nag_zunmrq (f08cxc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_zunmrq (f08cxc)

 Contents

    1  Purpose
    7  Accuracy
    10  Example

1  Purpose

nag_zunmrq (f08cxc) multiplies a general complex m by n matrix C by the complex unitary matrix Q from an RQ factorization computed by nag_zgerqf (f08cvc).

2  Specification

#include <nag.h>
#include <nagf08.h>
void  nag_zunmrq (Nag_OrderType order, Nag_SideType side, Nag_TransType trans, Integer m, Integer n, Integer k, Complex a[], Integer pda, const Complex tau[], Complex c[], Integer pdc, NagError *fail)

3  Description

nag_zunmrq (f08cxc) is intended to be used following a call to nag_zgerqf (f08cvc), which performs an RQ factorization of a complex matrix A and represents the unitary matrix Q as a product of elementary reflectors.
This function may be used to form one of the matrix products
QC ,   QHC ,   CQ ,   CQH ,  
overwriting the result on C, which may be any complex rectangular m by n matrix.
A common application of this function is in solving underdetermined linear least squares problems, as described in the f08 Chapter Introduction, and illustrated in Section 10 in nag_zgerqf (f08cvc).

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 http://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:     order Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by order=Nag_RowMajor. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2:     side Nag_SideTypeInput
On entry: indicates how Q or QH is to be applied to C.
side=Nag_LeftSide
Q or QH is applied to C from the left.
side=Nag_RightSide
Q or QH is applied to C from the right.
Constraint: side=Nag_LeftSide or Nag_RightSide.
3:     trans Nag_TransTypeInput
On entry: indicates whether Q or QH is to be applied to C.
trans=Nag_NoTrans
Q is applied to C.
trans=Nag_ConjTrans
QH is applied to C.
Constraint: trans=Nag_NoTrans or Nag_ConjTrans.
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: k, the number of elementary reflectors whose product defines the matrix Q.
Constraints:
  • if side=Nag_LeftSide, m k 0 ;
  • if side=Nag_RightSide, n k 0 .
7:     a[dim] ComplexInput/Output
Note: the dimension, dim, of the array a must be at least
  • max1,pda×m when side=Nag_LeftSide and order=Nag_ColMajor;
  • max1,k×pda when side=Nag_LeftSide and order=Nag_RowMajor;
  • max1,pda×n when side=Nag_RightSide and order=Nag_ColMajor;
  • max1,k×pda when side=Nag_RightSide and order=Nag_RowMajor.
The i,jth element of the matrix A is stored in
  • a[j-1×pda+i-1] when order=Nag_ColMajor;
  • a[i-1×pda+j-1] when order=Nag_RowMajor.
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 nag_zgerqf (f08cvc).
On exit: is modified by nag_zunmrq (f08cxc) but restored on exit.
8:     pda IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraints:
  • if order=Nag_ColMajor, pdamax1,k;
  • if order=Nag_RowMajor,
    • if side=Nag_LeftSide, pdamax1,m;
    • if side=Nag_RightSide, pdamax1,n.
9:     tau[dim] const ComplexInput
Note: the dimension, dim, of the array tau must be at least max1,k.
On entry: tau[i-1] must contain the scalar factor of the elementary reflector Hi, as returned by nag_zgerqf (f08cvc).
10:   c[dim] ComplexInput/Output
Note: the dimension, dim, of the array c must be at least
  • max1,pdc×n when order=Nag_ColMajor;
  • max1,m×pdc when order=Nag_RowMajor.
The i,jth element of the matrix C is stored in
  • c[j-1×pdc+i-1] when order=Nag_ColMajor;
  • c[i-1×pdc+j-1] when order=Nag_RowMajor.
On entry: the m by n matrix C.
On exit: c is overwritten by QC or QHC or CQ or CQH as specified by side and trans.
11:   pdc IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array c.
Constraints:
  • if order=Nag_ColMajor, pdcmax1,m;
  • if order=Nag_RowMajor, pdcmax1,n.
12:   fail NagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_ENUM_INT_3
On entry, side=value, m=value, n=value and k=value.
Constraint: if side=Nag_LeftSide, m k 0 ;
if side=Nag_RightSide, n k 0 .
On entry, side=value, pda=value, m=value and n=value.
Constraint: if side=Nag_LeftSide, pdamax1,m;
if side=Nag_RightSide, pdamax1,n.
NE_INT
On entry, m=value.
Constraint: m0.
On entry, n=value.
Constraint: n0.
On entry, pda=value.
Constraint: pda>0.
On entry, pdc=value.
Constraint: pdc>0.
NE_INT_2
On entry, pda=value and k=value.
Constraint: pdamax1,k.
On entry, pdc=value and m=value.
Constraint: pdcmax1,m.
On entry, pdc=value and n=value.
Constraint: pdcmax1,n.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

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

nag_zunmrq (f08cxc) is not threaded by NAG in any implementation.
nag_zunmrq (f08cxc) 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 function. 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 8nk2m-k if side=Nag_LeftSide and 8mk2n-k if side=Nag_RightSide.
The real analogue of this function is nag_dormrq (f08ckc).

10  Example

See Section 10 in nag_zgerqf (f08cvc).

nag_zunmrq (f08cxc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015