NAG FL Interface
f08kuf (zunmbr)

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1 Purpose

f08kuf multiplies an arbitrary complex m×n matrix C by one of the complex unitary matrices Q or P which were determined by f08ksf when reducing a complex matrix to bidiagonal form.

2 Specification

Fortran Interface
Subroutine f08kuf ( vect, 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
Complex (Kind=nag_wp), Intent (In) :: tau(*)
Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), c(ldc,*)
Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
Character (1), Intent (In) :: vect, side, trans
C Header Interface
#include <nag.h>
void  f08kuf_ (const char *vect, const char *side, const char *trans, const Integer *m, const Integer *n, const Integer *k, Complex a[], const Integer *lda, const Complex tau[], Complex c[], const Integer *ldc, Complex work[], const Integer *lwork, Integer *info, const Charlen length_vect, const Charlen length_side, const Charlen length_trans)
The routine may be called by the names f08kuf, nagf_lapackeig_zunmbr or its LAPACK name zunmbr.

3 Description

f08kuf is intended to be used after a call to f08ksf, which reduces a complex rectangular matrix A to real bidiagonal form B by a unitary transformation: A=QBPH. f08ksf represents the matrices Q and PH as products of elementary reflectors.
This routine may be used to form one of the matrix products
QC , QHC , CQ , CQH , PC , PHC , CP ​ or ​ CPH ,  
overwriting the result on C (which may be any complex rectangular matrix).

4 References

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

5 Arguments

Note: in the descriptions below, r denotes the order of Q or PH: if side='L', r=m and if side='R', r=n.
1: vect Character(1) Input
On entry: indicates whether Q or QH or P or PH is to be applied to C.
vect='Q'
Q or QH is applied to C.
vect='P'
P or PH is applied to C.
Constraint: vect='Q' or 'P'.
2: side Character(1) Input
On entry: indicates how Q or QH or P or PH is to be applied to C.
side='L'
Q or QH or P or PH is applied to C from the left.
side='R'
Q or QH or P or PH is applied to C from the right.
Constraint: side='L' or 'R'.
3: trans Character(1) Input
On entry: indicates whether Q or P or QH or PH is to be applied to C.
trans='N'
Q or P is applied to C.
trans='C'
QH or PH is applied to C.
Constraint: trans='N' or 'C'.
4: m Integer Input
On entry: m, the number of rows of the matrix C.
Constraint: m0.
5: n Integer Input
On entry: n, the number of columns of the matrix C.
Constraint: n0.
6: k Integer Input
On entry: if vect='Q', the number of columns in the original matrix A.
If vect='P', the number of rows in the original matrix A.
Constraint: k0.
7: a(lda,*) Complex (Kind=nag_wp) array Input
Note: the second dimension of the array a must be at least max(1,min(r,k)) if vect='Q' and at least max(1,r) if vect='P'.
On entry: details of the vectors which define the elementary reflectors, as returned by f08ksf.
8: lda Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08kuf is called.
Constraints:
  • if vect='Q', lda max(1,r) ;
  • if vect='P', lda max(1,min(r,k)) .
9: tau(*) Complex (Kind=nag_wp) array Input
Note: the dimension of the array tau must be at least max(1,min(r,k)).
On entry: further details of the elementary reflectors, as returned by f08ksf in its argument tauq if vect='Q', or in its argument taup if vect='P'.
10: c(ldc,*) Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array c must be at least max(1,n).
On entry: the matrix C.
On exit: c is overwritten by QC or QHC or CQ or CHQ or PC or PHC or CP or CHP as specified by vect, side and trans.
11: ldc Integer Input
On entry: the first dimension of the array c as declared in the (sub)program from which f08kuf is called.
Constraint: ldcmax(1,m).
12: work(max(1,lwork)) Complex (Kind=nag_wp) array Workspace
On exit: if info=0, the real part of work(1) contains the minimum value of lwork required for optimal performance.
13: lwork Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08kuf 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.
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.
f08kuf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kuf 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 real floating-point operations is approximately where k is the value of the argument k.
The real analogue of this routine is f08kgf.

10 Example

For this routine two examples are presented. Both illustrate how the reduction to bidiagonal form of a matrix A may be preceded by a QR or LQ factorization of A.
In the first example, m>n, and
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 ) .  
The routine first performs a QR factorization of A as A=QaR and then reduces the factor R to bidiagonal form B: R=QbBPH. Finally it forms Qa and calls f08kuf to form Q=QaQb.
In the second example, m<n, and
A = ( 0.28-0.36i 0.50-0.86i -0.77-0.48i 1.58+0.66i -0.50-1.10i -1.21+0.76i -0.32-0.24i -0.27-1.15i 0.36-0.51i -0.07+1.33i -0.75+0.47i -0.08+1.01i ) .  
The routine first performs an LQ factorization of A as A=LPaH and then reduces the factor L to bidiagonal form B: L=QBPbH. Finally it forms PbH and calls f08kuf to form PH=PbHPaH.

10.1 Program Text

Program Text (f08kufe.f90)

10.2 Program Data

Program Data (f08kufe.d)

10.3 Program Results

Program Results (f08kufe.r)