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16 Apr 1998 (Updated )

Analysis and Solution of Discrete Ill-Posed Problems.

tgsvd(U,sm,X,b,k)
function [x_k,rho,eta] = tgsvd(U,sm,X,b,k)
%TGSVD Truncated GSVD regularization.
%
% [x_k,rho,eta] = tgsvd(U,sm,X,b,k) ,  sm = [sigma,mu]
%
% Computes the truncated GSVD solution
%            [ 0              0                 0    ]
%    x_k = X*[ 0  inv(diag(sigma(p-k+1:p)))     0    ]*U'*b .
%            [ 0              0             eye(n-p) ]
% If k is a vector, then x_k is a matrix such that
%    x_k = [ x_k(1), x_k(2), ... ] .
%
% The solution seminorm and the residual norm are returned in eta and rho.

% Reference: P. C. Hansen, "Regularization, GSVD and truncated GSVD",
% BIT 29 (1989), 491-504.

% Per Christian Hansen, IMM, Feb. 24, 2008.

% Initialization
m = size(U,1);
n = size(X,1);
p = size(sm,1);
lk = length(k);
if (min(k)<0 | max(k)>p)
  error('Illegal truncation parameter k')
end
x_k = zeros(n,lk);
eta = zeros(lk,1); rho = zeros(lk,1);
beta = U'*b;
xi = beta(1:p)./sm(:,1);
if (nargout==3), mxi = sm(:,2).*xi; end

if (m>=n)

  % The overdetermined or square case. Treat each k separately.
  if (p==n)
    x_0 = zeros(n,1);
  else
    x_0 = X(:,p+1:n)*(U(:,p+1:n)'*b);
  end
  for j=1:lk
    i = k(j); pi1 = p-i+1;
    if(i==0)
      x_k(:,j) = x_0;
    else
      x_k(:,j) = X(:,pi1:p)*xi(pi1:p) + x_0;
    end
    if (nargout>1), rho(j) = norm(beta(1:p-i)); end
    if (nargout==3), eta(j) = norm(mxi(pi1:p)); end
  end

  if (nargout > 1 & size(U,1) > n)
    rho = sqrt(rho.^2 + norm(b - U(:,1:n)*beta(1:n))^2);
  end

else

  % The underdetermined case. Treat each k separately.
  if (p==m)
    x_0 = zeros(n,1);
  else
    x_0 = X(:,p+1:m)*(U(:,p+1:m)'*b);
  end
  for j=1:lk
    i = k(j); pi1 = p-i+1;
    if(i==0)
      x_k(:,j) = x_0;
    else
      x_k(:,j) = X(:,pi1:p)*xi(pi1:p) + x_0;
    end
    if (nargout>1), rho(j) = norm(beta(1:p-i)); end
    if (nargout==3), eta(j) = norm(mxi(pi1:p)); end
  end

end

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