nag_dormrq (f08ckc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

NAG Library Function Document

nag_dormrq (f08ckc)

+ Contents

    1  Purpose
    7  Accuracy
    10  Example

1  Purpose

nag_dormrq (f08ckc) multiplies a general real m by n matrix C by the real orthogonal matrix Q from an RQ factorization computed by nag_dgerqf (f08chc).

2  Specification

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

3  Description

nag_dormrq (f08ckc) is intended to be used following a call to nag_dgerqf (f08chc), which performs an RQ factorization of a real matrix A and represents the orthogonal matrix Q as a product of elementary reflectors.
This function may be used to form one of the matrix products
QC ,   QTC ,   CQ ,   CQT ,
overwriting the result on C, which may be any real 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_dgerqf (f08chc).

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:     orderNag_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:     sideNag_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:     transNag_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:     mIntegerInput
On entry: m, the number of rows of the matrix C.
Constraint: m0.
5:     nIntegerInput
On entry: n, the number of columns of the matrix C.
Constraint: n0.
6:     kIntegerInput
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]doubleInput/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_dgerqf (f08chc).
On exit: is modified by nag_dormrq (f08ckc) but restored on exit.
8:     pdaIntegerInput
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 doubleInput
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_dgerqf (f08chc).
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:   pdcIntegerInput
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:   failNagError *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.
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.

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_dormrq (f08ckc) is not threaded by NAG in any implementation.
nag_dormrq (f08ckc) 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 Users' Note for your implementation for any additional implementation-specific information.

9  Further Comments

The total number of floating-point operations is approximately 2nk2m-k if side=Nag_LeftSide and 2mk2n-k if side=Nag_RightSide.
The complex analogue of this function is nag_zunmrq (f08cxc).

10  Example

See Section 10 in nag_dgerqf (f08chc).

nag_dormrq (f08ckc) (PDF version)
f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014