NAG Library Routine Document

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

C Header Interface

#include <nagmk26.h>

void	f08zaf_ (const Integer m, const Integer n, const Integer p, double a[], const Integer lda, double b[], const Integer ldb, double c[], double d[], double x[], double work[], const Integer lwork, Integer *info)

The routine may be called by its LAPACK name dgglse.

3

Description

f08zaf (dgglse) solves the real linear equality-constrained least squares (LSE) problem

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

where

A

is an

m

n

matrix,

B

is a

p

n

matrix,

c

is an

m

element vector and

d

is a

p

element vector. It is assumed that

p \leq n \leq m + p

rank (B) = p

and

rank (E) = n

, where

E = (\begin{matrix} A \\ B \end{matrix})

. These conditions ensure that the LSE problem has a unique solution, which is obtained using a generalized

R Q

factorization of the matrices

B

and

A

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

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

Eldèn L (1980) Perturbation theory for the least squares problem with linear equality constraints SIAM J. Numer. Anal. 17 338–350

5

Arguments

1: $m$ – IntegerInput: On entry: $m$ , the number of rows of the matrix $A$ .

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

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

Constraint: $0 \leq p \leq n \leq m + p$ .
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: a is overwritten.
5: $lda$ – IntegerInput: On entry: the first dimension of the array a as declared in the (sub)program from which f08zaf (dgglse) is called.

Constraint: $lda \geq \max (1, m)$ .
6: $b (ldb *)$ – 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: b is overwritten.
7: $ldb$ – IntegerInput: On entry: the first dimension of the array b as declared in the (sub)program from which f08zaf (dgglse) is called.

Constraint: $ldb \geq \max (1, p)$ .
8: $c (m)$ – Real (Kind=nag_wp) arrayInput/Output: On entry: the right-hand side vector $c$ for the least squares part of the LSE problem.
On exit: the residual sum of squares for the solution vector $x$ is given by the sum of squares of elements $c (n - p + 1), c (n - p + 2), \dots, c (m)$ ; the remaining elements are overwritten.
9: $d (p)$ – Real (Kind=nag_wp) arrayInput/Output: On entry: the right-hand side vector $d$ for the equality constraints.

On exit: d is overwritten.
10: $x (n)$ – Real (Kind=nag_wp) arrayOutput: On exit: the solution vector $x$ of the LSE problem.
11: $work (\max (1, lwork))$ – Real (Kind=nag_wp) arrayWorkspace: On exit: if $info = 0$ , $work (1)$ contains the minimum value of lwork required for optimal performance.
12: $lwork$ – IntegerInput: On entry: the dimension of the array work as declared in the (sub)program from which f08zaf (dgglse) 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, $lwork \geq p + \min (m, n) + \max (m, n) \times nb$ , where $nb$ is the optimal block size.

Constraint: $lwork \geq \max (1, m + n + p)$ or $lwork = - 1$ .
13: $info$ – 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.

$info = 1$: The upper triangular factor $R$ associated with $B$ in the generalized $R Q$ factorization of the pair $(B, A)$ is singular, so that $rank (B) < p$ ; the least squares solution could not be computed.

$info = 2$: The $(N - P)$ by $(N - P)$ part of the upper trapezoidal factor $T$ associated with $A$ in the generalized $R Q$ factorization of the pair $(B, A)$ is singular, so that the rank of the matrix ( $E$ ) comprising the rows of $A$ and $B$ is less than $n$ ; the least squares solutions could not be computed.

7

Accuracy

For an error analysis, see Anderson et al. (1992) and Eldèn (1980). See also Section 4.6 of Anderson et al. (1999).

8

Parallelism and Performance

f08zaf (dgglse) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08zaf (dgglse) 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

When

m \geq n = p

, the total number of floating-point operations is approximately

\frac{2}{3} n^{2} (6 m + n)

; if

p ≪ n

, the number reduces to approximately

\frac{2}{3} n^{2} (3 m - n)

e04ncf/e04nca may also be used to solve LSE problems. It differs from f08zaf (dgglse) in that it uses an iterative (rather than direct) method, and that it allows general upper and lower bounds to be specified for the variables

x

and the linear constraints

B x

10

Example

This example solves the least squares problem

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

where

c = (\begin{array}{r} - 1.50 \\ - 2.14 \\ 1.23 \\ - 0.54 \\ - 1.68 \\ 0.82 \end{array}),

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{matrix} 1.0 & 0 & - 1.0 & 0 \\ 0 & 1.0 & 0 & - 1.0 \end{matrix})

and

d = (\begin{matrix} 0 \\ 0 \end{matrix}) .

The constraints

B x = d

correspond to

x_{1} = x_{3}

and

x_{2} = x_{4}

NAG Library Routine Document

f08zaf (dgglse)

▸▿ 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