Merge branch 'master' of https://github.com/ketch/finite-difference-c…

…ourse

* 'master' of https://github.com/ketch/finite-difference-course:
  Update first FD notebook.

ipython/Week_1_Finite_differences.ipynb

@@ -1,6 +1,6 @@
 {
  "metadata": {
-  "name": "Week_1_Finite_differences"
+  "name": ""
  },
  "nbformat": 3,
  "nbformat_minor": 0,
@@ -244,8 +244,8 @@
       "Here ${\\mathcal O}(h^4)$ indicates that the rest of the terms in the series vanish at least as quickly as $h^4$ when $h\\to 0$ (see Appendix A of the text).\n",
       "Substituting this series in our forward difference formula gives\n",
       "\\begin{align}\n",
-      "\\frac{f(x+h) - f(x)}{h} & = \\frac{f(x) + h f'(x) + \\frac{1}{2}h^2 f''(x) + \\frac{1}{6} h^3 f'''(x) + {\\mathcal O}(h^4) - f(x)}{h} \\\\\\\\\n",
-      "& = f'(x) + \\frac{1}{2}h f''(x) + \\frac{1}{6} h^2 f'''(x) + {\\mathcal O}(h^3) \\\\\\\\\n",
+      "\\frac{f(x+h) - f(x)}{h} & = \\frac{f(x) + h f'(x) + \\frac{1}{2}h^2 f''(x) + \\frac{1}{6} h^3 f'''(x) + {\\mathcal O}(h^4) - f(x)}{h} \\\\\n",
+      "& = f'(x) + \\frac{1}{2}h f''(x) + \\frac{1}{6} h^2 f'''(x) + {\\mathcal O}(h^3) \\\\\n",
       "& = f'(x) + \\frac{1}{2}h f''(x) + {\\mathcal O}(h^2).\n",
       "\\end{align}\n",
       "\n",
@@ -278,29 +278,29 @@
      "source": [
       "To begin, suppose we are given three function values: $f(x_0-h), f(x_0), f(x_0+h)$, which we'll denote by $(f_1,f_2,f_3)$.  We wish to find a polynomial that passes through these three points.  As you may know, a set of $n$ values uniquely defines a polynomial of degree $n-1$, so we will look for a quadratic polynomial.  To make the computation simpler, we'll write it this way: $p(x) = a + b (x-x_0) + c (x-x_0)^2$.  We know that $p$ and $f$ must agree at the three given points, which means\n",
       "\\begin{align}\n",
-      "a + b(-h) + c (-h)^2  & = f_1 \\\\\\\\\n",
-      "a + b(0) + c(0)^2 & = f_2 \\\\\\\\\n",
-      "a + b (h) + c (h)^2 & = f_3 \\\\\\\\\n",
+      "a + b(-h) + c (-h)^2  & = f_1 \\\\\n",
+      "a + b(0) + c(0)^2 & = f_2 \\\\\n",
+      "a + b (h) + c (h)^2 & = f_3 \\\\\n",
       "\\end{align}\n",
       "\n",
       "or simply\n",
       "\n",
       "\\begin{align}\n",
-      "a - hb + h^2c  & = f_1 \\\\\\\\\n",
-      "a & = f_2 \\\\\\\\\n",
-      "a + hb + h^2c & = f_3 \\\\\\\\\n",
+      "a - hb + h^2c  & = f_1 \\\\\n",
+      "a & = f_2 \\\\\n",
+      "a + hb + h^2c & = f_3 \\\\\n",
       "\\end{align}\n",
       "\n",
       "We can rewrite this system of equations in matrix form:\n",
       "\n",
       "\\begin{align}\n",
       "\\begin{pmatrix}\n",
-      "1 & -h & h^2 \\\\\\\\\n",
-      "1 & 0   & 0 \\\\\\\\\n",
-      "1 & h & h^2 \\\\\\\\\n",
+      "1 & -h & h^2 \\\\\n",
+      "1 & 0   & 0 \\\\\n",
+      "1 & h & h^2 \\\\\n",
       "\\end{pmatrix}\n",
-      "\\begin{pmatrix} a \\\\\\\\ b \\\\\\\\ c \\end{pmatrix}\n",
-      "& = \\begin{pmatrix} f_1 \\\\\\\\ f_2 \\\\\\\\ f_3 \\end{pmatrix}\n",
+      "\\begin{pmatrix} a \\\\ b \\\\ c \\end{pmatrix}\n",
+      "& = \\begin{pmatrix} f_1 \\\\ f_2 \\\\ f_3 \\end{pmatrix}\n",
       "\\end{align}\n",
       "\n",
       "This is a linear system that we can solve for the coefficients $a,b,c$ in terms of $h$, and the values of $f$.  The result is\n",
@@ -332,28 +332,25 @@
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "**Extra credit: show how this approach can be generalized to find finite difference formulas using values of $f(x)$ at arbitrary points and approximating any order derivative.  Write a short Python script that implements your approach.  Check that it gives the same results as the approach described in the text.**"
+      "<font color='red'>\n",
+      "Derive a formula for $f''(x)$ based on the values $f(x), f(x-h), f(x-3h)$ by determining the interpolating polynomial and differentiating it.  How accurate is your formula?\n",
+      "</font>"
      ]
     },
-    {
-     "cell_type": "code",
-     "collapsed": false,
-     "input": [],
-     "language": "python",
-     "metadata": {},
-     "outputs": []
-    },
     {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
-      "**One thing that is not clear to me after completing the reading and the notebooks is...** (fill in the blank)"
+      "**Can you think of a similar approach that would allow you to derive rules for numerical quadrature (integration)?**"
      ]
     },
     {
-     "cell_type": "markdown",
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [],
+     "language": "python",
      "metadata": {},
-     "source": []
+     "outputs": []
     }
    ],
    "metadata": {}