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## Pseudospectra of a Discretization of a Laser Integral Equation
This figure illustrates approximate pseudospectra of an integral equation
from laser theory investigated by Landau [Lan77]
in one of the earliest applications of pseudospectra.
This example discretizes the infinite dimensional problem
into a matrix of dimension Use the following MATLAB code compute a similar image using
EigTool.
To mimic the example above, set F = 12; N = 120; % calculate nodes and weights for Gaussian quadrature beta = 0.5*(1-(2*[1:N-1]).^(-2)).^(-0.5); T = diag(beta,1) + diag(beta,-1); [V D] = eig(T); [nodes index] = sort(diag(D)); weights = zeros(N,1); weights([1:N]) = (2*V(1,index).^2); % Construct matrix B B = zeros(N,N); for k=1:N B(k,:)= weights(:)'* sqrt(1i*F).*exp(-1i*pi*F*(nodes(k) - nodes(:)').^2); end % Weight matrix with Gaussian quadrature weights w = sqrt(weights); for j=1:N, B(:,j) = w.*B(:,j)/w(j); end clear D T V beta index j k nodes w weights % Compute Schur form and compress to interesting subspace: [U,T] = schur(B); eigB = diag(T); select = find(abs(eigB)>.01); n = length(select); for i = 1:n for k = select(i)-1:-1:i G([2 1],[2 1]) = planerot([T(k,k+1) T(k,k)-T(k+1,k+1)]')'; J = k:k+1; T(:,J) = T(:,J)*G; T(J,:) = G'*T(J,:); end end T = triu(T(1:n,1:n)); opts.npts=50; opts.ax = [-1 1.2 -.9 1.1]; opts.levels = -6:.5:-1; eigtool(T,opts)Download this code: landau.m. |