[Master Index] [Index for PublicToolbox/regutools]

# tikhonov

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

### Function Synopsis

    x_lambda = tikhonov(U,s,V,b,lambda)

### Help Text

    TIKHONOV Tikhonov regularization.

     x_lambda = tikhonov(U,s,V,b,lambda)
     x_lambda = tikhonov(U,sm,X,b,lambda) ,  sm = [sigma,mu]

     Computes the Tikhonov regularized solution x_lambda.  If the
     SVD is used, i.e. if U, s, and V are specified, then standard-
     form regularization is applied:
        min { || A x - b ||^2 + lambda^2 || x ||^2 } .
     If, on the other hand, the GSVD is used, i.e. if U, sm, and X
     are specified, then general-form regularization is applied:
        min { || A x - b ||^2 + lambda^2 || L x ||^2 } .

     If lambda is a vector, then x_lambda is a matrix such that
        x_lambda = [ x_lambda(1), x_lambda(2), ... ] .

### Cross-Reference Information

This function is called by

    minnorm                            C:\BrainStorm_2001\Toolbox\minnorm.m
    regudemo                           C:\BrainStorm_2001\PublicToolbox\regutools\regudemo.m
    sourceimaging                      C:\BrainStorm_2001\Toolbox\sourceimaging.m

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

    function x_lambda = tikhonov(U,s,V,b,lambda)
    %TIKHONOV Tikhonov regularization.
    %
    % x_lambda = tikhonov(U,s,V,b,lambda)
    % x_lambda = tikhonov(U,sm,X,b,lambda) ,  sm = [sigma,mu]
    %
    % Computes the Tikhonov regularized solution x_lambda.  If the
    % SVD is used, i.e. if U, s, and V are specified, then standard-
    % form regularization is applied:
    %    min { || A x - b ||^2 + lambda^2 || x ||^2 } .
    % If, on the other hand, the GSVD is used, i.e. if U, sm, and X
    % are specified, then general-form regularization is applied:
    %    min { || A x - b ||^2 + lambda^2 || L x ||^2 } .
    %
    % If lambda is a vector, then x_lambda is a matrix such that
    %    x_lambda = [ x_lambda(1), x_lambda(2), ... ] .

    % Per Christian Hansen, UNI-C, 03/10/90.

    % Reference: A. N. Tikhonov & V. Y. Arsenin, "Solutions of
    % Ill-Posed Problems", Wiley, 1977.

    % Initialization.
    if (min(lambda)<0)
      error('Illegal regularization parameter lambda')
    end
    [n,pv] = size(V); [p,ps] = size(s);
    eta = s(:,1).*(U(:,1:p)'*b);
    ll = length(lambda); x_lambda = zeros(n,ll);

    % Treat each lambda separately.
    if (ps==1)
      for i=1:ll
        x_lambda(:,i) = V(:,1:p)*(eta./(s.^2 + lambda(i)^2));
      end
    else
      x0 = V(:,p+1:n)*U(:,p+1:n)'*b;
      for i=1:ll
        x_lambda(:,i) = V(:,1:p)*(eta./(s(:,1).^2 + lambda(i)^2*s(:,2).^2)) + x0;
      end
    end

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