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Functions of matrices

MatrixPolynomials.FuncV — Type
FuncV{f,T}(s⁻¹Amc, c, s)

Structure for applying the action of a polynomial approximation of the function f with a matrix argument acting on a vector, i.e. w ← p(A)*v where p(z) ≈ f(z). Various properties of f may be used, such as shifting and scaling of the argument, to improve convergence and/or accuracy. s⁻¹Amc is the shifted linear operator $s⁻¹(A-c)$, c and s are the shift and scaling, respectively.

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MatrixPolynomials.FuncV — Type
FuncV(f, A, m[, t=1; distribution=:leja, kwargs...])

Create a FuncV that is used to compute f(t*A)*v using a polynomial approximation of f formed using m interpolation points.

kwargs... are passed on to spectral_range which estimates the range over which f has to be interpolated.

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LinearAlgebra.mul! — Method
mul!(w, funcv::FuncV, v)

Evaluate the action of the matrix polynomial funcv on v and store the result in w.

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Spectral ranges and shapes

MatrixPolynomials.spectral_range — Function
spectral_range(A[; ctol=√ε, verbosity=0])

Estimate the spectral range of the matrix/linear operator A using ArnoldiMethod.jl. If the spectral range along the real/imaginary axis is smaller than ctol, it is compressed into a line. Returns a spectral Shape.

Examples

julia> A = Diagonal(1.0:6)
6×6 Diagonal{Float64,StepRangeLen{Float64,Base.TwicePrecision{Float64},Base.TwicePrecision{Float64}}}:
 1.0   ⋅    ⋅    ⋅    ⋅    ⋅
  ⋅   2.0   ⋅    ⋅    ⋅    ⋅
  ⋅    ⋅   3.0   ⋅    ⋅    ⋅
  ⋅    ⋅    ⋅   4.0   ⋅    ⋅
  ⋅    ⋅    ⋅    ⋅   5.0   ⋅
  ⋅    ⋅    ⋅    ⋅    ⋅   6.0

julia> MatrixPolynomials.spectral_range(A, verbosity=2)
Converged: 1 of 1 eigenvalues in 6 matrix-vector products
Converged: 1 of 1 eigenvalues in 6 matrix-vector products
Converged: 1 of 1 eigenvalues in 6 matrix-vector products
Converged: 1 of 1 eigenvalues in 6 matrix-vector products
[ Info: Imaginary extent of spectral range 0.0 below tolerance 1.4901161193847656e-8, conflating.
[ Info: 0.0 below tolerance 1.4901161193847656e-8, truncating.
[ Info: 0.0 below tolerance 1.4901161193847656e-8, truncating.
MatrixPolynomials.Line{Float64}(0.9999999999999998 + 0.0im, 6.0 + 0.0im)

julia> MatrixPolynomials.spectral_range(exp(im*π/4)*A, verbosity=2)
Converged: 1 of 1 eigenvalues in 6 matrix-vector products
Converged: 1 of 1 eigenvalues in 6 matrix-vector products
Converged: 1 of 1 eigenvalues in 6 matrix-vector products
Converged: 1 of 1 eigenvalues in 6 matrix-vector products
MatrixPolynomials.Rectangle{Float64}(0.7071067811865468 + 0.7071067811865478im, 4.242640687119288 + 4.242640687119283im)

The second example should also be a Line, but the algorithm is not yet clever enough.

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spectral_range(t, A)

Finds the spectral range of t*A; if t is a vector, find the largest such range.

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MatrixPolynomials.hermitian_spectral_range — Function
hermitian_spectral_range(A;[ K=20])

Estimate the spectral range of a Hermitian operator A using Algorithm 1 of

  • Zhou, Y., & Li, R. (2011). Bounding the spectrum of large hermitian matrices. Linear Algebra and its Applications, 435(3), 480–493. http://dx.doi.org/10.1016/j.laa.2010.06.034
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Shapes

The shapes in the complex plane are mainly used to generate suitable distributions of Leja points, which are in turn used to generate Newton polynomials that efficiently approximate various functions on the field-of-values of a matrix $\mat{A}$ which is contained within the spectral shape.

MatrixPolynomials.Shape — Type
Shape

Abstract base type for different shapes in the complex plane encircling the spectra of linear operators.

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Lines

Base.range — Method
range(l::Line, n)

Generate a uniform distribution of n values along the Line l. If the line is on the real axis, the resulting values will be real as well.

Examples

julia> range(MatrixPolynomials.Line(0, 1.0im), 5)
5-element LinRange{Complex{Float64}}:
 0.0+0.0im,0.0+0.25im,0.0+0.5im,0.0+0.75im,0.0+1.0im

julia> range(MatrixPolynomials.Line(0, 1.0), 5)
0.0:0.25:1.0
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Base.union — Method
a::Line ∪ b::Line

Form the union of the Lines a and b, which need to be collinear.

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Rectangles

Base.range — Method
range(r::Rectangle, n)

Generate a uniform distribution of n values along the diagonal of the Rectangle r.

This assumes that the eigenvalues lie on the diagonal of the rectangle, i.e. that the spread is negligible. It would be more correct to instead generate samples along the sides of the rectangle, however, spectral_range needs to be modified to correctly identify spectral ranges falling on a line that is not lying on the real or imaginary axis.

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