NAG Library Function Document

nag_dormqr (f08agc)

 Contents

    1  Purpose
    7  Accuracy
    10  Example

1
Purpose

nag_dormqr (f08agc) multiplies an arbitrary real matrix C by the real orthogonal matrix Q from a QR factorization computed by nag_dgeqrf (f08aec), nag_dgeqpf (f08bec) or nag_dgeqp3 (f08bfc).

2
Specification

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

3
Description

nag_dormqr (f08agc) is intended to be used after a call to nag_dgeqrf (f08aec), nag_dgeqpf (f08bec) or nag_dgeqp3 (f08bfc) which perform a QR factorization of a real matrix A. The orthogonal matrix Q is represented as a product of elementary reflectors.
This function 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 function is in solving linear least squares problems, as described in the f08 Chapter Introduction and illustrated in Section 10 in nag_dgeqrf (f08aec).

4
References

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 3.3.1.3 in How to Use the NAG Library and its Documentation 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 QT is to be applied to C.
side=Nag_LeftSide
Q or QT is applied to C from the left.
side=Nag_RightSide
Q or QT is applied to C from the right.
Constraint: side=Nag_LeftSide or Nag_RightSide.
3:     trans Nag_TransTypeInput
On entry: indicates whether Q or QT is to be applied to C.
trans=Nag_NoTrans
Q is applied to C.
trans=Nag_Trans
QT is applied to C.
Constraint: trans=Nag_NoTrans or Nag_Trans.
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] const doubleInput
Note: the dimension, dim, of the array a must be at least
  • max1,pda×k when order=Nag_ColMajor;
  • max1,m×pda when order=Nag_RowMajor and side=Nag_LeftSide;
  • max1,n×pda when order=Nag_RowMajor and side=Nag_RightSide.
On entry: details of the vectors which define the elementary reflectors, as returned by nag_dgeqrf (f08aec), nag_dgeqpf (f08bec) or nag_dgeqp3 (f08bfc).
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,
    • if side=Nag_LeftSide, pda max1,m ;
    • if side=Nag_RightSide, pda max1,n ;
  • if order=Nag_RowMajor, pdamax1,k.
9:     tau[dim] const doubleInput
Note: the dimension, dim, of the array tau must be at least max1,k.
On entry: further details of the elementary reflectors, as returned by nag_dgeqrf (f08aec), nag_dgeqpf (f08bec) or nag_dgeqp3 (f08bfc).
10:   c[dim] doubleInput/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 QTC or CQ or CQT 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 3.7 in How to Use the NAG Library and its Documentation).

6
Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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, m=value, n=value and pda=value.
Constraint: if side=Nag_LeftSide, pda max1,m ;
if side=Nag_RightSide, pda max1,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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation 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_dormqr (f08agc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dormqr (f08agc) 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 2nk 2m-k  if side=Nag_LeftSide and 2mk 2n-k  if side=Nag_RightSide.
The complex analogue of this function is nag_zunmqr (f08auc).

10
Example

See Section 10 in nag_dgeqrf (f08aec).
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017