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

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

Get the quadrature points on the contour (ellipse or circle) ctr. Here we use the composite trapezoidal rule.

Arguments

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

Qep{T}

Construct quadratic eigenvlaue problems

$(\mathbf{A}_{0} + \alpha \mathbf{A}_{1} + \alpha^2 \mathbf{A}_{2}) \mathbf{u} = 0.$

Fields

• A₀: the matrix in the zero order term
• A₁: the matrix in the first order term
• A₂: the matrix in the second order term

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.

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

Use the contour integral method to solve quadratic eigenvalue problems.

hcim(pts::quadpts, NEP::Function, d::Int64, l::Int64, r::Int64, pbar::Int64)

Contour integral method with high-order moments

Arguments

• ctr: the contour
• nep: the nonlinear eigenvalue problem
• d: the scale of the nonlinear eigenvalue problem nep
• 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.

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

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

print_vec(vec; precision::Int=6)

Print the vector vec to the REPL.