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NAG Toolbox: nag_lapack_dtzrzf (f08bh)

▸▿ Contents

    1  Purpose
    2  Syntax
    3  Description
    4  References
▸▿ 5  Parameters
    6  Error Indicators and Warnings
    7  Accuracy
    8  Further Comments
    9  Example

Purpose

nag_lapack_dtzrzf (f08bh) reduces the m by n (m≤n) real upper trapezoidal matrix A to upper triangular form by means of orthogonal transformations.

Syntax

[a, tau, info] = f08bh(a, 'm', m, 'n', n)
[a, tau, info] = nag_lapack_dtzrzf(a, 'm', m, 'n', n)

Description

The m by n (m≤n) real upper trapezoidal matrix A given by
A = R1 R2 ,  
where R1 is an m by m upper triangular matrix and R2 is an m by n-m matrix, is factorized as
A = R 0 Z ,  
where R is also an m by m upper triangular matrix and Z is an n by n orthogonal matrix.

References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

Parameters

Compulsory Input Parameters

1:     alda: – double array
The first dimension of the array a must be at least max1,m.
The second dimension of the array a must be at least max1,n.
The leading m by n upper trapezoidal part of the array a must contain the matrix to be factorized.

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: the first dimension of the array a.
m, the number of rows of the matrix A.
Constraint: m≥0.
2:     n – int64int32nag_int scalar
Default: the second dimension of the array a.
n, the number of columns of the matrix A.
Constraint: n≥0.

Output Parameters

1:     alda: – double array
The first dimension of the array a will be max1,m.
The second dimension of the array a will be max1,n.
The leading m by m upper triangular part of a contains the upper triangular matrix R, and elements m+1 to n of the first m rows of a, with the array tau, represent the orthogonal matrix Z as a product of m elementary reflectors (see Representation of orthogonal or unitary matrices in the F08 Chapter Introduction).
2:     tau: – double array
The dimension of the array tau will be max1,m
The scalar factors of the elementary reflectors.
3:     info – int64int32nag_int scalar
info=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   info=-i
If info=-i, parameter i had an illegal value on entry. The parameters are numbered as follows:
1: m, 2: n, 3: a, 4: lda, 5: tau, 6: work, 7: lwork, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

The computed factorization is the exact factorization of a nearby matrix A+E, where
E2 = O⁡ε A2  
and ε is the machine precision.

Further Comments

The total number of floating-point operations is approximately 4m2n-m.
The complex analogue of this function is nag_lapack_ztzrzf (f08bv).

Example

This example solves the linear least squares problems
minx bj - Axj 2 ,   j=1,2  
for the minimum norm solutions x1 and x2, where bj is the jth column of the matrix B,
A = -0.09 0.14 -0.46 0.68 1.29 -1.56 0.20 0.29 1.09 0.51 -1.48 -0.43 0.89 -0.71 -0.96 -1.09 0.84 0.77 2.11 -1.27 0.08 0.55 -1.13 0.14 1.74 -1.59 -0.72 1.06 1.24 0.34   and   B= 7.4 2.7 4.2 -3.0 -8.3 -9.6 1.8 1.1 8.6 4.0 2.1 -5.7 .  
The solution is obtained by first obtaining a QR factorization with column pivoting of the matrix A, and then the RZ factorization of the leading k by k part of R is computed, where k is the estimated rank of A. A tolerance of 0.01 is used to estimate the rank of A from the upper triangular factor, R.
Note that the block size (NB) of 64 assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

Open in the MATLAB editor: f08bh_example

function f08bh_example


fprintf('f08bh example results\n\n');

% Going to solve Least squares problem Ax = b, m>n.
m = 6;
n = 5;
a = [-0.09,  0.14, -0.46,  0.68,  1.29;
     -1.56,  0.2,   0.29,  1.09,  0.51;
     -1.48, -0.43,  0.89, -0.71, -0.96;
     -1.09,  0.84,  0.77,  2.11, -1.27;
      0.08,  0.55, -1.13,  0.14,  1.74;
     -1.59, -0.72,  1.06,  1.24,  0.34];

b = [ 7.4,  2.7;
      4.2, -3.0;
     -8.3, -9.6;
      1.8,  1.1;
      8.6,  4.0;
      2.1, -5.7];

% QR factorization of A with column pivoting = Q*(R1 R2 )*(P^T)
%                                                (0  R22)

[qr, jpvt, tau, info] = f08bf( ...
			       a, zeros(n,1,'int64'));

% QRP'X = B, => RP'X = Q'B = C
% Compute C = Q'B
[c, info] = f08ag( ...
		   'Left', 'Transpose', qr, tau, b);

% Determine the rank, k, of R relative to tol;
% Choose tol to reflect the relative accuracy of the input data
tol = 0.01;
k = find(abs(diag(qr)) <= tol*abs(qr(1,1)));
if numel(k) == 0
  k = numel(diag(qr));
else
  k = k(1)-1;
end

fprintf('Tolerance used to estimate the rank of A\n     %11.2e\n', tol);
fprintf('Estimated rank of A\n        %d\n', k);

% Compute the RZ (TZ) factorization of the first k rows of (R1 R2)
[rz, taurz, info] = f08bh( ...
			   qr(1:k,:));

% Now, (TZ)P'X = C on first k rows of C
% Let ZP'X = T^{-1}C = Y (on first k rows)
y = zeros(n, 2);
y(1:k, :) = inv(triu(rz(1:k,1:k)))*c(1:k,:);

% ZP'X = Y => P'X = Z^T Y = W; Form W = Z^TY.
[w, info] = f08bk( ...
		   'Left', 'Transpose', int64(n-k), rz, taurz, y);

% P'X = W => X = PW, '
x = zeros(n, 2);
for i=1:n
   x(jpvt(i), :) = w(i, :);
end
fprintf('\nLeast-squares solution(s)\n');
disp(x);

% Compute estimates of the square roots of the residual sums of squares
rnorm = [norm(c(k+1:6,1)), norm(c(k+1:6,2))];
fprintf('Square root(s) of the residual sum(s) of squares\n');
disp(rnorm);



f08bh example results

Tolerance used to estimate the rank of A
        1.00e-02
Estimated rank of A
        4

Least-squares solution(s)
    0.6344    3.6258
    0.9699    1.8284
   -1.4402   -1.6416
    3.3678    2.4307
    3.3992    0.2818

Square root(s) of the residual sum(s) of squares
    0.0254    0.0365
PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox
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