void	nag_dggrqf (Nag_OrderType order, Integer m, Integer p, Integer n, double a[], Integer pda, double taua[], double b[], Integer pdb, double taub[], NagError *fail)

3

Description

nag_dggrqf (f08zfc) forms the generalized

R Q

factorization of an

m

n

matrix

A

and a

p

n

matrix

B

A = R Q, B = Z T Q,

where

Q

is an

n

n

orthogonal matrix,

Z

is a

p

p

orthogonal matrix and

R

and

T

are of the form

R = \{\begin{matrix} \begin{array}{rc} n - m & m \\ m & 0 & R_{12} \\ ( & ) \end{array};   if ​ m \leq n, \\ \begin{array}{rc} n \\ m - n & R_{11} \\ n & R_{21} \\ ( & ) \end{array};   if ​ m > n, \end{matrix}

with

R_{12}

R_{21}

upper triangular,

T = \{\begin{matrix} \begin{array}{rc} n \\ n & T_{11} \\ p - n & 0 \\ ( & ) \end{array};   if ​ p \geq n, \\ \begin{array}{rc} p & n - p \\ p & T_{11} & T_{12} \\ ( & ) \end{array};   if ​ p < n, \end{matrix}

with

T_{11}

upper triangular.

In particular, if

B

is square and nonsingular, the generalized

R Q

factorization of

A

and

B

implicitly gives the

R Q

factorization of

A B^{- 1}

A B^{- 1} = (R T^{- 1}) Z^{T} .

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

Anderson E, Bai Z and Dongarra J (1992) Generalized QR factorization and its applications Linear Algebra Appl. (Volume 162–164) 243–271

Hammarling S (1987) The numerical solution of the general Gauss-Markov linear model Mathematics in Signal Processing (eds T S Durrani, J B Abbiss, J E Hudson, R N Madan, J G McWhirter and T A Moore) 441–456 Oxford University Press

Paige C C (1990) Some aspects of generalized

Q R

factorizations . In Reliable Numerical Computation (eds M G Cox and S Hammarling) 73–91 Oxford University Press

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: $m$ – IntegerInput

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: $p$ – IntegerInput

On entry:

p

, the number of rows of the matrix

B

Constraint:

p \geq 0

4: $n$ – IntegerInput

On entry:

n

, the number of columns of the matrices

A

and

B

Constraint:

n \geq 0

5: $a [\dim]$ – doubleInput/Output

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

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

Where

A (i, j)

appears in this document, it refers to the array element

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

On entry: the

m

n

matrix

A

On exit: if

m \leq n

, the upper triangle of the subarray

A (1 : m, n - m + 1 : n)

contains the

m

m

upper triangular matrix

R_{12}

m \geq n

, the elements on and above the

(m - n)

th subdiagonal contain the

m

n

upper trapezoidal matrix

R

; the remaining elements, with the array taua, represent the orthogonal matrix

Q

as a product of

\min (m, n)

elementary reflectors (see Section 3.3.6 in the f08 Chapter Introduction).

6: $pda$ – IntegerInput

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

Constraints:

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

7: $taua [\min (m, n)]$ – doubleOutput

On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix

Q

8: $b [\dim]$ – doubleInput/Output

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

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

The

(i, j)

th element of the matrix

B

is stored in

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

On entry: the

p

n

matrix

B

On exit: the elements on and above the diagonal of the array contain the

\min (p, n)

n

upper trapezoidal matrix

T

(

T

is upper triangular if

p \geq n

); the elements below the diagonal, with the array taub, represent the orthogonal matrix

Z

as a product of elementary reflectors (see Section 3.3.6 in the f08 Chapter Introduction).

9: $pdb$ – IntegerInput

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

Constraints:

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

10: $taub [\min (p, n)]$ – doubleOutput

On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix

Z

11: $fail$ – 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_INT: On entry, $m = 〈value〉$ .
Constraint: $m \geq 0$ .

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

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

On entry, $pda = 〈value〉$ .
Constraint: $pda > 0$ .

On entry, $pdb = 〈value〉$ .
Constraint: $pdb > 0$ .
NE_INT_2: On entry, $pda = 〈value〉$ and $m = 〈value〉$ .
Constraint: $pda \geq \max (1, m)$ .

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

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

On entry, $pdb = 〈value〉$ and $p = 〈value〉$ .
Constraint: $pdb \geq \max (1, p)$ .
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 generalized

R Q

factorization is the exact factorization for nearby matrices

(A + E)

and

(B + F)

, where

{‖E‖}_{2} = O ε {‖A‖}_{2} and {‖F‖}_{2} = O ε {‖B‖}_{2},

and

ε

is the machine precision.

8

Parallelism and Performance

nag_dggrqf (f08zfc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_dggrqf (f08zfc) 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 orthogonal matrices

Q

and

Z

may be formed explicitly by calls to nag_dorgrq (f08cjc) and nag_dorgqr (f08afc) respectively. nag_dormrq (f08ckc) may be used to multiply

Q

by another matrix and nag_dormqr (f08agc) may be used to multiply

Z

by another matrix.

The complex analogue of this function is nag_zggrqf (f08ztc).

10

Example

This example solves the least squares problem

\underset{x}{minimize} {‖c - A x‖}_{2} subject to B x = d

where

A = (\begin{array}{r} - 0.57 & - 1.28 & - 0.39 & 0.25 \\ - 1.93 & 1.08 & - 0.31 & - 2.14 \\ 2.30 & 0.24 & 0.40 & - 0.35 \\ - 1.93 & 0.64 & - 0.66 & 0.08 \\ 0.15 & 0.30 & 0.15 & - 2.13 \\ - 0.02 & 1.03 & - 1.43 & 0.50 \end{array}), B = (\begin{array}{r} 1 & 0 & - 1 & 0 \\ 0 & 1 & 0 & - 1 \end{array}),

c = (\begin{array}{r} - 1.50 \\ - 2.14 \\ 1.23 \\ - 0.54 \\ - 1.68 \\ 0.82 \end{array}) and d = (\begin{matrix} 0 \\ 0 \end{matrix}) .

The constraints

B x = d

correspond to

x_{1} = x_{3}

and

x_{2} = x_{4}

The solution is obtained by first computing a generalized

R Q

factorization of the matrix pair

(B, A)

. The example illustrates the general solution process.

NAG Library Function Document

nag_dggrqf (f08zfc)

▸▿ 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

10.1 Program Text

10.2 Program Data

10.3 Program Results

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

10.1

Program Text

10.2

Program Data

10.3

Program Results