NAG Library Function Document

nag_dtpmqrt (f08bcc)

void	nag_dtpmqrt (Nag_OrderType order, Nag_SideType side, Nag_TransType trans, Integer m, Integer n, Integer k, Integer l, Integer nb, const double v[], Integer pdv, const double t[], Integer pdt, double c1[], Integer pdc1, double c2[], Integer pdc2, NagError *fail)

3 Description

nag_dtpmqrt (f08bcc) is intended to be used after a call to nag_dtpqrt (f08bbc) which performs a

Q R

factorization of a triangular-pentagonal matrix containing an upper triangular matrix

A

over a pentagonal matrix

B

. The orthogonal matrix

Q

is represented as a product of elementary reflectors.

This function may be used to form the matrix products

Q C, Q^{T} C, C Q ​ or ​ C Q^{T},

where the real rectangular

m_{c}

n_{c}

matrix

C

is split into component matrices

C_{1}

and

C_{2}

Q

is being applied from the left (

Q C

Q^{T} C

) then

C = (\begin{matrix} C_{1} \\ C_{2} \end{matrix})

where

C_{1}

k

n_{c}

C_{2}

m_{v}

n_{c}

m_{c} = k + m_{v}

is fixed and

m_{v}

is the number of rows of the matrix

V

containing the elementary reflectors (i.e., m as passed to nag_dtpqrt (f08bbc)); the number of columns of

V

n_{v}

(i.e., n as passed to nag_dtpqrt (f08bbc)).

Q

is being applied from the right (

C Q

C Q^{T}

) then

C = (\begin{matrix} C_{1} & C_{2} \end{matrix})

where

C_{1}

m_{c}

k

, and

C_{2}

m_{c}

m_{v}

and

n_{c} = k + m_{v}

is fixed.

The matrices

C_{1}

and

C_{2}

are overwriten by the result of the matrix product.

A common application of this routine is in updating the solution of a linear least squares problem as illustrated in Section 10 in nag_dtpqrt (f08bbc).

4 References

Golub G H and Van Loan C F (2012) Matrix Computations (4th 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

Nag_ColMajor

2: $side$ – Nag_SideTypeInput

On entry: indicates how

Q

Q^{T}

is to be applied to

C

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

Constraint:

side = Nag_LeftSide

Nag_RightSide

3: $trans$ – Nag_TransTypeInput

On entry: indicates whether

Q

Q^{T}

is to be applied to

C

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

Constraint:

trans = Nag_NoTrans

Nag_Trans

4: $m$ – IntegerInput

On entry: the number of rows of the matrix

C_{2}

, that is,

if $side = Nag_LeftSide$: then $m_{v}$ , the number of rows of the matrix $V$ ;
if $side = Nag_RightSide$: then $m_{c}$ , the number of rows of the matrix $C$ .

Constraint:

m \geq 0

5: $n$ – IntegerInput

On entry: the number of columns of the matrix

C_{2}

, that is,

if $side = Nag_LeftSide$: then $n_{c}$ , the number of columns of the matrix $C$ ;
if $side = Nag_RightSide$: then $n_{v}$ , the number of columns of the matrix $V$ .

Constraint:

n \geq 0

6: $k$ – IntegerInput

On entry:

k

, the number of elementary reflectors whose product defines the matrix

Q

Constraint:

k \geq 0

7: $l$ – IntegerInput

On entry:

l

, the number of rows of the upper trapezoidal part of the pentagonal composite matrix

V

, passed (as b) in a previous call to nag_dtpqrt (f08bbc). This must be the same value used in the previous call to nag_dtpqrt (f08bbc) (see l in nag_dtpqrt (f08bbc)).

Constraint:

0 \leq l \leq k

8: $nb$ – IntegerInput

On entry:

nb

, the blocking factor used in a previous call to nag_dtpqrt (f08bbc) to compute the

Q R

factorization of a triangular-pentagonal matrix containing composite matrices

A

and

B

Constraints:

$nb \geq 1$ ;
if $k > 0$ , $nb \leq k$ .

9: $v [\dim]$ – const doubleInput

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

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

The

(i, j)

th element of the matrix

V

is stored in

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

On entry: the

m_{v}

n_{v}

matrix

V

; this should remain unchanged from the array b returned by a previous call to nag_dtpqrt (f08bbc).

10: $pdv$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array v.

Constraints:

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

11: $t [\dim]$ – const doubleInput

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

$\max (1, pdt \times k)$ when $order = Nag_ColMajor$ ;
$\max (1, nb \times pdt)$ when $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

T

is stored in

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

On entry: this must remain unchanged from a previous call to nag_dtpqrt (f08bbc) (see t in nag_dtpqrt (f08bbc)).

12: $pdt$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array t.

Constraints:

if $order = Nag_ColMajor$ , $pdt \geq nb$ ;
if $order = Nag_RowMajor$ , $pdt \geq \max (1, k)$ .

13: $c1 [\dim]$ – doubleInput/Output

Note: the dimension, dim, of the array c1 must be at least

$\max (1, pdc1 \times n)$ when $side = Nag_LeftSide$ and $order = Nag_ColMajor$ ;
$\max (1, k \times pdc1)$ when $side = Nag_LeftSide$ and $order = Nag_RowMajor$ ;
$\max (1, pdc1 \times k)$ when $side = Nag_RightSide$ and $order = Nag_ColMajor$ ;
$\max (1, m \times pdc1)$ when $side = Nag_RightSide$ and $order = Nag_RowMajor$ .

On entry:

C_{1}

, the first part of the composite matrix

C

if $side = Nag_LeftSide$: then c1 contains the first $k$ rows of $C$ ;
if $side = Nag_RightSide$: then c1 contains the first $k$ columns of $C$ .

On exit: c1 is overwritten by the corresponding block of

Q C

Q^{T} C

C Q

C Q^{T}

14: $pdc1$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array c1.

Constraints:

if order=Nag_ColMajor,
- if $side = Nag_LeftSide$ , $pdc1 \geq \max (1, k)$ ;
- if $side = Nag_RightSide$ , $pdc1 \geq \max (1, m)$ ;
if order=Nag_RowMajor,
- if $side = Nag_LeftSide$ , $pdc1 \geq \max (1, n)$ ;
- if $side = Nag_RightSide$ , $pdc1 \geq \max (1, k)$ .

15: $c2 [\dim]$ – doubleInput/Output

Note: the dimension, dim, of the array c2 must be at least

$\max (1, pdc2 \times n)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pdc2)$ when $order = Nag_RowMajor$ .

On entry:

C_{2}

, the second part of the composite matrix

C

if $side = Nag_LeftSide$: then c2 contains the remaining $m_{v}$ rows of $C$ ;
if $side = Nag_RightSide$: then c2 contains the remaining $m_{v}$ columns of $C$ ;

On exit: c2 is overwritten by the corresponding block of

Q C

Q^{T} C

C Q

C Q^{T}

16: $pdc2$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array c2.

Constraints:

if $order = Nag_ColMajor$ , $pdc2 \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pdc2 \geq \max (1, n)$ .

17: $fail$ – 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, $side = 〈value〉$ , $k = 〈value〉$ , $m = 〈value〉$ and $pdc1 = 〈value〉$ .
Constraint: if $side = Nag_LeftSide$ , $pdc1 \geq \max (1, k)$ ;
if $side = Nag_RightSide$ , $pdc1 \geq \max (1, m)$ .

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

On entry, $side = 〈value〉$ , $pdc1 = 〈value〉$ , $n = 〈value〉$ and $k = 〈value〉$ .
Constraint: if $side = Nag_LeftSide$ , $pdc1 \geq \max (1, n)$ ;
if $side = Nag_RightSide$ , $pdc1 \geq \max (1, k)$ .
NE_INT: On entry, $k = 〈value〉$ .
Constraint: $k \geq 0$ .

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

On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
NE_INT_2: On entry, $l = 〈value〉$ and $k = 〈value〉$ .
Constraint: $0 \leq l \leq k$ .

On entry, $m = 〈value〉$ and $pdc2 = 〈value〉$ .
Constraint: $pdc2 \geq \max (1, m)$ .

On entry, $nb = 〈value〉$ and $k = 〈value〉$ .
Constraint: $nb \geq 1$ and
if $k > 0$ , $nb \leq k$ .

On entry, $pdc2 = 〈value〉$ and $n = 〈value〉$ .
Constraint: $pdc2 \geq \max (1, n)$ .

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

On entry, $pdt = 〈value〉$ and $nb = 〈value〉$ .
Constraint: $pdt \geq nb$ .

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

{‖E‖}_{2} = O (ε) {‖C‖}_{2},

where

ε

is the machine precision.

8 Parallelism and Performance

nag_dtpmqrt (f08bcc) is not threaded by NAG in any implementation.

nag_dtpmqrt (f08bcc) 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

2 n k (2 m - k)

side = Nag_LeftSide

and

2 m k (2 n - k)

side = Nag_RightSide

The complex analogue of this function is nag_ztpmqrt (f08bqc).

10 Example

See Section 10 in nag_dtpqrt (f08bbc).

nag_dtpmqrt (f08bcc) (PDF version)

f08 Chapter Contents

f08 Chapter Introduction

NAG Library Manual

NAG Library Function Documentnag_dtpmqrt (f08bcc)

▸▿ 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

NAG Library Function Document

nag_dtpmqrt (f08bcc)