NAG FL Interface
f08acf (dgemqrt)

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1 Purpose

f08acf multiplies an arbitrary real matrix C by the real orthogonal matrix Q from a QR factorization computed by f08abf.

2 Specification

Fortran Interface
Subroutine f08acf ( side, trans, m, n, k, nb, v, ldv, t, ldt, c, ldc, work, info)
Integer, Intent (In) :: m, n, k, nb, ldv, ldt, ldc
Integer, Intent (Out) :: info
Real (Kind=nag_wp), Intent (In) :: t(ldt,*)
Real (Kind=nag_wp), Intent (Inout) :: v(ldv,*), c(ldc,*), work(*)
Character (1), Intent (In) :: side, trans
C Header Interface
#include <nag.h>
void  f08acf_ (const char *side, const char *trans, const Integer *m, const Integer *n, const Integer *k, const Integer *nb, double v[], const Integer *ldv, const double t[], const Integer *ldt, double c[], const Integer *ldc, double work[], Integer *info, const Charlen length_side, const Charlen length_trans)
The routine may be called by the names f08acf, nagf_lapackeig_dgemqrt or its LAPACK name dgemqrt.

3 Description

f08acf is intended to be used after a call to f08abf which performs a QR factorization of a real matrix A. The orthogonal matrix Q is represented as a product of elementary reflectors.
This routine may be used to form one of the matrix products
QC , QTC , CQ ​ or ​ CQT ,  
overwriting the result on C (which may be any real rectangular matrix).
A common application of this routine is in solving linear least squares problems, as described in the F08 Chapter Introduction and illustrated in Section 10 in f08abf.

4 References

Golub G H and Van Loan C F (2012) Matrix Computations (4th 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. Usually k=min(mA,nA) where mA, nA are the dimensions of the matrix A supplied in a previous call to f08abf.
Constraints:
  • if side='L', m k 0 ;
  • if side='R', n k 0 .
6: nb Integer Input
On entry: the block size used in the QR factorization performed in a previous call to f08abf; this value must remain unchanged from that call.
Constraints:
  • nb1;
  • if k>0, nbk.
7: v(ldv,*) Real (Kind=nag_wp) array Input
Note: the second dimension of the array v must be at least max(1,k).
On entry: details of the vectors which define the elementary reflectors, as returned by f08abf in the first k columns of its array argument a.
8: ldv Integer Input
On entry: the first dimension of the array v as declared in the (sub)program from which f08acf is called.
Constraints:
  • if side='L', ldv max(1,m) ;
  • if side='R', ldv max(1,n) .
9: t(ldt,*) Real (Kind=nag_wp) array Input
Note: the second dimension of the array t must be at least max(1,k).
On entry: further details of the orthogonal matrix Q as returned by f08abf. The number of blocks is b=knb, where k=min(m,n) and each block is of order nb except for the last block, which is of order k-(b-1)×nb. For the b blocks the upper triangular block reflector factors T1,T2,,Tb are stored in the nb×n matrix T as T=[T1|T2||Tb].
10: ldt Integer Input
On entry: the first dimension of the array t as declared in the (sub)program from which f08acf is called.
Constraint: ldtnb.
11: 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.
12: ldc Integer Input
On entry: the first dimension of the array c as declared in the (sub)program from which f08acf is called.
Constraint: ldcmax(1,m).
13: work(*) Real (Kind=nag_wp) array Workspace
Note: the dimension of the array work must be at least n×nb if side='L' and at least m×nb if side='R'.
14: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. 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 = O(ε) C2 ,  
where ε is the machine precision.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08acf 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 f08aqf.

10 Example

See f08abf.