NAG Library Routine Document

F08ZNF (ZGGLSE)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F08ZNF (ZGGLSE) solves a complex linear equality-constrained least squares problem.

2 Specification

SUBROUTINE F08ZNF (

M, N, P, A, LDA, B, LDB, C, D, X, WORK, LWORK, INFO)

INTEGER	M, N, P, LDA, LDB, LWORK, INFO
COMPLEX (KIND=nag_wp)	A(LDA,), B(LDB,), C(M), D(P), X(N), WORK(max(1,LWORK))

The routine may be called by its LAPACK name zgglse.

3 Description

F08ZNF (ZGGLSE) solves the complex 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 Parameters

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, $*$ ) – COMPLEX (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 F08ZNF (ZGGLSE) is called.
Constraint: $LDA \geq \max (1, M)$ .
6: B(LDB, $*$ ) – COMPLEX (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 F08ZNF (ZGGLSE) is called.
Constraint: $LDB \geq \max (1, P)$ .
8: C(M) – COMPLEX (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) – COMPLEX (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) – COMPLEX (KIND=nag_wp) arrayOutput: On exit: the solution vector $x$ of the LSE problem.
11: WORK( $\max (1, LWORK)$ ) – COMPLEX (KIND=nag_wp) arrayWorkspace: On exit: if $INFO = 0$ , the real part of $WORK (1)$ 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 F08ZNF (ZGGLSE) 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

Errors or warnings detected by the routine:

$INFO < 0$: 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 Further Comments

When

m \geq n = p

, the total number of real floating point operations is approximately

\frac{8}{3} n^{2} (6 m + n)

; if

p ≪ n

, the number reduces to approximately

\frac{8}{3} n^{2} (3 m - n)

9 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} - 2.54 + 0.09 i \\ 1.65 - 2.26 i \\ - 2.11 - 3.96 i \\ 1.82 + 3.30 i \\ - 6.41 + 3.77 i \\ 2.07 + 0.66 i \end{array}),

and

A = (\begin{array}{r} 0.96 - 0.81 i & - 0.03 + 0.96 i & - 0.91 + 2.06 i & - 0.05 + 0.41 i \\ - 0.98 + 1.98 i & - 1.20 + 0.19 i & - 0.66 + 0.42 i & - 0.81 + 0.56 i \\ 0.62 - 0.46 i & 1.01 + 0.02 i & 0.63 - 0.17 i & - 1.11 + 0.60 i \\ 0.37 + 0.38 i & 0.19 - 0.54 i & - 0.98 - 0.36 i & 0.22 - 0.20 i \\ 0.83 + 0.51 i & 0.20 + 0.01 i & - 0.17 - 0.46 i & 1.47 + 1.59 i \\ 1.08 - 0.28 i & 0.20 - 0.12 i & - 0.07 + 1.23 i & 0.26 + 0.26 i \end{array}),

B = (\begin{array}{r} 1.0 + 0.0 i & 0 & - 1.0 + 0.0 i & 0 \\ 0 & 1.0 + 0.0 i & 0 & - 1.0 + 0.0 i \end{array})

and

d = (\begin{array}{r} 0 \\ 0 \end{array}) .

The constraints

B x = d

correspond to

x_{1} = x_{3}

and

x_{2} = x_{4}

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 DocumentF08ZNF (ZGGLSE)