SpecialMatrices.jl

A Julia package for working with special matrix types.

This Julia package extends the LinearAlgebra library with support for special matrices that are used in linear algebra. Every special matrix has its own type and is stored efficiently. The full matrix is accessed by the command Matrix(A).

Installation

julia> ] add SpecialMatrices

Related packages

ToeplitzMatrices.jl supports Toeplitz, Hankel, and circulant matrices.

Currently supported special matrices

Cauchy matrix

Cauchy(x,y)[i,j]=1/(x[i]+y[j])
Cauchy(x)=Cauchy(x,x)
cauchy(k::Number)=Cauchy(collect(1:k))

julia> Cauchy([1,2,3],[3,4,5])
3x3 Cauchy{Int64}:
 0.25      0.2       0.166667
 0.2       0.166667  0.142857
 0.166667  0.142857  0.125

julia> Cauchy([1,2,3])
3x3 Cauchy{Int64}:
 0.5       0.333333  0.25
 0.333333  0.25      0.2
 0.25      0.2       0.166667

julia> Cauchy(pi)
3x3 Cauchy{Float64}:
 0.5       0.333333  0.25
 0.333333  0.25      0.2
 0.25      0.2       0.166667

Companion matrix

julia> A=Companion([3,2,1])
3x3 Companion{Int64}:
 0  0  -3
 1  0  -2
 0  1  -1

Also, directly from a polynomial:

julia> using Polynomials

julia> P=Polynomial([2.0,3,4,5])
Polynomial(2 + 3x + 4x^2 + 5x^3)

julia> C=Companion(P)
3×3 Companion{Float64}:
 0.0  0.0  -0.4
 1.0  0.0  -0.6
 0.0  1.0  -0.8

Frobenius matrix

Example

julia> using SpecialMatrices

julia> F=Frobenius(3, [1.0,2.0,3.0]) #Specify subdiagonals of column 3
6x6 Frobenius{Float64}:
 1.0  0.0  0.0  0.0  0.0  0.0
 0.0  1.0  0.0  0.0  0.0  0.0
 0.0  0.0  1.0  0.0  0.0  0.0
 0.0  0.0  1.0  1.0  0.0  0.0
 0.0  0.0  2.0  0.0  1.0  0.0
 0.0  0.0  3.0  0.0  0.0  1.0

julia> inv(F) #Special form of inverse
6x6 Frobenius{Float64}:
 1.0  0.0   0.0  0.0  0.0  0.0
 0.0  1.0   0.0  0.0  0.0  0.0
 0.0  0.0   1.0  0.0  0.0  0.0
 0.0  0.0  -1.0  1.0  0.0  0.0
 0.0  0.0  -2.0  0.0  1.0  0.0
 0.0  0.0  -3.0  0.0  0.0  1.0

julia> F*F #Special form preserved if the same column has the subdiagonals
6x6 Frobenius{Float64}:
 1.0  0.0  0.0  0.0  0.0  0.0
 0.0  1.0  0.0  0.0  0.0  0.0
 0.0  0.0  1.0  0.0  0.0  0.0
 0.0  0.0  2.0  1.0  0.0  0.0
 0.0  0.0  4.0  0.0  1.0  0.0
 0.0  0.0  6.0  0.0  0.0  1.0

julia> F*Frobenius(2, [5.0,4.0,3.0,2.0]) #Promotes to Matrix
6x6 Array{Float64,2}:
 1.0   0.0  0.0  0.0  0.0  0.0
 0.0   1.0  0.0  0.0  0.0  0.0
 0.0   5.0  1.0  0.0  0.0  0.0
 0.0   9.0  1.0  1.0  0.0  0.0
 0.0  13.0  2.0  0.0  1.0  0.0
 0.0  17.0  3.0  0.0  0.0  1.0

julia> F*[10.0,20,30,40,50,60.0]
6-element Array{Float64,1}:
  10.0
  20.0
  30.0
  70.0
 110.0
 150.0

Hilbert matrix

julia> A=Hilbert(5)
Hilbert{Rational{Int64}}(5,5)

julia> Matrix(A)
5x5 Array{Rational{Int64},2}:
 1//1  1//2  1//3  1//4  1//5
 1//2  1//3  1//4  1//5  1//6
 1//3  1//4  1//5  1//6  1//7
 1//4  1//5  1//6  1//7  1//8
 1//5  1//6  1//7  1//8  1//9

julia> Matrix(Hilbert(5))
5x5 Array{Rational{Int64},2}:
 1//1  1//2  1//3  1//4  1//5
 1//2  1//3  1//4  1//5  1//6
 1//3  1//4  1//5  1//6  1//7
 1//4  1//5  1//6  1//7  1//8
 1//5  1//6  1//7  1//8  1//9

Inverses are also integer matrices:

julia> inv(A)
5x5 Array{Rational{Int64},2}:
    25//1    -300//1     1050//1    -1400//1     630//1
  -300//1    4800//1   -18900//1    26880//1  -12600//1
  1050//1  -18900//1    79380//1  -117600//1   56700//1
 -1400//1   26880//1  -117600//1   179200//1  -88200//1
   630//1  -12600//1    56700//1   -88200//1   44100//1

Kahan matrix

julia> A=Kahan(5,5,1,35)
5x5 Kahan{Int64,Int64}:
 1.0  -0.540302  -0.540302  -0.540302  -0.540302
 0.0   0.841471  -0.454649  -0.454649  -0.454649
 0.0   0.0        0.708073  -0.382574  -0.382574
 0.0   0.0        0.0        0.595823  -0.321925
 0.0   0.0        0.0        0.0        0.501368

julia> A=Kahan(5,3,0.5,0)
5x3 Kahan{Float64,Int64}:
 1.0  -0.877583  -0.877583
 0.0   0.479426  -0.420735
 0.0   0.0        0.229849
 0.0   0.0        0.0
 0.0   0.0        0.0

julia> A=Kahan(3,5,0.5,1e-3)
3x5 Kahan{Float64,Float64}:
 1.0  -0.877583  -0.877583  -0.877583  -0.877583
 0.0   0.479426  -0.420735  -0.420735  -0.420735
 0.0   0.0        0.229849  -0.201711  -0.201711

For more details see N. J. Higham (1987).

Riemann matrix

Riemann matrix is defined as A = B[2:N+1, 2:N+1], where B[i,j] = i-1 if i divides j, and -1 otherwise. Riemann hypothesis holds if and only if det(A) = O( N! N^(-1/2+epsilon)) for every epsilon > 0.

julia> Riemann(7)
7x7 Riemann{Int64}:
  1  -1   1  -1   1  -1   1
 -1   2  -1  -1   2  -1  -1
 -1  -1   3  -1  -1  -1   3
 -1  -1  -1   4  -1  -1  -1
 -1  -1  -1  -1   5  -1  -1
 -1  -1  -1  -1  -1   6  -1
 -1  -1  -1  -1  -1  -1   7

For more details see F. Roesler (1986).

Strang matrix

A special SymTridiagonal matrix named after Gilbert Strang

julia> Strang(6)
6x6 Strang{Float64}:
  2.0  -1.0   0.0   0.0   0.0   0.0
 -1.0   2.0  -1.0   0.0   0.0   0.0
  0.0  -1.0   2.0  -1.0   0.0   0.0
  0.0   0.0  -1.0   2.0  -1.0   0.0
  0.0   0.0   0.0  -1.0   2.0  -1.0
  0.0   0.0   0.0   0.0  -1.0   2.0

Vandermonde matrix

julia> a = 1:5
julia> A = Vandermonde(a)
5×5 Vandermonde{Int64}:
 1  1   1    1    1
 1  2   4    8   16
 1  3   9   27   81
 1  4  16   64  256
 1  5  25  125  625

Adjoint Vandermonde:

julia> A'
5×5 adjoint(::Vandermonde{Int64}) with eltype Int64:
 1   1   1    1    1
 1   2   3    4    5
 1   4   9   16   25
 1   8  27   64  125
 1  16  81  256  625

The backslash operator \ is overloaded to solve Vandermonde and adjoint Vandermonde systems in O(n^2) time using the algorithm of Björck & Pereyra (1970):

julia> A \ a
5-element Vector{Float64}:
 0.0
 1.0
 0.0
 0.0
 0.0

julia> A' \ A[2,:]
5-element Vector{Float64}:
 0.0
 1.0
 0.0
 0.0
 0.0