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# nu

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

### Function Synopsis

    [X,rho,eta,F] = nu(A,b,k,nu,s)

### Help Text

    NU Brakhage's nu-method.

     [X,rho,eta,F] = nu(A,b,k,nu,s)

     Performs k steps of Brakhage's nu-method for the problem
        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 nu is not specified, nu = .5 is the default value, which gives
     the Chebychev method of Nemirovskii and Polyak.

     If the singular values s are also provided, nu computes the
     filter factors associated with each step and stores them
     columnwise in the matrix F.

### Cross-Reference Information

This function is called by

    fil_fac                            C:\BrainStorm_2001\PublicToolbox\regutools\fil_fac.m
    lsolve                             C:\BrainStorm_2001\PublicToolbox\regutools\lsolve.m
    ltsolve                            C:\BrainStorm_2001\PublicToolbox\regutools\ltsolve.m
    pinit                              C:\BrainStorm_2001\PublicToolbox\regutools\pinit.m
    pnu                                C:\BrainStorm_2001\PublicToolbox\regutools\pnu.m
    std_form                           C:\BrainStorm_2001\PublicToolbox\regutools\std_form.m

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

    function [X,rho,eta,F] = nu(A,b,k,nu,s)
    %NU Brakhage's nu-method.
    %
    % [X,rho,eta,F] = nu(A,b,k,nu,s)
    %
    % Performs k steps of Brakhage's nu-method for the problem
    %    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 nu is not specified, nu = .5 is the default value, which gives
    % the Chebychev method of Nemirovskii and Polyak.
    %
    % If the singular values s are also provided, nu computes the
    % filter factors associated with each step and stores them
    % columnwise in the matrix F.

    % Reference: H. Brakhage, "On ill-posed problems and the method of
    % conjugate gradients"; in H. W. Engl & G. W. Groetsch, "Inverse and
    % Ill-Posed Problems", Academic Press, 1987.

    % Martin Hanke, Institut fuer Praktische Mathematik, Universitaet
    % Karlsruhe and Per Christian Hansen, UNI-C, 03/21/92.

    % Set parameter.
    l_steps = 3;      % Number of Lanczos steps for est. of || A ||.
    fudge   = 0.99;   % Scale A and b by fudge/|| A*L_p ||.
    fudge_thr = 1e-4; % Used to prevent filter factors from exploding.

    % Initialization.
    if (k < 1), error('Number of steps k must be positive'), end
    if (nargin==3), nu = .5; end
    [m,n] = size(A); X = zeros(n,k);
    if (nargout > 1)
      rho = zeros(k,1); eta = rho;
    end;
    if (nargin==5)
      F = zeros(n,k); Fd = zeros(n,1); s = s.^2;
    end
    V = zeros(n,l_steps); B = zeros(l_steps+1,l_steps);
    v = zeros(n,1); eta = zeros(l_steps+1,1);

    % Compute a rough estimate of the norm of A by means of a few
    % steps of Lanczos bidiagonalization, and scale A and b such
    % that || A || is slightly less than one.
    beta = norm(b); u = b/beta;
    for i=1:l_steps
      r = A'*u - beta*v;
      alpha = norm(r); v = r/alpha;
      B(i,i) = alpha; V(:,i) = v;
      p = A*v - alpha*u;
      beta = norm(p); u = p/beta;
      B(i+1,i) = beta;
    end
    scale = fudge/norm(B); A = scale*A; b = scale*b;
    if (nargin==5), s = scale^2*s; end

    % Prepare for iteration.
    x = zeros(n,1);
    d = A'*b;
    r = d;
    if (nargout>1), z = b; end

    % Iterate.
    for j=0:k-1

      alpha = 4*(j+nu)*(j+nu+0.5)/(j+2*nu)/(j+2*nu+0.5);
      beta  = (j+nu)*(j+1)*(j+0.5)/(j+2*nu)/(j+2*nu+0.5)/(j+nu+1);
      Ad = A*d; AAd = A'*Ad;
      x  = x + alpha*d;
      r  = r - alpha*AAd;
      d  = r + beta*d;
      X(:,j+1) = x;
      if (nargout>1)
        z = z - alpha*Ad; rho(j+1) = norm(z)/scale;
      end;
      if (nargout>2), eta(j+1) = norm(x); end;

      if (nargin==5)
        if (j==0)
          F(:,1) = alpha*s;
          Fd = s - s.*F(:,1) + beta*s;
        else
          F(:,j+1) = F(:,j) + alpha*Fd;
          Fd = s - s.*F(:,j+1) + beta*Fd;
        end
        if (j > 1)
          f = find(abs(F(:,j)-1) < fudge_thr & abs(F(:,j-1)-1) < fudge_thr);
          if (length(f) > 0), F(f,j+1) = ones(length(f),1); end
        end
      end

    end

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