95 lines
3.1 KiB
Matlab
Executable File
95 lines
3.1 KiB
Matlab
Executable File
%%***************************************************************************
|
|
%% lanczos: find the largest eigenvalue of
|
|
%% invXchol'*dX*invXchol via the lanczos iteration.
|
|
%%
|
|
%% [lam,delta] = lanczos(Xchol,dX,maxit,tol,v)
|
|
%%
|
|
%% lam: an estimate of the largest eigenvalue.
|
|
%% lam2: an estimate of the second largest eigenvalue.
|
|
%% res: residual norm of the largest eigen-pair.
|
|
%% res2: residual norm of the second largest eigen-pair.
|
|
%%***************************************************************************
|
|
|
|
function [lam,delta,res] = lanczos(Xchol,dX,maxit,tol,v)
|
|
|
|
if (norm(dX,'fro') < 1e-13)
|
|
lam = 0; delta = 0; res = 0;
|
|
return;
|
|
end
|
|
n = length(dX);
|
|
if (nargin < 5);
|
|
state = randn('state');
|
|
randn('state',0);
|
|
v = randn(n,1);
|
|
randn('state',state);
|
|
end
|
|
if (nargin < 4); maxit = 30; end
|
|
if (nargin < 3); tol = 1e-3; end
|
|
V = zeros(n,maxit+1); H = zeros(maxit+1,maxit);
|
|
v = v/norm(v);
|
|
V(:,1) = v;
|
|
if issparse(Xchol); Xcholtransp = Xchol'; end
|
|
%%
|
|
%% lanczos iteration.
|
|
%%
|
|
for k = 1:maxit
|
|
if issparse(Xchol)
|
|
w = dX*mextriangsp(Xcholtransp,v,1);
|
|
w = mextriangsp(Xchol,w,2);
|
|
else
|
|
w = dX*mextriang(Xchol,v,1);
|
|
w = mextriang(Xchol,w,2);
|
|
end
|
|
wold = w;
|
|
if (k > 1);
|
|
w = w - H(k,k-1)*V(:,k-1);
|
|
end;
|
|
alp = w'*V(:,k);
|
|
w = w - alp*V(:,k);
|
|
H(k,k) = alp;
|
|
%%
|
|
%% one step of iterative refinement if necessary.
|
|
%%
|
|
if (norm(w) <= 0.8*norm(wold));
|
|
s = (w'*V(:,1:k))';
|
|
w = w - V(:,1:k)*s;
|
|
H(1:k,k) = H(1:k,k) + s;
|
|
end;
|
|
nrm = norm(w);
|
|
v = w/nrm;
|
|
V(:,k+1) = v;
|
|
H(k+1,k) = nrm; H(k,k+1) = nrm;
|
|
%%
|
|
%% compute ritz pairs and test for convergence
|
|
%%
|
|
if (rem(k,5) == 0) | (k == maxit);
|
|
Hk = H(1:k,1:k); Hk = 0.5*(Hk+Hk');
|
|
[Y,D] = eig(Hk);
|
|
eigH = real(diag(D));
|
|
[dummy,idx] = sort(eigH);
|
|
res_est = abs(H(k+1,k)*Y(k,idx(k)));
|
|
if (res_est <= 0.1*tol) | (k == maxit);
|
|
lam = eigH(idx(k));
|
|
lam2 = eigH(idx(k-1));
|
|
z = V(:,1:k)*Y(:,idx(k));
|
|
z2 = V(:,1:k)*Y(:,idx(k-1));
|
|
if issparse(Xchol)
|
|
tmp = dX*mextriangsp(Xcholtransp,z,1);
|
|
res = norm(mextriangsp(Xchol,tmp,2) -lam*z);
|
|
tmp = dX*mextriangsp(Xcholtransp,z2,1);
|
|
res2 = norm(mextriangsp(Xchol,tmp,2) -lam*z2);
|
|
else
|
|
tmp = dX*mextriang(Xchol,z,1);
|
|
res = norm(mextriang(Xchol,tmp,2) -lam*z);
|
|
tmp = dX*mextriang(Xchol,z2,1);
|
|
res2 = norm(mextriang(Xchol,tmp,2) -lam*z2);
|
|
end
|
|
tmp = lam-lam2 -res2;
|
|
if (tmp > 0); beta = tmp; else; beta = eps; end;
|
|
delta = min(res,res^2/beta);
|
|
if (delta <= tol); break; end;
|
|
end
|
|
end
|
|
end
|
|
%%***************************************************************************
|