NAG Library Routine Document

f08bkf (dormrz)


    1  Purpose
    7  Accuracy
    10  Example


f08bkf (dormrz) multiplies a general real m by n matrix C by the real orthogonal matrix Z from an RZ factorization computed by f08bhf (dtzrzf).


Fortran Interface
Subroutine f08bkf ( side, trans, m, n, k, l, a, lda, tau, c, ldc, work, lwork, info)
Integer, Intent (In):: m, n, k, l, 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 nagmk26.h
void  f08bkf_ (const char *side, const char *trans, const Integer *m, const Integer *n, const Integer *k, const Integer *l, 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 its LAPACK name dormrz.


f08bkf (dormrz) is intended to be used following a call to f08bhf (dtzrzf), which performs an RZ factorization of a real upper trapezoidal matrix A and represents the orthogonal matrix Z as a product of elementary reflectors.
This routine may be used to form one of the matrix products
ZC ,   ZTC ,   CZ ,   CZT ,  
overwriting the result on C, which may be any real rectangular m by n matrix.


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


1:     side – Character(1)Input
On entry: indicates how Z or ZT is to be applied to C.
Z or ZT is applied to C from the left.
Z or ZT is applied to C from the right.
Constraint: side='L' or 'R'.
2:     trans – Character(1)Input
On entry: indicates whether Z or ZT is to be applied to C.
Z is applied to C.
ZT is applied to C.
Constraint: trans='N' or 'T'.
3:     m – IntegerInput
On entry: m, the number of rows of the matrix C.
Constraint: m0.
4:     n – IntegerInput
On entry: n, the number of columns of the matrix C.
Constraint: n0.
5:     k – IntegerInput
On entry: k, the number of elementary reflectors whose product defines the matrix Z.
  • if side='L', m k 0 ;
  • if side='R', n k 0 .
6:     l – IntegerInput
On entry: l, the number of columns of the matrix A containing the meaningful part of the Householder reflectors.
  • if side='L', m l 0 ;
  • if side='R', n l 0 .
7:     alda* – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array a must be at least max1,m if side='L' and at least max1,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 f08bhf (dtzrzf).
8:     lda – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08bkf (dormrz) is called.
Constraint: ldamax1,k.
9:     tau* – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array tau must be at least max1,k.
On entry: taui must contain the scalar factor of the elementary reflector Hi, as returned by f08bhf (dtzrzf).
10:   cldc* – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array c must be at least max1,n.
On entry: the m by n matrix C.
On exit: c is overwritten by ZC or ZTC or CZ or ZTC as specified by side and trans.
11:   ldc – IntegerInput
On entry: the first dimension of the array c as declared in the (sub)program from which f08bkf (dormrz) is called.
Constraint: ldcmax1,m.
12:   workmax1,lwork – Real (Kind=nag_wp) arrayWorkspace
On exit: if info=0, 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 f08bkf (dormrz) 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

f08bkf (dormrz) 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 floating-point operations is approximately 4nlk if side='L' and 4mlk if side='R'.
The complex analogue of this routine is f08bxf (zunmrz).


See Section 10 in f08bhf (dtzrzf).
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017