[Master Index] [Index for PublicToolbox/regutools]

# picard

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

### Function Synopsis

    xi = picard(U,s,b,d)

### Help Text

    PICARD Visual inspection of the Picard condition.

     xi = picard(U,s,b,d)
     xi = picard(U,sm,b,d)  ,  sm = [sigma,mu]

     Plots the singular values, s(i), the abs. value of the Fourier
     coefficients, |U(:,i)'*b|, and a (possibly smoothed) curve of
     the solution coefficients xi(i) = |U(:,i)'*b|/s(i).

     If s = [sigma,mu], where gamma = sigma./mu are the generalized
     singular values, then this routine plots gamma(i), |U(:,i)'*b|,
     and (smoothed) xi(i) = |U(:,i)'*b|/gamma(i).

     The smoothing is a geometric mean over 2*d+1 points, centered
     at point # i. If nargin = 3, then d = 0 (i.e, no smothing).

### Cross-Reference Information

This function is called by

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

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

    function xi = picard(U,s,b,d)
    %PICARD Visual inspection of the Picard condition.
    %
    % xi = picard(U,s,b,d)
    % xi = picard(U,sm,b,d)  ,  sm = [sigma,mu]
    %
    % Plots the singular values, s(i), the abs. value of the Fourier
    % coefficients, |U(:,i)'*b|, and a (possibly smoothed) curve of
    % the solution coefficients xi(i) = |U(:,i)'*b|/s(i).
    %
    % If s = [sigma,mu], where gamma = sigma./mu are the generalized
    % singular values, then this routine plots gamma(i), |U(:,i)'*b|,
    % and (smoothed) xi(i) = |U(:,i)'*b|/gamma(i).
    %
    % The smoothing is a geometric mean over 2*d+1 points, centered
    % at point # i. If nargin = 3, then d = 0 (i.e, no smothing).

    % Reference: P. C. Hansen, "The discrete Picard condition for
    % discrete ill-posed problems", BIT 30 (1990), 658-672.

    % Per Christian Hansen, UNI-C, 04/11/90.

    % Initialization.
    [n,ps] = size(s); beta = abs(U(:,1:n)'*b); eta = zeros(n,1);
    if (nargin==3), d = 0; end;
    if (ps==2), s = s(:,1)./s(:,2); end
    d21 = 2*d+1; keta = 1+d:n-d;
    for i=keta
      eta(i) = (prod(beta(i-d:i+d))^(1/d21))/s(i);
    end

    % Plot the data.
    semilogy(1:n,s,'-',1:n,beta,'x',keta,eta(keta),'o')
    xlabel('i')
    if (ps==1)
      title('--- = sing. values     x = rhs. coef.     o = solution coef.')
    else
      title('--- = gen. sing. values     x = rhs. coef.     o = sol. coef.')
    end

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