NAG FL Interface
f08ckf (dormrq)

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1 Purpose

f08ckf multiplies a general real m×n matrix C by the real orthogonal matrix Q from an RQ factorization computed by f08chf.

2 Specification

Fortran Interface
Subroutine f08ckf ( side, trans, m, n, k, a, lda, tau, c, ldc, work, lwork, info)
Integer, Intent (In) :: m, n, k, lda, ldc, lwork
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (In) :: tau(*)
Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), c(ldc,*)
Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
Character (1), Intent (In) :: side, trans
C Header Interface
#include <nag.h>
void  f08ckf_ (const char *side, const char *trans, const Integer *m, const Integer *n, const Integer *k, double a[], const Integer *lda, const double tau[], double c[], const Integer *ldc, double work[], const Integer *lwork, Integer *info, const Charlen length_side, const Charlen length_trans)
The routine may be called by the names f08ckf, nagf_lapackeig_dormrq or its LAPACK name dormrq.

3 Description

f08ckf is intended to be used following a call to f08chf, which performs an RQ factorization of a real matrix A and represents the orthogonal matrix Q as a product of elementary reflectors.
This routine may be used to form one of the matrix products
QC ,   QTC ,   CQ ,   CQT ,  
overwriting the result on C, which may be any real rectangular m×n matrix.
A common application of this routine is in solving underdetermined linear least squares problems, as described in the F08 Chapter Introduction, and illustrated in Section 10 in f08chf.

4 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 https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: side Character(1) Input
On entry: indicates how Q or QT is to be applied to C.
side='L'
Q or QT is applied to C from the left.
side='R'
Q or QT is applied to C from the right.
Constraint: side='L' or 'R'.
2: trans Character(1) Input
On entry: indicates whether Q or QT is to be applied to C.
trans='N'
Q is applied to C.
trans='T'
QT is applied to C.
Constraint: trans='N' or 'T'.
3: m Integer Input
On entry: m, the number of rows of the matrix C.
Constraint: m0.
4: n Integer Input
On entry: n, the number of columns of the matrix C.
Constraint: n0.
5: k Integer Input
On entry: 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 .
6: a(lda,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least max(1,m) if side='L' and at least max(1,n) if side='R'.
On entry: the ith row of a must contain the vector which defines the elementary reflector Hi, for i=1,2,,k, as returned by f08chf.
On exit: is modified by f08ckf but restored on exit.
7: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08ckf is called.
Constraint: ldamax(1,k).
8: tau(*) Real (Kind=nag_wp) array Input
Note: the dimension of the array tau must be at least max(1,k).
On entry: tau(i) must contain the scalar factor of the elementary reflector Hi, as returned by f08chf.
9: c(ldc,*) Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array c must be at least max(1,n).
On entry: the m×n matrix C.
On exit: c is overwritten by QC or QTC or CQ or CQT as specified by side and trans.
10: ldc Integer Input
On entry: the first dimension of the array c as declared in the (sub)program from which f08ckf is called.
Constraint: ldcmax(1,m).
11: work(max(1,lwork)) Real (Kind=nag_wp) array Workspace
On exit: if info=0, work(1) contains the minimum value of lwork required for optimal performance.
12: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08ckf is called.
If lwork=−1, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance, lworkn×nb if side='L' and at least m×nb if side='R', where nb is the optimal block size.
Constraints:
  • if side='L', lworkmax(1,n) or lwork=−1;
  • if side='R', lworkmax(1,m) or lwork=−1.
13: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

-999<info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
info=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
An explanatory message is output, and execution of the program is terminated.

7 Accuracy

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

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08ckf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The total number of floating-point operations is approximately 2nk(2m-k) if side='L' and 2mk(2n-k) if side='R'.
The complex analogue of this routine is f08cxf.

10 Example

See f08chf.