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# lsolve

## (PublicToolbox/regutools/lsolve.m in BrainStorm 2.0 (Alpha))

### Function Synopsis

    x = lsolve(L,y,W,NAA)

### Help Text

    LSOLVE Utility routine for "preconditioned" iterative methods.

     x = lsolve(L,y,W,NAA)

     Computes the vector
        x = L_p*y
     where L_p is the A-weighted generalized inverse of L.

     Typically, L is a p-by-n band matrix with bandwidth n-p+1, W holds
     a basis for the null space of L, and NAA is a utility matrix which
     should be computed by routine pinit.

     Alternatively, L is square and dense, and W and NAA are not needed.

     Notice that x and y may be matrices, in which case
        x(:,i) = L_p*y(:,i) .

### Cross-Reference Information

This function calls

    nu                                 C:\BrainStorm_2001\PublicToolbox\regutools\nu.m

This function is called by

    pcgls                              C:\BrainStorm_2001\PublicToolbox\regutools\pcgls.m
    plsqr                              C:\BrainStorm_2001\PublicToolbox\regutools\plsqr.m
    pnu                                C:\BrainStorm_2001\PublicToolbox\regutools\pnu.m

### Listing of function C:\BrainStorm_2001\PublicToolbox\regutools\lsolve.m

    function x = lsolve(L,y,W,NAA)
    %LSOLVE Utility routine for "preconditioned" iterative methods.
    %
    % x = lsolve(L,y,W,NAA)
    %
    % Computes the vector
    %    x = L_p*y
    % where L_p is the A-weighted generalized inverse of L.
    %
    % Typically, L is a p-by-n band matrix with bandwidth n-p+1, W holds
    % a basis for the null space of L, and NAA is a utility matrix which
    % should be computed by routine pinit.
    %
    % Alternatively, L is square and dense, and W and NAA are not needed.
    %
    % Notice that x and y may be matrices, in which case
    %    x(:,i) = L_p*y(:,i) .

    % Reference: M. Hanke, "Regularization with differential operators.
    % An iterative approach", J. Numer. Funct. Anal. Optim. 13 (1992),
    % 523-540.

    % Per Christian Hansen, UNI-C, and Martin Hanke, Institut fuer
    % Praktische Mathematik, Universitaet Karlsruhe, 05/26/93.

    % Initialization.
    [p,n] = size(L); nu = n-p; [py,ly] = size(y);

    % Special treatment of square L.
    if (nu==0), x = L\y; return; end

    % Compute a particular solution
    x = [[eye(nu),zeros(nu,p)];L]\[zeros(nu,ly);y];

    % Perform the necessary projection.
    x = x - W*(NAA*x);

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