lsqr_b.m

A comparison of methods for inverting helioseismic data

function [X,rho,eta,F] = lsqr_b(A,b,k,reorth,s)
%LSQR_B Solution of least squares problems by Lanczos bidiagonalization.
%
% [X,rho,eta,F] = lsqr_b(A,b,k,reorth,s)
%
% Performs k steps of the LSQR Lanczos bidiagonalization algorithm
% applied to the system
%    min || A x - b || .
% The routine returns all k solutions, stored as columns of
% the matrix X.  The solution norm and residual norm are returned
% in eta and rho, respectively.
%
% If the singular values s are also provided, lsqr computes the
% filter factors associated with each step and stores them columnwise
% in the matrix F.
%
% Reorthogonalization is controlled by means of reorth:
%    reorth = 0 : no reorthogonalization (default),
%    reorth = 1 : reorthogonalization by means of MGS.

% Reference: C. C. Paige & M. A. Saunders, "LSQR: an algorithm for
% sparse linear equations and sparse least squares", ACM Trans.
% Math. Software 8 (1982), 43-71.

% Per Christian Hansen, IMM, April 8, 2001.

% The fudge threshold is used to prevent filter factors from exploding.
fudge_thr = 1e-4;

% Initialization.
if (k < 1), error('Number of steps k must be positive'), end
if (nargin==3), reorth = 0; end
if (nargout==4 & nargin<5), error('Too few input arguments'), end
[m,n] = size(A); X = zeros(n,k);
if (reorth==0)
  UV = 0;
elseif (reorth==1)
  U = zeros(m,k); V = zeros(n,k); UV = 1;
  if (k>=n), error('No. of iterations must satisfy k < n'), end
else
  error('Illegal reorth')
end
if (nargout > 1)
  eta = zeros(k,1); rho = eta;
  c2 = -1; s2 = 0; xnorm = 0; z = 0;
end
if (nargin==5)
  ls = length(s);
  F = zeros(ls,k); Fv = zeros(ls,1); Fw = Fv;
  s = s.^2;
end

% Prepare for LSQR iteration.
v = zeros(n,1); x = v; beta = norm(b);
if (beta==0), error('Right-hand side must be nonzero'), end
u = b/beta; if (UV), U(:,1) = u; end
r = (u'*A)'; alpha = norm(r);   % A'*u;
v = r/alpha; if (UV), V(:,1) = v; end
phi_bar = beta; rho_bar = alpha; w = v;
if (nargin==5), Fv = s/(alpha*beta); Fw = Fv; end

% Perform Lanczos bidiagonalization with/without reorthogonalization.
for i=2:k+1

  alpha_old = alpha; beta_old = beta;

  % Compute A*v - alpha*u.
  p = A*v - alpha*u;
  if (reorth==0)
    beta = norm(p); u = p/beta;
  else
    for j=1:i-1, p = p - (U(:,j)'*p)*U(:,j); end
    beta = norm(p); u = p/beta;
  end

  % Compute A'*u - beta*v.
  r = A'*u - beta*v;
  if (reorth==0)
    alpha = norm(r); v = r/alpha;
  else
    for j=1:i-1, r = r - (V(:,j)'*r)*V(:,j); end
    alpha = norm(r); v = r/alpha;
  end

  % Store U and V if necessary.
  if (UV), U(:,i) = u; V(:,i) = v; end

  % Construct and apply orthogonal transformation.
  rrho = norm([rho_bar,beta]); c1 = rho_bar/rrho;
  s1 = beta/rrho; theta = s1*alpha; rho_bar = -c1*alpha;
  phi = c1*phi_bar; phi_bar = s1*phi_bar;

  % Compute solution norm and residual norm if necessary;
  if (nargout > 1)
    delta = s2*rrho; gamma_bar = -c2*rrho; rhs = phi - delta*z;
    z_bar = rhs/gamma_bar; eta(i-1) = norm([xnorm,z_bar]);
    gamma = norm([gamma_bar,theta]);
    c2 = gamma_bar/gamma; s2 = theta/gamma;
    z = rhs/gamma; xnorm = norm([xnorm,z]);
    rho(i-1) = abs(phi_bar);
  end

  % If required, compute the filter factors.
  if (nargin==5)

    if (i==2)
      Fv_old = Fv;
      Fv = Fv.*(s - beta^2 - alpha_old^2)/(alpha*beta);
      F(:,i-1) = (phi/rrho)*Fw;
    else
      tmp = Fv;
      Fv = (Fv.*(s - beta^2 - alpha_old^2) - ...
                 Fv_old*alpha_old*beta_old)/(alpha*beta);
      Fv_old = tmp;
      F(:,i-1) = F(:,i-2) + (phi/rrho)*Fw;
    end
    if (i > 3)
      f = find(abs(F(:,i-2)-1) < fudge_thr & abs(F(:,i-3)-1) < fudge_thr);
      if ~isempty(f), F(f,i-1) = ones(length(f),1); end
    end
    Fw = Fv - (theta/rrho)*Fw;

  end

  % Update the solution.
  x = x + (phi/rrho)*w; w = v - (theta/rrho)*w;
  X(:,i-1) = x;

end