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

R Z

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

Z C, Z^{T} C, C Z, C Z^{T},

overwriting the result on

C

, which may be any real rectangular

m

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

Nag_ColMajor

2: $side$ – Nag_SideTypeInput

On entry: indicates how

Z

Z^{T}

is to be applied to

C

$side = Nag_LeftSide$: $Z$ or $Z^{T}$ is applied to $C$ from the left.
$side = Nag_RightSide$: $Z$ or $Z^{T}$ is applied to $C$ from the right.

Constraint:

side = Nag_LeftSide

Nag_RightSide

3: $trans$ – Nag_TransTypeInput

On entry: indicates whether

Z

Z^{T}

is to be applied to

C

$trans = Nag_NoTrans$: $Z$ is applied to $C$ .
$trans = Nag_Trans$: $Z^{T}$ is applied to $C$ .

Constraint:

trans = Nag_NoTrans

Nag_Trans

4: $m$ – IntegerInput

On entry:

m

, the number of rows of the matrix

C

Constraint:

m \geq 0

5: $n$ – IntegerInput

On entry:

n

, the number of columns of the matrix

C

Constraint:

n \geq 0

6: $k$ – IntegerInput

On entry:

k

, the number of elementary reflectors whose product defines the matrix

Z

Constraints:

if $side = Nag_LeftSide$ , $m \geq k \geq 0$ ;
if $side = Nag_RightSide$ , $n \geq k \geq 0$ .

7: $l$ – IntegerInput

On entry:

l

, the number of columns of the matrix

A

containing the meaningful part of the Householder reflectors.

Constraints:

if $side = Nag_LeftSide$ , $m \geq l \geq 0$ ;
if $side = Nag_RightSide$ , $n \geq l \geq 0$ .

8: $a [\dim]$ – const doubleInput

Note: the dimension, dim, of the array a must be at least

$\max (1, pda \times m)$ when $side = Nag_LeftSide$ and $order = Nag_ColMajor$ ;
$\max (1, k \times pda)$ when $side = Nag_LeftSide$ and $order = Nag_RowMajor$ ;
$\max (1, pda \times n)$ when $side = Nag_RightSide$ and $order = Nag_ColMajor$ ;
$\max (1, k \times pda)$ when $side = Nag_RightSide$ and $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

A

is stored in

$a [(j - 1) \times pda + i - 1]$ when $order = Nag_ColMajor$ ;
$a [(i - 1) \times pda + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

i

th row of a must contain the vector which defines the elementary reflector

H_{i}

, for

i = 1, 2, \dots, 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 $order = Nag_ColMajor$ , $pda \geq \max (1, k)$ ;
if order=Nag_RowMajor,
- if $side = Nag_LeftSide$ , $pda \geq \max (1, m)$ ;
- if $side = Nag_RightSide$ , $pda \geq \max (1, n)$ .

10: $tau [\dim]$ – const doubleInput

Note: the dimension, dim, of the array tau must be at least

\max (1, k)

On entry:

tau [i - 1]

must contain the scalar factor of the elementary reflector

H_{i}

, as returned by nag_dtzrzf (f08bhc).

11: $c [\dim]$ – doubleInput/Output

Note: the dimension, dim, of the array c must be at least

$\max (1, pdc \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pdc)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

C

is stored in

$c [(j - 1) \times pdc + i - 1]$ when $order = Nag_ColMajor$ ;
$c [(i - 1) \times pdc + j - 1]$ when $order = Nag_RowMajor$ .

On entry: the

m

n

matrix

C

On exit: c is overwritten by

Z C

Z^{T} C

C Z

Z^{T} C

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 $order = Nag_ColMajor$ , $pdc \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pdc \geq \max (1, n)$ .

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 $side = Nag_LeftSide$ , $m \geq k \geq 0$ ;
if $side = Nag_RightSide$ , $n \geq k \geq 0$ .

On entry, $side = 〈value〉$ , $m = 〈value〉$ , $n = 〈value〉$ and $l = 〈value〉$ .
Constraint: if $side = Nag_LeftSide$ , $m \geq l \geq 0$ ;
if $side = Nag_RightSide$ , $n \geq l \geq 0$ .

On entry, $side = 〈value〉$ , $pda = 〈value〉$ , $m = 〈value〉$ and $n = 〈value〉$ .
Constraint: if $side = Nag_LeftSide$ , $pda \geq \max (1, m)$ ;
if $side = Nag_RightSide$ , $pda \geq \max (1, n)$ .
NE_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 0$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .

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: $pda \geq \max (1, k)$ .

On entry, $pdc = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pdc \geq \max (1, m)$ .

On entry, $pdc = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdc \geq \max (1, n)$ .
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

{‖E‖}_{2} = O ε {‖C‖}_{2}

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

4 n l k

side = Nag_LeftSide

and

4 m l k

side = Nag_RightSide

The complex analogue of this function is nag_zunmrz (f08bxc).

10

Example

See Section 10 in nag_dtzrzf (f08bhc).

nag_dormrz (f08bkc) (PDF version)

f08 Chapter Contents

f08 Chapter Introduction

NAG Library Manual

NAG Library Function Document

nag_dormrz (f08bkc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example