 1, \quad i = j$ -2. Siccome $R_n[x]$ ha dimensione n+1 è sufficiente dimostrare che sono linearmente indipendenti +2. Siccome $R_n[x]$ ha dimensione $n+1$ è sufficiente dimostrare che sono linearmente indipendenti $$\sum {\alpha_i L_i(x)} = 0 \implies \alpha_i = 0\quad i=0,...,n$$ valutiamo la combinazione lineare in $x_j \quad j = 0,..,n$ $$0=\sum_{i=0}^n {\alpha_iL_i(x_j)} = \alpha_jL_j(x_j) = \alpha_j$$ -3. $p(x) = \sum_{i=0}^n{y_iL_i(x)}$ dobbiamo dimostrare che $p(x_j) = y_j$ poiché p ha grado al più n e i nodi sono distinti +3. $p(x) = \sum_{i=0}^n{y_iL_i(x)}$ dobbiamo dimostrare che $p(x_j) = y_j$ poiché $p$ ha grado al più $n$ e i nodi sono distinti $$ p(x_j) = \sum_{i=0}^n {y_i L_i(x_j)} = y_jL_j(x_j) = y_j$$ ``` diff --git a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.pdf b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.pdf index 3f1f17f8852d3b3dfa9a104c2b16fc7a63855b04..fab36303cd73c20ebe0ba98c8d672ceb2bbed666 100644 GIT binary patch delta 239653 zcmYhiQ*b6sv~WGKZQHh;Ol;e>ohP=R*qqq5Z9AFR$;3{+_k2I<-0j-cRoyq;y?XD} z%ayn(r#Oj+(5%eNJRCgi>`9k+|9{+s2r^0yVC5DRBzARkF*mk{^~&kd*s{T6LH6IP 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Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Fri, 6 Oct 2023 11:54:36 +0200 Subject: [PATCH 30/37] Update CalcNum-merged.md --- triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md | 1 - 1 file changed, 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md index f399d1b10..0c285590e 100644 --- a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md +++ b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md @@ -1611,7 +1611,6 @@ $$p(x) = \sum_{i=1}^n {y_iL_i(x)}$$ ```ad-done title: Dimostrazione -collapse: true 1. $L_i(x_j) = \frac{(x_j-x_0) ... (x_j-x_j)...(x_j - x_{i+1})}{...} = 0, \quad i \neq j$ From b7ba0cd7507b324ffbdca1523b1a79ebb6536005 Mon Sep 17 00:00:00 2001 From: Michele <56927140+michelevantaggi02@users.noreply.github.com> Date: Fri, 6 Oct 2023 11:56:46 +0200 Subject: [PATCH 31/37] Apply suggestions from code review MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Nicolò Vescera --- triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md index 0c285590e..e48141af3 100644 --- a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md +++ b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md @@ -712,9 +712,9 @@ $$\begin{cases}a_{1,1}x_1 +. .. a_{1,n}x_n=b_1 \\ a_{2,1}x_1 +. .. a_{2,n}x_n=b_ # Teorema Di Rouché-Capelli -Il sistema lineare $Ax = b$ ammette soluzione se e solo se rango(A) = rango(A|b). +Il sistema lineare $Ax = b$ ammette soluzione se e solo se $rango(A) = rango(A|b)$. -Se esiste almeno una soluzione allora l'insieme delle soluzioni è un sottospazio affine di dimensione n-rango(A). +Se esiste almeno una soluzione allora l'insieme delle soluzioni è un sottospazio affine di dimensione $n-rango(A)$. $K^n$ spazio vettoriale @@ -736,7 +736,7 @@ L'insieme $W=\{v+v_0, v \in V, v_0 \in K^n\}$ è un sottospazio affine di $K^n$ - I sottospazi affini di dimensione 0 sono detti punti (non sono sottospazi) ``` -$Ax=b \iff c_1x_1+c_2x_2+...+c_nx_n=b \iff$ $b$ è combinazione lineare delle colonne di $A$ $\iff$ rango(A)=rango(A|b) +$Ax=b \iff c_1x_1+c_2x_2+...+c_nx_n=b \iff$ $b$ è combinazione lineare delle colonne di $A$ $\iff rango(A)=rango(A|b)$ $$|c_1|...|c_n|=\begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix}$$ From a705273be20afc4b683c20e7ae09538b1386df1b Mon Sep 17 00:00:00 2001 From: michelevantaggi02 Date: Mon, 23 Oct 2023 10:17:54 +0200 Subject: [PATCH 32/37] Modifiche varie --- .../Anno 2/Calcolo Numerico/CalcNum-merged.md | 51 +++++++++++------- .../Calcolo Numerico/CalcNum-merged.pdf | Bin 422804 -> 465134 bytes .../argomenti/2. Error Analysis.md | 8 +-- .../3. Errore Per Le Funzioni Razionali.md | 4 +- .../5. Derivata di pi\303\271 variabili.md" | 4 +- .../6. Valutazione di un polinomio.md | 8 +-- .../argomenti/a0. Sistemi lineari.md | 6 +-- .../argomenti/a1. Algoritmo di Gauss.md | 4 +- .../argomenti/a2. Matrici tridiagonali.md | 16 +++++- .../argomenti/a7. Interpolazione.md | 2 +- 10 files changed, 66 insertions(+), 37 deletions(-) diff --git a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md index e48141af3..d2072811a 100644 --- a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md +++ b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md @@ -151,12 +151,12 @@ Limiti di sequenze di numeri razionali che possono essere approssimate. Data una base di numerazione $\beta \geq 2$ posso prendere un numero reale tra 0 e 1 e le sequenze di cifre -$$R \iff \{d_i\}_{i=1,2,3....}, \quad d_i \in \{0,...,\beta -1\}$$ +$$\mathbb{R} \iff \{d_i\}_{i=1,2,3....}, \quad d_i \in \{0,...,\beta -1\}$$ ```ad-important title: Teorema Della Rappresentazione In Basi -Dato $x \in R \backslash \{0\}$ e data una base di numerazione $\beta \geq 2$ esiste un unico $p \in Z$ e una sequenza $\{d_i\}_{i=1,2,3,...}$ tali che: +Dato $x \in\mathbb{R}\backslash \{0\}$ e data una base di numerazione $\beta \geq 2$ esiste un unico $p \in Z$ e una sequenza $\{d_i\}_{i=1,2,3,...}$ tali che: 1. $d_i \in \{0,1,...,\beta - 1\}$ 2. $d_1 \neq 0$ @@ -199,8 +199,8 @@ Questi numeri __non sono uniformi__. Tra $\frac 12$ e $\frac 14$ e tra $\frac 12 ![[Pasted image 20230125111411.png]] -Data $S = \{x \in R : \omega \leq x \leq \Omega\}$ costruiamo una __funzione di rappresentazione__ -$$fl : R \to F \cup\{\pm \infty\}$$ +Data $S = \{x \in\mathbb{R}: \omega \leq x \leq \Omega\}$ costruiamo una __funzione di rappresentazione__ +$$fl :\mathbb{R}\to F \cup\{\pm \infty\}$$ Con una delle due regole, dato $x=\beta^p \sum_{i=1}^\infty {\beta^{-i}d_i} \in S$: @@ -258,7 +258,7 @@ Non seguono infatti: -Data una funzione razionale $f: R^2 \to R$ che sia $f = \frac pq$ con $p$ e $q$ polinomi. +Data una funzione razionale $f: \mathbb{R}^2 \to \mathbb{R}$ che sia $f = \frac pq$ con $p$ e $q$ polinomi. Dall'analisi matematica sappiamo che $f$ è definita e differenziabile per $q \neq 0$ (assumendo che $p$ e $q$ siano primi). @@ -294,7 +294,7 @@ La situazione ideale è $|\varepsilon_{tot}| < u$, ma in pratica è sufficiente ```ad-important title: Teorema -Dato $x \in R^n \backslash \{0\}$ e $f : R^n \to R$ razionale con $f(x) \neq 0$, dove $\tilde x = fl(x)$, allora +Dato $x \in \mathbb{R}^n \backslash \{0\}$ e $f : \mathbb{R}^n \to \mathbb{R}$ razionale con $f(x) \neq 0$, dove $\tilde x = fl(x)$, allora $$\varepsilon_{TOT} = \varepsilon_{IN} + \varepsilon_{ALG} + \varepsilon_{IN}\varepsilon_{ALG}$$ @@ -392,8 +392,8 @@ La dipendenza continua dai dati è meno ovviamente importante: # Derivata Di Fréchet -$$f:R^n \to R$$ -$$Df(x_0) : R^2 \to R$$ +$$f:\mathbb{R}^n \to \mathbb{R}$$ +$$Df(x_0) : \mathbb{R}^2 \to \mathbb{R}$$ Se $x = (x_1,...,x_n), \quad x_i \neq 0, \quad f(x) \neq 0$ l'[[3. Errore Per Le Funzioni Razionali#Errore Inerente|errore inerente]] è: $$\varepsilon_{IN} \doteq c_1\varepsilon_1 + ... + c_n\varepsilon_n, \quad c_i = \frac {x_i}{f(x)}\frac {\partial f} {\partial x_i}(x), \quad \varepsilon_i = \frac {\tilde x_i - x_i}{x_i}$$ @@ -440,10 +440,10 @@ $$|\varepsilon_{IN}| = |\frac{x\varepsilon_x + y\varepsilon_y}{x + y}| \leq \fra Dati $p$ e $x$, calcolare $p(x)$ -- $R^n(K^n)$ vettori con $n$ componenti in $R(K)$ -- $R_n(x)(K_n(x))$ polinomi di grado al più $n$ con coefficienti reali -- $R^{nxm}(K^{nxm})$ matrici $n \times m$ con elementi in $R$ -- $R^{n_1 x n_2 x ... x n_l}$ tensore di $l$ dimensioni +- $\mathbb{R}^n(K^n)$ vettori con $n$ componenti in $\mathbb{R}(K)$ +- $\mathbb{R}_n(x)(K_n(x))$ polinomi di grado al più $n$ con coefficienti reali +- $\mathbb{R}^{nxm}(K^{nxm})$ matrici $n \times m$ con elementi in $\mathbb{R}$ +- $\mathbb{R}^{n_1 x n_2 x ... x n_l}$ tensore di $l$ dimensioni - $p(x) = a_0 + a_1 x + ... + a_n x^n$ In computer grafica quasi tutto dipende da delle curve, le superfici possono essere curve, ecc... @@ -712,9 +712,9 @@ $$\begin{cases}a_{1,1}x_1 +. .. a_{1,n}x_n=b_1 \\ a_{2,1}x_1 +. .. a_{2,n}x_n=b_ # Teorema Di Rouché-Capelli -Il sistema lineare $Ax = b$ ammette soluzione se e solo se $rango(A) = rango(A|b)$. +Il sistema lineare $Ax = b$ ammette soluzione se e solo se rango$rango(A) = rango(A|b)$. -Se esiste almeno una soluzione allora l'insieme delle soluzioni è un sottospazio affine di dimensione $n-rango(A)$. +Se esiste almeno una soluzione allora l'insieme delle soluzioni è un sottospazio affine di dimensione n-rango(A). $K^n$ spazio vettoriale @@ -759,7 +759,7 @@ $mn \quad rango(A)=rango(A|b)=n \iff$ la soluzione non è unica, anche se il sistema ammette una soluzione unica per un dato $b$, esiste $c: \forall \varepsilon >0$ il sistema $Ax=c+\varepsilon c$ non ammette soluzioni $\implies$ la soluzione non può dipendere in modo continuo dai dati -$$Im(A) =\{v \in K^n, v=Ax, x\in R^n\} \subseteq K^n$$ +$$Im(A) =\{v \in K^n, v=Ax, x\in \mathbb{R}^n\} \subseteq K^n$$ $$dim(Im(A))=rango(A)\leq n0$$ @@ -906,9 +906,9 @@ Ma $det(M_h)=det(N_h)$ per quanto detto prima, quindi il metodo non è applicabi Se $A$ è una matrice definita positiva allora la condizione del teorema è verificata -$A \in R^{nxn}$ è definita positiva se: +$A \in \mathbb{R}^{nxn}$ è definita positiva se: - $A^T=A$ (simmetrica) -- $v^T Av > 0$ se $v \in R^n \setminus \{0\}$ +- $v^T Av > 0$ se $v \in \mathbb{R}^n \setminus \{0\}$ Se $A$ è a dominanza diagonale allora la condizione del teorema è verificata @@ -1014,11 +1014,25 @@ $$ \end{bmatrix} -= \begin{bmatrix} +\begin{bmatrix} + +1&1&0\\ +1&1&1\\ + +0&1&1 + +\end{bmatrix} + += \begin{bmatrix} +2 & 2 & 1\\ +2 & 3 & 2\\ +1 & 2 & 2 \end{bmatrix} $$ + +Il risultato non è una matrice tridiagonale, quindi non è una sottoalgebra ``` ```ad-question @@ -1612,6 +1626,7 @@ $$p(x) = \sum_{i=1}^n {y_iL_i(x)}$$ ```ad-done title: Dimostrazione + 1. $L_i(x_j) = \frac{(x_j-x_0) ... 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zR!91xB{u*Ma0$wXu~6DZI;8~$Vyne07v%s=t>NpMIdMcLL{Hkzb;bGWWYz%upYg{b z{s;q1T!8E=ElTd%A&)nh6!3p$^#->8uobuX5B<>3CB#P%W!3`(siW~V@}U4_+CM1K zh>_y+e%Eq+xkUX>juT%zc(5Q33=K17v{gec_=f5SRU4~@MI~p(3!(q_00biI(?Lqn z<3tdQ4^%#0sjvxB{}*Xd8cNCRJbw@^SBKU$FO0g98W3neis)oGowkr>AL{wT1Rnuq zk1<=+Mzur208u#pQ!znM&zcU6=SH+G8NE#~YPZ4n^(6kVAKExjwRI(+qi84<`9}Tq z|Ht=#P+EET{Qc$Sm3K@~toYl^mw*3x`NWDQdh~yn \quad rango(A)=rango(A|b)=n \iff$ la soluzione non è unica, anche se il sistema ammette una soluzione unica per un dato $b$, esiste $c: \forall \varepsilon >0$ il sistema $Ax=c+\varepsilon c$ non ammette soluzioni $\implies$ la soluzione non può dipendere in modo continuo dai dati -$$Im(A) =\{v \in K^n, v=Ax, x\in R^n\} \subseteq K^n$$ +$$Im(A) =\{v \in K^n, v=Ax, x\in \mathbb{R}^n\} \subseteq K^n$$ $$dim(Im(A))=rango(A)\leq n0$$ diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a1. Algoritmo di Gauss.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a1. Algoritmo di Gauss.md index 57c6d0be1..4659e82e5 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a1. Algoritmo di Gauss.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a1. Algoritmo di Gauss.md @@ -65,9 +65,9 @@ Ma $det(M_h)=det(N_h)$ per quanto detto prima, quindi il metodo non è applicabi Se $A$ è una matrice definita positiva allora la condizione del teorema è verificata -$A \in R^{nxn}$ è definita positiva se: +$A \in \mathbb{R}^{nxn}$ è definita positiva se: - $A^T=A$ (simmetrica) -- $v^T Av > 0$ se $v \in R^n \setminus \{0\}$ +- $v^T Av > 0$ se $v \in \mathbb{R}^n \setminus \{0\}$ Se $A$ è a dominanza diagonale allora la condizione del teorema è verificata diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a2. Matrici tridiagonali.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a2. Matrici tridiagonali.md index 546027500..951e40c1c 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a2. Matrici tridiagonali.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a2. Matrici tridiagonali.md @@ -25,11 +25,25 @@ $$ \end{bmatrix} -= \begin{bmatrix} +\begin{bmatrix} + +1&1&0\\ + +1&1&1\\ +0&1&1 + +\end{bmatrix} + += \begin{bmatrix} +2 & 2 & 1\\ +2 & 3 & 2\\ +1 & 2 & 2 \end{bmatrix} $$ + +Il risultato non è una matrice tridiagonale, quindi non è una sottoalgebra ``` ```ad-question diff --git a/triennale/Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md index 5517f89ab..af4d9392d 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a7. Interpolazione.md @@ -112,7 +112,7 @@ $$p(x) = \sum_{i=1}^n {y_iL_i(x)}$$ ```ad-done title: Dimostrazione -collapse: true + 1. $L_i(x_j) = \frac{(x_j-x_0) ... (x_j-x_j)...(x_j - x_{i+1})}{...} = 0, \quad i \neq j$ From 75406006e1af94b6ea3671b2d6d4499be8b1f89c Mon Sep 17 00:00:00 2001 From: michelevantaggi02 Date: Mon, 23 Oct 2023 10:27:59 +0200 Subject: [PATCH 33/37] altri fix --- triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md | 4 ++-- .../Anno 2/Calcolo Numerico/argomenti/2. Error Analysis.md | 2 +- .../Anno 2/Calcolo Numerico/argomenti/9. Altre Strutture.md | 2 +- 3 files changed, 4 insertions(+), 4 deletions(-) diff --git a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md index d2072811a..17e257fa4 100644 --- a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md +++ b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md @@ -125,7 +125,7 @@ Selezionare un numero finito di valori che possano rappresentare i numeri reali. ## Distribuzione uniforme Scegliere $\varepsilon > 0$, N pari: -$$\varepsilon = \{\varepsilon k: - \frac N2 < k \leq \frac N2, k \in Z\}$$ +$$\varepsilon = \{\varepsilon k: - \frac N2 < k \leq \frac N2, k \in \mathbb{Z}\}$$ Possiamo rappresentare i numeri reali con l'insieme: $$ S = \{-\varepsilon\frac N2 0$, N pari: -$$\varepsilon = \{\varepsilon k: - \frac N2 < k \leq \frac N2, k \in Z\}$$ +$$\varepsilon = \{\varepsilon k: - \frac N2 < k \leq \frac N2, k \in \mathbb{Z}\}$$ Possiamo rappresentare i numeri reali con l'insieme: $$ S = \{-\varepsilon\frac N2 Date: Mon, 23 Oct 2023 10:30:09 +0200 Subject: [PATCH 34/37] pdf finale --- .../Calcolo Numerico/CalcNum-merged.pdf | Bin 465134 -> 465733 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.pdf b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.pdf index 339f06b0f3f9d7d94da1e4d48de54795f9573880..3369e3f67c3aea2bf06821beedfe38754953cb4f 100644 GIT binary patch delta 13965 zcmZu%30O^C7j{c0k|9HpCVWYu?iueniI5>gWGER*C}Slh89IdWkw!V95Gq5)s0@YB zAVm>UN+kNsL?z{4H%0C9_wZEfuD#dVYkc?GeK@CeJ4q|#O?!f)snB=2P2Ur{N_rg? zp0kqn!uv3Zb~qPn%gaWL&|B^4v3S8sm9ZpbkgtNWoy~z_r?=;_A#Gp zK_xz7FMj%Y`BuqZiv(x=G0S@YC>Gp1H#Tgn`xLwN=(wW zX^a|TeRH?dgFWGs^EWNfnzJwdHhDLb{Oy%}Xyh-7!kM9UmnBl2&%W1ZgBs#R)eZN3e ze&tqn$x zlr+ds|6u#0{=JwoCh=kC71L)%@uSBpIqXrpej+1E^91AUcRtJ6?De8uQtOzjl^0^& zI>(0iZPPV96_d92Zu+q-|5Yv@3;VvwAykIfF675%m@RytM?4jLDn|TRu;k}#bD!h} zoksf|uGTlB4;LJc&^zgz5Ikp&li$T&liiP-o0h--Rjl`I^Vzg-amCRo&o-{JA7OUb 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zmZW9aEU5=HOo`A+DZT{yt(jkQdRPYt6lE-j>q%&Gw9}$#Ig4Uk-~sPg(7gfL2NPxi>shZ6o;^57W#7r>Il!P(#UZPg#w8W%P@Fmj;8Zy7+ x{T-A8iMD{$GIAD#c=^xu34#BmVm-k>%t!g32Ra&QT4=%GK&ww5N0$*={|6A}L>2%5 From 0d8da767764d93ca0cc03f72b642a51485ba753f Mon Sep 17 00:00:00 2001 From: michelevantaggi02 Date: Tue, 24 Oct 2023 11:06:46 +0200 Subject: [PATCH 35/37] sistemate formule --- .../Anno 2/Calcolo Numerico/CalcNum-merged.md | 6 +++--- .../Calcolo Numerico/CalcNum-merged.pdf | Bin 465733 -> 465799 bytes .../argomenti/a0. Sistemi lineari.md | 4 ++-- 3 files changed, 5 insertions(+), 5 deletions(-) diff --git a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md index 17e257fa4..6dbe1bd1e 100644 --- a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md +++ b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md @@ -151,7 +151,7 @@ Limiti di sequenze di numeri razionali che possono essere approssimate. Data una base di numerazione $\beta \geq 2$ posso prendere un numero reale tra 0 e 1 e le sequenze di cifre -$$\mathbb{R} \iff \{d_i\}_{i=1,2,3....}, \quad d_i \in \{0,...,\beta -1\}$$ +$$\mathbb{R}32 \iff \{d_i\}_{i=1,2,3....}, \quad d_i \in \{0,...,\beta -1\}$$ ```ad-important title: Teorema Della Rappresentazione In Basi @@ -712,9 +712,9 @@ $$\begin{cases}a_{1,1}x_1 +. .. a_{1,n}x_n=b_1 \\ a_{2,1}x_1 +. .. a_{2,n}x_n=b_ # Teorema Di Rouché-Capelli -Il sistema lineare $Ax = b$ ammette soluzione se e solo se rango$rango(A) = rango(A|b)$. +Il sistema lineare $Ax = b$ ammette soluzione se e solo se $rango(A) = rango(A|b)$. -Se esiste almeno una soluzione allora l'insieme delle soluzioni è un sottospazio affine di dimensione n-rango(A). +Se esiste almeno una soluzione allora l'insieme delle soluzioni è un sottospazio affine di dimensione $n-rango(A)$. $K^n$ spazio vettoriale diff --git a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.pdf b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.pdf index 3369e3f67c3aea2bf06821beedfe38754953cb4f..fefc520fb0323c2ad596a9def30b2da0ab812ae8 100644 GIT binary patch delta 14921 zcmZu%2|QI>7cWAHBAG&lBtrMxd+xdCk|`20mmwr7WGI9p!)Y`%M`S8v5i$>z29lC8 z6`~?VW+hSyeOvF#n``+W)PrD9=aVrP3fPzsYmr()-Z#eV&7 z(c<|X`79V3HB=x$NFMTOm22P>+}-sW-!T8G*VK>0XGOQIb}gp8Ox+gPruzJ&aR|TP z^q%j=BVVmH4DC-l+VSJ7-}j;IlU-$=348{&@)@17v#usx@?KSMVtH?Q)j!Czc%?zS z-gR&+_VVPrSdEaE>eCCpd;aJQ33{!*PDw#aWcB)yyzSV9POX+D#Qu@#*(ty85*cAD zCfkU;2Hg2lgLv_u&^uC-K&uG*8Q`8x{Kb8#s(XZ0REkCrd{c4+$zevHy zc9Wd1ap5f=3f!ZY?f-G}#%k%{0Uge-+9Nu&dX-dWiek;h-CDXri$$$HPBv5zAK~p- znR+19zQ{aVGcQ1@KuEkP|4hZ^h?o#Hf7kYlIvVX_zA^WnGjH?9v(_F~dHarSev@Nr7O#BMn!R3=-|E>sxq(7v!lP!TsJr@B z%UEl-*A;7h66E!3S*7W@%}|1{cxklH+Rr|IB3q^0ecdADn?R?`?p6As%B7)-josmU zo*vOJ9ZYhmi$>#RPRZ70VQ$N|7+bD2GL({+)uU_4MX@$GMc)YD5ndR*g}FR2J<| zNI!DXuVS%4<;mpmXikCo&Zat|7hWHV)XOMjJ$=F`-UE^%J{6@7HeA(*W#5N*R|xRYDa#?@pMO1 z?roI{5^qCYl|;PGqX|A8S~BLVb}vv{FJu15u}fRVTyFOQy0%QSaWY?)8Y=C0L^nN2 z^QY8;nx(RPrOKMQuiGjkH$829Vd&g@tv5@pTg^sji>SJxY~! 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Sistemi lineari.md b/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md index a7e0be3e3..7ece692ac 100644 --- a/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md +++ b/triennale/Anno 2/Calcolo Numerico/argomenti/a0. Sistemi lineari.md @@ -6,9 +6,9 @@ $$\begin{cases}a_{1,1}x_1 +. .. a_{1,n}x_n=b_1 \\ a_{2,1}x_1 +. .. a_{2,n}x_n=b_ # Teorema Di Rouché-Capelli -Il sistema lineare $Ax = b$ ammette soluzione se e solo se rango$rango(A) = rango(A|b)$. +Il sistema lineare $Ax = b$ ammette soluzione se e solo se $rango(A) = rango(A|b)$. -Se esiste almeno una soluzione allora l'insieme delle soluzioni è un sottospazio affine di dimensione n-rango(A). +Se esiste almeno una soluzione allora l'insieme delle soluzioni è un sottospazio affine di dimensione $n-rango(A)$. $K^n$ spazio vettoriale From fbe1f574f142800c45c80fbb3c47c57cb92c8755 Mon Sep 17 00:00:00 2001 From: michelevantaggi02 Date: Tue, 24 Oct 2023 12:16:35 +0200 Subject: [PATCH 36/37] fix typo --- .../Anno 2/Calcolo Numerico/CalcNum-merged.md | 2 +- .../Calcolo Numerico/CalcNum-merged.pdf | Bin 465799 -> 465784 bytes 2 files changed, 1 insertion(+), 1 deletion(-) diff --git a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md index 6dbe1bd1e..5f5f29c7f 100644 --- a/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md +++ b/triennale/Anno 2/Calcolo Numerico/CalcNum-merged.md @@ -151,7 +151,7 @@ Limiti di sequenze di numeri razionali che possono essere approssimate. 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From f92f0a07f09855e44ef4f5faceb9d80332ddc8db Mon Sep 17 00:00:00 2001 From: michelevantaggi02 Date: Tue, 24 Oct 2023 12:17:07 +0200 Subject: [PATCH 37/37] Rimosse dispense del prof --- .../dispense/dispense_cn_2021.pdf | Bin 710134 -> 0 bytes .../dispense/esercizi_cn_2021.pdf | Bin 689964 -> 0 bytes .../dispense/lez_erroranalysis_2122.pdf | Bin 688432 -> 0 bytes .../dispense/lezione1_cn__published.pdf | Bin 249462 -> 0 bytes .../dispense/lezionenetworks_2021.pdf | Bin 527782 -> 0 bytes 5 files changed, 0 insertions(+), 0 deletions(-) delete mode 100644 triennale/Anno 2/Calcolo Numerico/dispense/dispense_cn_2021.pdf delete mode 100644 triennale/Anno 2/Calcolo Numerico/dispense/esercizi_cn_2021.pdf delete mode 100644 triennale/Anno 2/Calcolo Numerico/dispense/lez_erroranalysis_2122.pdf delete mode 100644 triennale/Anno 2/Calcolo Numerico/dispense/lezione1_cn__published.pdf delete mode 100644 triennale/Anno 2/Calcolo Numerico/dispense/lezionenetworks_2021.pdf diff --git a/triennale/Anno 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