NAG Library Function Document

nag_dormrz (f08bkc)

 Contents

    1  Purpose
    7  Accuracy
    10  Example

1
Purpose

nag_dormrz (f08bkc) multiplies a general real m by n matrix C by the real orthogonal matrix Z from an RZ factorization computed by nag_dtzrzf (f08bhc).

2
Specification

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

3
Description

nag_dormrz (f08bkc) is intended to be used following a call to nag_dtzrzf (f08bhc), 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 function 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.

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

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 Z or ZT is to be applied to C.
side=Nag_LeftSide
Z or ZT is applied to C from the left.
side=Nag_RightSide
Z or ZT is applied to C from the right.
Constraint: side=Nag_LeftSide or Nag_RightSide.
3:     trans Nag_TransTypeInput
On entry: indicates whether Z or ZT is to be applied to C.
trans=Nag_NoTrans
Z is applied to C.
trans=Nag_Trans
ZT 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 Z.
Constraints:
  • if side=Nag_LeftSide, m k 0 ;
  • if side=Nag_RightSide, n k 0 .
7:     l IntegerInput
On entry: l, the number of columns of the matrix A containing the meaningful part of the Householder reflectors.
Constraints:
  • if side=Nag_LeftSide, m l 0 ;
  • if side=Nag_RightSide, n l 0 .
8:     a[dim] const doubleInput
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_dtzrzf (f08bhc).
9:     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.
10:   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_dtzrzf (f08bhc).
11:   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 ZC or ZTC or CZ or ZTC as specified by side and trans.
12:   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.
13:   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 l=value.
Constraint: if side=Nag_LeftSide, m l 0 ;
if side=Nag_RightSide, n l 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.
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_dormrz (f08bkc) 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 4nlk if side=Nag_LeftSide and 4mlk if side=Nag_RightSide.
The complex analogue of this function is nag_zunmrz (f08bxc).

10
Example

See Section 10 in nag_dtzrzf (f08bhc).
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017