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The Matrix Computation Toolbox

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The Matrix Computation Toolbox


Nick Higham (view profile)


11 Sep 2002 (Updated )

A collection of M-files for carrying out various numerical linear algebra tasks.

function [Q, T] = trap2tri(L)
%TRAP2TRI  Unitary reduction of trapezoidal matrix to triangular form.
%          [Q, T] = TRAP2TRI(L), where L is an m-by-n lower trapezoidal
%          matrix with m >= n, produces a unitary Q such that Q*L = [T; 0],
%          where T is n-by-n and lower triangular.
%          Q is a product of Householder transformations.

%          Called by COD.
%          Reference:
%          G. H. Golub and C. F. Van Loan, Matrix Computations, third
%          edition, Johns Hopkins University Press, Baltimore, Maryland,
%          1996; P5.2.5.

[n, r] = size(L);

if r > n  | ~isequal(L,tril(L))
   error('Matrix must be lower trapezoidal and m-by-n with m >= n.')

Q = eye(n);  % To hold product of Householder transformations.

if r ~= n

   % Reduce nxr L =   r  [L1]  to lower triangular form: QL = [T].
   %                 n-r [L2]                                 [0]

   for j=r:-1:1
       % x is the vector to be reduced, which we overwrite with the H.T. vector.
       x = L(j:n,j);
       x(2:r-j+1) = zeros(r-j,1);  % These elts of column left unchanged.
       [v,beta,s] = gallery('house',x);

       % Nothing to do if x is zero (or x=a*e_1, but we don't check for that).
       if s ~= 0

          %  Implicitly apply H.T. to pivot column.
          % L(r+1:n,j) = zeros(n-r,1); % We throw these elts away at the end.
          L(j,j) = s;

          % Apply H.T. to rest of matrix.
          if j > 1
             y = v'*L(j:n, 1:j-1);
             L(j:n, 1:j-1) = L(j:n, 1:j-1) - beta*v*y;

          % Update H.T. product.
          y = v'*Q(j:n,:);
          Q(j:n,:) = Q(j:n,:) - beta*v*y;

T = L(1:r,:);   % Rows r+1:n have been zeroed out.

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