Code for the TLS Problem

This archive contains four algorithms for solving large-scale total least squares problems.

This archive contains four algorithms for solving large-scale total least squares problems：

[fx,x,err] = PSLB_TLS(A,b,kmax,tol): PSLB algorithm for solving TLS problems (without restarted)

[fx1,x,err1] = Restart_PSLB_TLS(A,b,int_it,out_it,tol,flag): a restarted block Lanczos bidiagonalization method for TLS problem

[V, D, flag] = tls_eigs(varargin): This algorithm is built on the eigs.m, with only the stopping criterion adjusted

[S, V, flag] = tls_svds(A,varargin): This algorithm is built on the svds.m, with only the stopping criterion adjusted

[x, f_hist] = TLS_RQI(A, b, max_iter, tol): Rayleigh quotient iteration (RQI) iteration for Total Least Squares

%test:

% (i) Test the generic case

A = sprand(m,n,0.005);

b = ones(size(A,1),1); b = b/norm(b);

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t = tic; [fx1,x,err1] = Restart_PSLB_TLS(A,b,50,200,1e-13,1); t = toc(t)

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opts.maxit = 200; opts.p = 50; opts.tol = 1e-13;

t = tic; Ab = [A,b]'*[A,b]; Ab = 0.5*(Ab'+Ab); [V, fx] = tls_eigs(Ab,1,'sr',opts); t = toc(t)

x = -V(1:end-1)/V(end);

t = tic; [S, V] = tls_svds([A,b],1,'smallest',opts);tsvds = toc(t)

fx = S*S; x = -V(1:end-1)/V(end);

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t = tic; [x, f_hist] = TLS_RQI(A, b, 200, 1e-13); t = toc(t)

% (ii) Test the generic case

n = 3000;

m = 10000;

Q1 = randn(n);

Q1(n, 1:2) = zeros(1, 2);

[Q, ~] = qr(Q1);

D = (1:n)/1000;

D(end) = 3;

S = eye(n) + diag(D);

P = [S; zeros(m - n, n)] * Q';

A = P(:, 1:2999);

b = P(:, 3000);

[fx1,x,err1] = Restart_PSLB_TLS(A,b,50,200,1e-8,1);

[fx,x,err] = PSLB_TLS(A,b,n,1e-8);