NAG Library Routine Document

f08kuf (zunmbr)


    1  Purpose
    7  Accuracy


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


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 nagmk26.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 its LAPACK name zunmbr.


f08kuf (zunmbr) is intended to be used after a call to f08ksf (zgebrd), which reduces a complex rectangular matrix A to real bidiagonal form B by a unitary transformation: A=QBPH. f08ksf (zgebrd) 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).


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


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.
Q or QH is applied to C.
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.
Q or QH or P or PH is applied to C from the left.
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.
Q or P is applied to C.
QH or PH is applied to C.
Constraint: trans='N' or 'C'.
4:     m – IntegerInput
On entry: m, the number of rows of the matrix C.
Constraint: m0.
5:     n – IntegerInput
On entry: n, the number of columns of the matrix C.
Constraint: n0.
6:     k – IntegerInput
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:     alda* – Complex (Kind=nag_wp) arrayInput
Note: the second dimension of the array a must be at least max1,minr,k  if vect='Q' and at least max1,r if vect='P'.
On entry: details of the vectors which define the elementary reflectors, as returned by f08ksf (zgebrd).
8:     lda – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08kuf (zunmbr) is called.
  • if vect='Q', lda max1,r ;
  • if vect='P', lda max1,minr,k .
9:     tau* – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array tau must be at least max1,minr,k.
On entry: further details of the elementary reflectors, as returned by f08ksf (zgebrd) in its argument tauq if vect='Q', or in its argument taup if vect='P'.
10:   cldc* – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array c must be at least max1,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 – IntegerInput
On entry: the first dimension of the array c as declared in the (sub)program from which f08kuf (zunmbr) is called.
Constraint: ldcmax1,m.
12:   workmax1,lwork – Complex (Kind=nag_wp) arrayWorkspace
On exit: if info=0, the real part of work1 contains the minimum value of lwork required for optimal performance.
13:   lwork – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08kuf (zunmbr) 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.
  • if side='L', lworkmax1,n or lwork=-1;
  • if side='R', lworkmax1,m or lwork=-1.
14:   info – IntegerOutput
On exit: info=0 unless the routine detects an error (see Section 6).

Error Indicators and Warnings

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.


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

Parallelism and Performance

f08kuf (zunmbr) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08kuf (zunmbr) 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.

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 (dormbr).


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 (zunmbr) 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 (zunmbr) to form PH=PbHPaH.

Program Text

Program Text (f08kufe.f90)

Program Data

Program Data (f08kufe.d)

Program Results

Program Results (f08kufe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017