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

Anderson E, Bai Z and Dongarra J (1992) Generalized QR factorization and its applications Linear Algebra Appl. (Volume 162–164) 243–271

Parameters

Compulsory Input Parameters

1: $a (lda :)$ – complex array: The first dimension of the array a must be at least $\max (1, m)$ .
The second dimension of the array a must be at least $\max (1, n)$ .

The $m$ by $n$ matrix $A$ .
2: $b (ldb :)$ – complex array: The first dimension of the array b must be at least $\max (1, m)$ .
The second dimension of the array b must be at least $\max (1, p)$ .

The $m$ by $p$ matrix $B$ .
3: $d (m)$ – complex array: The left-hand side vector $d$ of the GLM equation.

Optional Input Parameters

1: $m$ – int64int32nag_int scalar: Default: the dimension of the array d and the first dimension of the array b. (An error is raised if these dimensions are not equal.)
$m$ , the number of rows of the matrices $A$ and $B$ .

Constraint: $m \geq 0$ .
2: $n$ – int64int32nag_int scalar: Default: the second dimension of the array a.
$n$ , the number of columns of the matrix $A$ .

Constraint: $0 \leq n \leq m$ .
3: $p$ – int64int32nag_int scalar: Default: the second dimension of the array b.
$p$ , the number of columns of the matrix $B$ .

Constraint: $p \geq m - n$ .

Output Parameters

1: $a (lda :)$ – complex array: The first dimension of the array a will be $\max (1, m)$ .
The second dimension of the array a will be $\max (1, n)$ .
2: $b (ldb :)$ – complex array: The first dimension of the array b will be $\max (1, m)$ .
The second dimension of the array b will be $\max (1, p)$ .
3: $d (m)$ – complex array
4: $x (n)$ – complex array: The solution vector $x$ of the GLM problem.
5: $y (p)$ – complex array: The solution vector $y$ of the GLM problem.
6: $info$ – 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: p, 4: a, 5: lda, 6: b, 7: ldb, 8: d, 9: x, 10: y, 11: work, 12: lwork, 13: 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.

$info = 1$: The upper triangular factor $R$ associated with $A$ in the generalized $R Q$ factorization of the pair $(A, B)$ is singular, so that $rank (A) < m$ ; the least squares solution could not be computed.

$info = 2$: The bottom $(N - M)$ by $(N - M)$ part of the upper trapezoidal factor $T$ associated with $B$ in the generalized $Q R$ factorization of the pair $(A, B)$ is singular, so that $rank (\begin{matrix} A & B \end{matrix}) < N$ ; the least squares solutions could not be computed.

Accuracy

For an error analysis, see Anderson et al. (1992). See also Section 4.6 of Anderson et al. (1999).

Further Comments

When

p = m \geq n

, the total number of real floating-point operations is approximately

\frac{8}{3} (2 m^{3} - n^{3}) + 16 n m^{2}

; when

p = m = n

, the total number of real floating-point operations is approximately

\frac{56}{3} m^{3}

Example

This example solves the weighted least squares problem

\underset{x}{minimize} {‖B^{- 1} (d - A x)‖}_{2},

where

B = (\begin{array}{r} 0.5 - 1.0 i \\ 1.0 - 2.0 i \\ 2.0 - 3.0 i \\ 5.0 - 4.0 i \end{array}),

d = (\begin{array}{r} 6.00 - 0.40 i \\ - 5.27 + 0.90 i \\ 2.72 - 2.13 i \\ - 1.30 - 2.80 i \end{array})

and

A = (\begin{array}{r} 0.96 - 0.81 i & - 0.03 + 0.96 i & - 0.91 + 2.06 i \\ - 0.98 + 1.98 i & - 1.20 + 0.19 i & - 0.66 + 0.42 i \\ 0.62 - 0.46 i & 1.01 + 0.02 i & 0.63 - 0.17 i \\ 1.08 - 0.28 i & 0.20 - 0.12 i & - 0.07 + 1.23 i \end{array}) .

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: f08zp_example

function f08zp_example


fprintf('f08zp example results\n\n');

a = [ 0.96 - 0.81i, -0.03 + 0.96i, -0.91 + 2.06i;
     -0.98 + 1.98i, -1.20 + 0.19i, -0.66 + 0.42i;
      0.62 - 0.46i,  1.01 + 0.02i,  0.63 - 0.17i;
      1.08 - 0.28i,  0.20 - 0.12i, -0.07 + 1.23i];
b = [ 0.5 - 1i,  0 + 0i,  0 + 0i,  0 + 0i;
      0   + 0i,  1 - 2i,  0 + 0i,  0 + 0i;
      0   + 0i,  0 + 0i,  2 - 3i,  0 + 0i;
      0   + 0i,  0 + 0i,  0 + 0i,  5 - 4i];
d = [ 6.00 - 0.40i;
     -5.27 + 0.90i;
      2.72 - 2.13i;
     -1.30 - 2.80i];

%Solve complex general Guass-Markov linear model
[~, ~, ~, x, y, info] = f08zp( ...
                               a, b, d);


disp('Weighted least-squares solution');
disp(x);
ncols  = int64(80);
indent = int64(0);
[ifail] = x04db( ...
                 'Gen', ' ', y, 'B', '1P,E10.2', 'Residual vector', ...
                 'N', 'N', ncols, indent);
sqres = norm(y,2);
fprintf('\nSquare root of the residual sum of squares\n%11.2e\n', ...
        sqres);

f08zp example results

Weighted least-squares solution
  -0.9846 + 1.9950i
   3.9929 - 4.9748i
  -3.0026 + 0.9994i

 Residual vector
   (  1.26E-04, -4.66E-04)
   (  1.11E-03, -8.61E-04)
   (  3.84E-03, -1.82E-03)
   (  2.03E-03,  3.02E-03)

Square root of the residual sum of squares
   5.79e-03

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox