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

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1: $trans$ – string (length ≥ 1): If $trans ='N'$ , the linear system involves $A$ .
If $trans ='C'$ , the linear system involves $A^{H}$ .

Constraint: $trans ='N'$ or $'C'$ .
2: $a (lda :)$ – 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 $A$ .
3: $b (ldb :)$ – complex array: The first dimension of the array b must be at least $\max (1, m, n)$ .
The second dimension of the array b must be at least $\max (1, nrhs_p)$ .

The matrix $B$ of right-hand side vectors, stored in columns; b is $m$ by $r$ if $trans ='N'$ , or $n$ by $r$ if $trans ='C'$ .

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

Constraint: $n \geq 0$ .
3: $nrhs_p$ – int64int32nag_int scalar: Default: the second dimension of the array b.
$r$ , the number of right-hand sides, i.e., the number of columns of the matrices $B$ and $X$ .

Constraint: $nrhs_p \geq 0$ .

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)

m \geq n

, a stores details of its

Q R

factorization, as returned by nag_lapack_zgeqrf (f08as).

m < n

, a stores details of its

L Q

factorization, as returned by nag_lapack_zgelqf (f08av).

2: $b (ldb :)$ – complex array

The first dimension of the array b will be

\max (1, m, n)

The second dimension of the array b will be

\max (1, nrhs_p)

b stores the solution vectors,

x

, stored in columns:

if $trans ='N'$ and $m \geq n$ , or $trans ='C'$ and $m < n$ , elements $1$ to $\min (m, n)$ in each column of b contain the least squares solution vectors; the residual sum of squares for the solution is given by the sum of squares of the modulus of elements $(\min (m, n) + 1)$ to $\max (m, n)$ in that column;
otherwise, elements $1$ to $\max (m, n)$ in each column of b contain the minimum norm solution vectors.

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: trans, 2: m, 3: n, 4: nrhs_p, 5: a, 6: lda, 7: b, 8: ldb, 9: work, 10: lwork, 11: 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.

$info > 0$: If $info = i$ , diagonal element $i$ of the triangular factor of $A$ is zero, so that $A$ does not have full rank; the least squares solution could not be computed.

Accuracy

See Section 4.5 of Anderson et al. (1999) for details of error bounds.

Further Comments

The total number of floating-point operations required to factorize

A

is approximately

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

m \geq n

and

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

otherwise. Following the factorization the solution for a single vector

x

requires

O (\min (m^{2}, n^{2}))

operations.

The real analogue of this function is nag_lapack_dgels (f08aa).

Example

This example solves the linear least squares problem

\min_{x} {‖b - A x‖}_{2},

where

A = (\begin{array}{r} 0.96 - 0.81 i & - 0.03 + 0.96 i & - 0.91 + 2.06 i & - 0.05 + 0.41 i \\ - 0.98 + 1.98 i & - 1.20 + 0.19 i & - 0.66 + 0.42 i & - 0.81 + 0.56 i \\ 0.62 - 0.46 i & 1.01 + 0.02 i & 0.63 - 0.17 i & - 1.11 + 0.60 i \\ - 0.37 + 0.38 i & 0.19 - 0.54 i & - 0.98 - 0.36 i & 0.22 - 0.20 i \\ 0.83 + 0.51 i & 0.20 + 0.01 i & - 0.17 - 0.46 i & 1.47 + 1.59 i \\ 1.08 - 0.28 i & 0.20 - 0.12 i & - 0.07 + 1.23 i & 0.26 + 0.26 i \end{array})

and

b = (\begin{array}{r} - 2.09 + 1.93 i \\ 3.34 - 3.53 i \\ - 4.94 - 2.04 i \\ 0.17 + 4.23 i \\ - 5.19 + 3.63 i \\ 0.98 + 2.53 i \end{array}) .

The square root of the residual sum of squares is also output.

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: f08an_example

function f08an_example


fprintf('f08an example results\n\n');

a = [ 0.96 - 0.81i, -0.03 + 0.96i,  -0.91 + 2.06i, -0.05 + 0.41i;
     -0.98 + 1.98i, -1.20 + 0.19i,  -0.66 + 0.42i, -0.81 + 0.56i;
      0.62 - 0.46i,  1.01 + 0.02i,   0.63 - 0.17i, -1.11 + 0.60i;
     -0.37 + 0.38i,  0.19 - 0.54i,  -0.98 - 0.36i,  0.22 - 0.20i;
      0.83 + 0.51i,  0.20 + 0.01i,  -0.17 - 0.46i,  1.47 + 1.59i;
      1.08 - 0.28i,  0.20 - 0.12i,  -0.07 + 1.23i,  0.26 + 0.26i];
b = [-2.09 + 1.93i;
      3.34 - 3.53i;
     -4.94 - 2.04i;
      0.17 + 4.23i;
     -5.19 + 3.63i;
      0.98 + 2.53i];
[m,n] = size(a);

% Solve the least squares problem min( norm2(b - Ax) ) for x
trans = 'No transpose';
[af, x, info] = f08an( ...
		       trans, a, b);

% Print Solution
fprintf('\nLeast Squares Solution:\n');
disp(transpose(x(1:n)));
fprintf('Square root of the residual sum of squares\n');
disp(norm(x(n+1:m),2));

f08an example results


Least Squares Solution:
  -0.5044 - 1.2179i  -2.4281 + 2.8574i   1.4872 - 2.1955i   0.4537 + 2.6904i

Square root of the residual sum of squares
    0.0688

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

Chapter Contents

Chapter Introduction

NAG Toolbox