API

contours that we plan to support: 1. ellipse 2. circle 3. rectangle (not supported yet)

quadpts

Quadrature points of the trapezoidal rule on the contours.

Properties

• N: the number of the quadrature nodes
• nodes: quadrature nodes size of N x 2
• nodes_prime: derivative of the parametrization size of N x 2

cim(ctr::AbstractContour, nep::Function, D, l::Int64; n=50, tol=1e-12)

Contour integral method to calculate the eigenvalues inside the contour ctr

Arugments

• ctr: the contour
• nep: the nonlinear eigenvalue problem
• D: the scale of the nep
• l: the number of columns of random matrix
• n: the number of the quadrature points (here we use the trapezoid rule)
• tol: tolerance to determine the number of nonzero singular values

Reference

• Wolf-Jurgen Beyn, An integral method for solving NEPs, Linear Algebra Appl., 2012.

contr_int_ho()

Contour integral method with high-order moments

Arguments

In this function, we follow the notations in Stefan Guttel, Francoise Tisseur, Acta Numerica, 2017

• ctr:
• nep:
• D:
• r, l: size of the probing matrices
• pbar: the number of the moments (for p = 0, ..., pbar)

Reference

• Stefan Guttel, Francoise Tisseur, The NEP, Acta Numerica, 2017.

get_quadpts(ctr::circle, num_quadpts::Int64)

Get the quadrature points on the circle ctr. Here we use trapezoidal rule.

Arguments

• ctr: the contour that we discretize
• num_quadpts: the number of the quadrature nodes

TODO: note that rectangle is different with ellipse and circle

get_quadpts(ctr::ellipse, num_quadpts::Int64)

Get the quadrature points on the ellipse ctr. Here we use trapezoidal rule.

Arguments

• ctr`: the contour that we discretize
• num_quadpts: the number of the quadrature nodes

TODO: note that rectangle is different with ellipse and circle

is_inside(λ, ctr::ellipse)
is_inside(λ, ctr::circle)

Check if the eigenvalue λ is in the contour ctr to avoid spurious eigenvalues.

show_contour!(ax, ctr::circle)

Plot the contour ctr::circle on Axis ax by using CairoMakie.

show_contour!(ax, ctr::ellipse)

Plot the contour ctr::ellipse on Axis ax by using CairoMakie.

show_eigenvalues(ax, eigvals::AbstractArray)

Plot the eigenvalues on complex plane on Axis ax by using CairoMakie.

show_quadpts!(ax, pts::quadpts)

Plot the quadrature points pts on Axis ax by using CairoMakie.