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# bidiag

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

### Function Synopsis

    [U,B_n,V] = bidiag(A)

### Help Text

    BIDIAG Bidiagonalization of an m-times-n matrix with m >= n.

     B_n = bidiag(A)
     [U,B_n,V] = bidiag(A)

     Computes the bidiagonalization of the m-times-n matrix A
     with m >= n:
         A = U*B*V' ,
     where B is an upper bidiagonal n-times-n matrix, and U and
     V have orthogonal columns.

     The matrix B is stored in compact form in B_n as follows:
             [b_11 b_12            ]         [b_11    b_12]
         B = [     b_22 b_23       ]   B_n = [b_22    b_23] .
             [            .    .   ]         [ .       .  ]
             [                b_nn ]         [b_nn    NaN ]
     The NaN in B_n(n,2) is used to distinguish a "compact" upper
     bidiagonal matrix from a "compact" lower bidiagonal one.

### Cross-Reference Information

This function calls

    app_hh                             C:\BrainStorm_2001\PublicToolbox\regutools\app_hh.m
    gen_hh                             C:\BrainStorm_2001\PublicToolbox\regutools\gen_hh.m

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

    function [U,B_n,V] = bidiag(A)
    %BIDIAG Bidiagonalization of an m-times-n matrix with m >= n.
    %
    % B_n = bidiag(A)
    % [U,B_n,V] = bidiag(A)
    %
    % Computes the bidiagonalization of the m-times-n matrix A
    % with m >= n:
    %     A = U*B*V' ,
    % where B is an upper bidiagonal n-times-n matrix, and U and
    % V have orthogonal columns.
    %
    % The matrix B is stored in compact form in B_n as follows:
    %         [b_11 b_12            ]         [b_11    b_12]
    %     B = [     b_22 b_23       ]   B_n = [b_22    b_23] .
    %         [            .    .   ]         [ .       .  ]
    %         [                b_nn ]         [b_nn    NaN ]
    % The NaN in B_n(n,2) is used to distinguish a "compact" upper
    % bidiagonal matrix from a "compact" lower bidiagonal one.

    % Reference: L. Elden, "Algorithms for regularization of ill-
    % conditioned least-squares problems", BIT 17 (1977), 134-145.

    % Per Christian Hansen, UNI-C, 03/11/92.

    % Initialization.
    [m,n] = size(A);
    if (m<n), error('Illegal dimensions of A'), end
    B_n = zeros(n,2);
    if (nargout> 1), U = [eye(n);zeros(m-n,n)]; betaU = zeros(n,1); end
    if (nargout==3), V = eye(n); betaV = zeros(n,1); end

    % Bidiagonalization; save Householder quantities.
    if (m > n), k_last = n; else k_last = n-1; end
    for k=1:k_last

      [B_n(k,1),beta,A(k:m,k)] = gen_hh(A(k:m,k));
      if (k < n), A(k:m,k+1:n) = app_hh(A(k:m,k+1:n),beta,A(k:m,k)); end
      if (nargout>1), betaU(k) = beta; end

      if (k < n-1)
        [B_n(k,2),beta,v] = gen_hh(A(k,k+1:n)'); A(k,k+1:n) = v';
        A(k+1:m,k+1:n) = app_hh(A(k+1:m,k+1:n)',beta,A(k,k+1:n)')';
        if (nargout==3), betaV(k) = beta;, end
      elseif (k == n-1)
        B_n(n-1,2) = A(n-1,n);
      end

    end

    % Save bottom element if A is square.
    if (k_last < n), B_n(n,1) = A(n,n); end

    % Put a NaN in bottom element of B_n.
    B_n(n,2) = NaN;

    % Compute U if wanted.
    if (nargout>1)
      for k=k_last:-1:1
        U(k:m,k:n) = app_hh(U(k:m,k:n),betaU(k),A(k:m,k));
      end
    end

    % Compute V if wanted.
    if (nargout==3)
      for k=n-2:-1:1
        V(k+1:n,k:n) = app_hh(V(k+1:n,k:n),betaV(k),A(k,k+1:n)');
      end
    end

    if (nargout==1), U = B_n; end

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