Integer, Intent (In)	::	m, p, n, lda, ldb, lwork
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Inout)	::	a(lda,), b(ldb,)
Real (Kind=nag_wp), Intent (Out)	::	taua(min(m,n)), taub(min(p,n)), work(max(1,lwork))

C Header Interface

#include nagmk26.h

void	f08zff_ ( const Integer m, const Integer p, const Integer n, double a[], const Integer lda, double taua[], double b[], const Integer ldb, double taub[], double work[], const Integer lwork, Integer *info)

The routine may be called by its LAPACK name dggrqf.

3

Description

f08zff (dggrqf) 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: $m$ – IntegerInput: On entry: $m$ , the number of rows of the matrix $A$ .

Constraint: $m \geq 0$ .
2: $p$ – IntegerInput: On entry: $p$ , the number of rows of the matrix $B$ .

Constraint: $p \geq 0$ .
3: $n$ – IntegerInput: On entry: $n$ , the number of columns of the matrices $A$ and $B$ .

Constraint: $n \geq 0$ .
4: $a (lda *)$ – Real (Kind=nag_wp) arrayInput/Output: Note: the second dimension of the array a must be at least $\max (1, n)$ .

On entry: the $m$ by $n$ matrix $A$ .

On exit: if $m \leq n$ , the upper triangle of the subarray $a (1 : m, n - m + 1 : n)$ contains the $m$ by $m$ upper triangular matrix $R_{12}$ .
If $m \geq n$ , the elements on and above the $(m - n)$ th subdiagonal contain the $m$ by $n$ 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).
5: $lda$ – IntegerInput: On entry: the first dimension of the array a as declared in the (sub)program from which f08zff (dggrqf) is called.

Constraint: $lda \geq \max (1, m)$ .
6: $taua (\min (m, n))$ – Real (Kind=nag_wp) arrayOutput: On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix $Q$ .
7: $b (ldb *)$ – Real (Kind=nag_wp) arrayInput/Output: Note: the second dimension of the array b must be at least $\max (1, n)$ .

On entry: the $p$ by $n$ matrix $B$ .

On exit: the elements on and above the diagonal of the array contain the $\min (p, n)$ by $n$ 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).
8: $ldb$ – IntegerInput: On entry: the first dimension of the array b as declared in the (sub)program from which f08zff (dggrqf) is called.

Constraint: $ldb \geq \max (1, p)$ .
9: $taub (\min (p, n))$ – Real (Kind=nag_wp) arrayOutput: On exit: the scalar factors of the elementary reflectors which represent the orthogonal matrix $Z$ .
10: $work (\max (1, lwork))$ – Real (Kind=nag_wp) arrayWorkspace: On exit: if $info = 0$ , $work (1)$ contains the minimum value of lwork required for optimal performance.
11: $lwork$ – IntegerInput: On entry: the dimension of the array work as declared in the (sub)program from which f08zff (dggrqf) is called.
If $lwork = - 1$ , a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.

Suggested value: for optimal performance, $lwork \geq \max (n, m, p) \times \max (nb1, nb2, nb3)$ , where $nb1$ is the optimal block size for the $R Q$ factorization of an $m$ by $n$ matrix by f08chf (dgerqf), $nb2$ is the optimal block size for the $Q R$ factorization of a $p$ by $n$ matrix by f08aef (dgeqrf), and $nb3$ is the optimal block size for a call of f08ckf (dormrq).

Constraint: $lwork \geq \max (1, n, m, p)$ or $lwork = - 1$ .
12: $info$ – IntegerOutput: On exit: $info = 0$ unless the routine detects an error (see Section 6).

6

Error Indicators and Warnings

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

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

f08zff (dggrqf) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08zff (dggrqf) 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 routine. 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 f08cjf (dorgrq) and f08aff (dorgqr) respectively. f08ckf (dormrq) may be used to multiply

Q

by another matrix and f08agf (dormqr) may be used to multiply

Z

by another matrix.

The complex analogue of this routine is f08ztf (zggrqf).

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.

Note that the block size (NB) of

64

assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

NAG Library Routine Document

f08zff (dggrqf)

▸▿ 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