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).
julia> ] add SpecialMatricesToeplitzMatrices.jl supports Toeplitz, Hankel, and circulant matrices.
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 0.166667julia> A=Companion([3,2,1])
3×3 Companion{Int64}:
0 0 -3
1 0 -2
0 1 -1Also, 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 -0.8Example
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
150.0julia> 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//9Inverses 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//1julia> 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 -0.201711For more details see N. J. Higham (1987).
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 7For more details see F. Roesler (1986).
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 2julia> 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 625Adjoint 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 625The 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
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