SpecialMatrices.jl

https://github.com/JuliaLinearAlgebra/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::Int) = Cauchy(1:k)

julia> Cauchy([1,2,3],[3,4,5])
3×3 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])
3×3 Cauchy{Int64}:
 0.5       0.333333  0.25
 0.333333  0.25      0.2
 0.25      0.2       0.166667

julia> Cauchy(3)
3×3 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])
3×3 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
6×6 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
6×6 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
6×6 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
6×6 Matrix{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   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 Vector{Float64}:
  10.0
  20.0
  30.0
  70.0
 110.0
 150.0

`Hilbert` matrix

julia> A=Hilbert(5)
5×5 Hilbert{Rational{Int64}}:
 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)
5×5 InverseHilbert{Rational{Int64}}:
    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> Kahan(5,5,1,35)
5×5 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> Kahan(5,3,0.5,0)
5×3 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> Kahan(3,5,0.5,1e-3)
3×5 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+ϵ)) for every ϵ > 0.

julia> Riemann(7)
7×7 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 symmetric, tridiagonal, Toeplitz matrix named after Gilbert Strang.

julia> Strang(6)
6×6 Strang{Int64}:
  2  -1   0   0   0   0
 -1   2  -1   0   0   0
  0  -1   2  -1   0   0
  0   0  -1   2  -1   0
  0   0   0  -1   2  -1
  0   0   0   0  -1   2

`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