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BlMRes.m
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BlMRes.m
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% A block minimum residual Code.
% With householder reflections and optimal storage.
%
% Uses a Ruhe variation to calculate lanczos.
function [X] = BlMRes(A,B,setup)
% Set up the arguments
if nargin <2
disp('You must include at least 2 arguments: A and b');
X = ensureSetup(struct(),B);
return;
elseif nargin <3
setup = ensureSetup(struct(),B);
else
setup = ensureSetup(setup,B);
end
% Set up Arguments
p = rank(full(B));
w = zeros(p+1,setup.maxits);%This p,k vector holds all the reflection info
% Allocate space for T
% There are only four entries per column.
% This has the main diag on 3, with one subdiag
% T is changed in place to R, and the last row of T can be ignored
% To look at T use:
%
% T = spdiags(T(1:3,1:k)',[2 1 0],k,k);
%
T = zeros(p*3+1,setup.maxits+p);
% p*2+1 is the diagonal
d = p*2+1;
symInd = sub2ind([size(T,1),inf],d-(1:p),1+(1:p));
symIndUpd = p*3+1;
R = zeros(p*3+1,setup.maxits);
% Set up Q and P
% You will only need 3 at a time, so cycle through them
Q = zeros(length(B),1+2*p);
P = Q;
cShift = [2:(2*p+1) 1];
log_nrm_rk = inf;
logTol = log10(setup.tol);
cnt = 1;
Xk = setup.x0;
G = zeros(setup.maxits+1,size(B,2));
if setup.precond%left preconditioning!
%if you are using preconditioning, do a quick forward and back
%solve to find your R_0* = M^-1 * R_0
[q r] = qr(setup.U\(setup.L\(B-A*X0)),0);
else
[q r] = qr(B-A*setup.x0,0);
end
Q(:,p+1:end-1) = q(:,1:p);
G = G.*0;
G(1:p,1:size(B,2)) = r(1:p,1:size(B,2));
skip = 0;
% Loop Until Convergence or Max Iterations
for k = p:setup.maxits
j = k-p+1;
% If you have preconditioning
if setup.precond
z = setup.U\(setup.L\(A*Q(:,p+1)));
else
z = A*Q(:,p+1);
end
% Subtract off the info we know
z = z - Q(:,1:p)*T((1:p)+p,j);
% Create the info We don't know
for i = p+1:(2*p)
T(i+p,j) = Q(:,i)'*z;
z = z - T(i+p,j).*Q(:,i);
end
T(i+p+1,j) = norm(z);
T(symInd) = T(d+1:end,j);
symInd = symInd + symIndUpd;
if abs(T(i+p+1,j)) < setup.tol;
fprintf('Rank Deficient\n');
skip = skip + 1;
end
%if B(j) ~= 0 divide z by it to get your new orth. vec
Q(:,2*p+1) = z./T(i+p+1,j);
% QR Factorize H using Housholder Reflections
%
% Qj = I - 2*w*w';
%
% Where:
% w = z./norm(z);
%
% z = X(:,j); z(1) = z(1) + beta;
%
% beta = sign(X(j,j))*norm(X(:,j));
%
R(:,j) = T(:,j);
% Apply all the previous reflections to the k-th column of T.
% this is a simplified matrix-vector product Qj'*R_k
for ref = max(1,j-2*(p)):j-1
rind = max(1,d-j+ref):d-j+ref+p;
wind = length(rind)-1;
% pad = zeros(p+1-length(rind),1);
% v = 2*([pad;R(rind,j)]'*w(:,ref));%This is a scalar!
% R(rind,j) = R(rind,j) - w(:,ref).*v;
v = 2*(R(rind,j)'*w(end-wind:end,ref));%This is a scalar!
R(rind,j) = R(rind,j) - w(end-wind:end,ref).*v;
end
% Solve for the current Householder Reflection in short form
% We are getting wj in:
%
% Qj = I - 2*wj*wj';
%
beta = sign(R(d,j))*norm(R(d:end,j));% Go to the p+1 below the diag
w(1,j) = beta+R(d,j);%z_j,j Add beta to first element
w(2:p+1,j) = R(d+1:d+p,j);% Here there are only p elements
w(:,j) = w(:,j)./norm(w(:,j));% Normalize by the length
% Apply the reflection to the sub-diagonal elements in column j
% Using:
%
% (I - 2ww')X = X-wv'
%
% v = 2X'w
%
v = 2*R(d:end,j)'*w(:,j);%This is a scalar!
R(d:end,j) = R(d:end,j) - w(:,j).*v;
% Apply the reflection to G
v = 2*G(j:j+p,:)'*w(:,j);%This is not a scalar.
G(j:j+p,:) = G(j:j+p,:) - w(:,j)*v';
% Update P = QR^-1
P(:,2*p+1) = Q(:,p+1);
% Do the forward solve on the transpose of P
% R'P' = Q'
for f = 1:2*p
P(:,2*p+1) = P(:,2*p+1) - R(f,j).*P(:,f);
end
P(:,2*p+1) = P(:,2*p+1)./R(d,j);
if rem(j,p) == 0%Update all of X at one time.
Xk = Xk + P(:,p+2:2*p+1)*G(j-p+1:j,:);
end
Q = Q(:,cShift);
P = P(:,cShift);
% Approximate the Norm
% Only do this if you are close or evey p iterations
if log_nrm_rk-logTol< 0.1 || rem(cnt-1,p) == 0
nrm_rk = norm(G(j+1:end,:),'fro');
log_nrm_rk = log10(nrm_rk);
end
cnt = cnt+1;
if setup.showComments;fprintf('norm(b-A*x) = %e\n',nrm_rk);end
% Stopping Criteria
if(nrm_rk < setup.tol)
break;
end
end %loop
% Output a few things
if(nrm_rk < setup.tol)
fprintf('minres CONVERGED at iteration %i. \nnorm(b-A*x) = %e\n',k,norm(B-A*Xk));
else
fprintf('\n\n*****************************\n\nminres DID NOT converge by iteration %i. \nnorm(b-A*x) = %e\n\n*****************************\n\n',k,norm(B-A*Xk));
end
X = Xk;
end
%% Ensure Setup Structure is Correct
% A setup structure can be passed in to switch between internal methods.
% The setup structure has fields:
%
% SETUP:
% maxits: 10
% tol: 1.000000e-06
% showComments: false
% record: false
% precond: false
% L: 0
% U: 0
% M: 0
% x0: 0
function setup = ensureSetup(setup,b)
% This function ensures the correct setup of your structure
names = fieldnames(setup);
trueNames = {'maxits','tol','showComments','record','precond','L','U','M','x0'};
for i =1:length(names)
if ~any(strcmp(names{i},trueNames))
warning('Setup:FieldNR','Field not recognized: ''%s''',names{i});
end
end
if ~isfield(setup, 'maxits')
setup.maxits = 10;
end
if ~isfield(setup, 'tol')
setup.tol = 1.000000e-06;
end
if ~isfield(setup, 'showComments')
setup.showComments = false;
end
if ~isfield(setup, 'record')
setup.record = false;
end
if ~isfield(setup, 'precond')
setup.precond = false;
end
if ~isfield(setup, 'L')
setup.L = 0;
end
if ~isfield(setup, 'U')
setup.U = 0;
end
if ~isfield(setup, 'M')
setup.M = 0;
end
if ~isfield(setup, 'x0')
setup.x0 = zeros(size(b));
end
end