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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_lapack_zunmqr (f08au)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_lapack_zunmqr (f08au) multiplies an arbitrary complex matrix C by the complex unitary matrix Q from a QR factorization computed by nag_lapack_zgeqrf (f08as), nag_lapack_zgeqpf (f08bs) or nag_lapack_zgeqp3 (f08bt).

Syntax

[c, info] = f08au(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)
[c, info] = nag_lapack_zunmqr(side, trans, a, tau, c, 'm', m, 'n', n, 'k', k)

Description

nag_lapack_zunmqr (f08au) is intended to be used after a call to nag_lapack_zgeqrf (f08as), nag_lapack_zgeqpf (f08bs) or nag_lapack_zgeqp3 (f08bt), which perform a QR factorization of a complex matrix A. The unitary matrix Q is represented as a product of elementary reflectors.
This function may be used to form one of the matrix products
QC , QHC , CQ ​ or ​ CQH ,  
overwriting the result on c (which may be any complex rectangular matrix).
A common application of this function is in solving linear least squares problems, as described in the F08 Chapter Introduction and illustrated in Example in nag_lapack_zgeqrf (f08as).

References

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

Parameters

Compulsory Input Parameters

1:     side – string (length ≥ 1)
Indicates how Q or QH is to be applied to C.
side='L'
Q or QH is applied to C from the left.
side='R'
Q or QH is applied to C from the right.
Constraint: side='L' or 'R'.
2:     trans – string (length ≥ 1)
Indicates whether Q or QH is to be applied to C.
trans='N'
Q is applied to C.
trans='C'
QH is applied to C.
Constraint: trans='N' or 'C'.
3:     alda: – complex array
The first dimension, lda, of the array a must satisfy
  • if side='L', lda max1,m ;
  • if side='R', lda max1,n .
The second dimension of the array a must be at least max1,k.
Details of the vectors which define the elementary reflectors, as returned by nag_lapack_zgeqrf (f08as), nag_lapack_zgeqpf (f08bs) or nag_lapack_zgeqp3 (f08bt).
4:     tau: – complex array
The dimension of the array tau must be at least max1,k
Further details of the elementary reflectors, as returned by nag_lapack_zgeqrf (f08as), nag_lapack_zgeqpf (f08bs) or nag_lapack_zgeqp3 (f08bt).
5:     cldc: – complex array
The first dimension of the array c must be at least max1,m.
The second dimension of the array c must be at least max1,n.
The m by n matrix C.

Optional Input Parameters

1:     m int64int32nag_int scalar
Default: the first dimension of the array c.
m, the number of rows of the matrix C.
Constraint: m0.
2:     n int64int32nag_int scalar
Default: the second dimension of the array c.
n, the number of columns of the matrix C.
Constraint: n0.
3:     k int64int32nag_int scalar
Default: the second dimension of the arrays a, tau.
k, the number of elementary reflectors whose product defines the matrix Q.
Constraints:
  • if side='L', m k 0 ;
  • if side='R', n k 0 .

Output Parameters

1:     cldc: – complex array
The first dimension of the array c will be max1,m.
The second dimension of the array c will be max1,n.
c stores QC or QHC or CQ or CQH as specified by side and trans.
2:     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: side, 2: trans, 3: m, 4: n, 5: k, 6: a, 7: lda, 8: tau, 9: c, 10: ldc, 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.

Accuracy

The computed result differs from the exact result by a matrix E such that
E2 = Oε C2 ,  
where ε is the machine precision.

Further Comments

The total number of real floating-point operations is approximately 8nk 2m-k  if side='L' and 8mk 2n-k  if side='R'.
The real analogue of this function is nag_lapack_dormqr (f08ag).

Example

See Example in nag_lapack_zgeqrf (f08as).
function f08au_example


fprintf('f08au 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];

% Compute the QR factorization of A
[qr, tau, info] = f08as(a);

c = [-2.09 + 1.93i,  3.26 - 2.70i;
      3.34 - 3.53i, -6.22 + 1.16i;
     -4.94 - 2.04i,  7.94 - 3.13i;
      0.17 + 4.23i,  1.04 - 4.26i;
     -5.19 + 3.63i, -2.31 - 2.12i;
      0.98 + 2.53i, -1.39 - 4.05i];

% Perform B = Q^H*C
side = 'Left';
trans = 'Conjugate transpose';
[b, info] = f08au( ...
		   side, trans, qr, tau, c);

disp('B = Q^H*C:');
disp(b);


f08au example results

B = Q^H*C:
   5.3510 - 0.1638i  -4.7626 - 2.8427i
  -5.7559 - 0.2004i   6.3325 - 2.7406i
  -2.5366 + 4.0215i   6.4835 - 3.8629i
  -1.0677 - 6.3316i   6.4968 - 0.7809i
  -0.0381 - 0.0273i  -0.1320 - 0.0612i
  -0.0144 + 0.0483i   0.0906 - 0.0740i


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