v2007.09.17 - Convex Optimization

F.5. STURM’S PROCEDURE 665F.5 Sturm’s procedureThis is a demonstration program that can easily be transformed to asubroutine for decomposing positive semidefinite matrix X . This procedureprovides a nonorthogonal alternative (A.7.5.0.1) to eigen decomposition.That particular decomposition obtained is dependent on choice of matrix A .% Sturm procedure to find dyad-decomposition of X -Jon Dattorroclear allN = 4;r = 2;X = 2*(rand(r,N)-0.5);X = X’*X;t = null(svect(X)’);A = svectinv(t(:,1));% Suppose given matrix A is positive semidefinite%[v,d] = signeig(X);%d(1,1)=0; d(2,2)=0; d(3,3)=pi;%A = v*d*v’;tol = 1e-8;Y = X;y = zeros(size(X,1),r);rho = r;for k=1:r[v,d] = signeig(Y);v = v*sqrt(chop(d,1e-14));viol = 0;j = [ ];for i=2:rhoif chop((v(:,1)’*A*v(:,1))*(v(:,i)’*A*v(:,i)),tol) ~= 0viol = 1;endif (v(:,1)’*A*v(:,1))*(v(:,i)’*A*v(:,i)) < 0j = i;