void	f08zac (Nag_OrderType order, Integer m, Integer n, Integer p, double a[], Integer pda, double b[], Integer pdb, double c[], double d[], double x[], NagError *fail)

The function may be called by the names: f08zac, nag_lapackeig_dgglse or nag_dgglse.

3 Description

f08zac 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 \times n

matrix,

B

is a

p \times 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

Q R

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: $order$ – Nag_OrderType Input

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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $m$ – Integer Input

On entry:

m

, the number of rows of the matrix

A

Constraint:

m \geq 0

3: $n$ – Integer Input

On entry:

n

, the number of columns of the matrices

A

and

B

Constraint:

n \geq 0

4: $p$ – Integer Input

On entry:

p

, the number of rows of the matrix

B

Constraint:

0 \leq p \leq n \leq m + p

5: $a [\dim]$ – double Input/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 $order = Nag_RowMajor$ .

The

(i, j)

th element of the matrix

A

is stored in

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

On entry: the

m \times n

matrix

A

On exit: a is overwritten.

6: $pda$ – Integer Input

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: $b [\dim]$ – double Input/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 \times n

matrix

B

On exit: b is overwritten.

8: $pdb$ – Integer Input

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 $order = Nag_RowMajor$ , $pdb \geq \max (1, n)$ .

9: $c [m]$ – double Input/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], c [n - p + 1], \dots, c [m - 1]

; the remaining elements are overwritten.

10: $d [p]$ – double Input/Output

On entry: the right-hand side vector

d

for the equality constraints.

On exit: d is overwritten.

11: $x [n]$ – double Output

On exit: the solution vector

x

of the LSE problem.

12: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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, $pda = ⟨ value ⟩$ .
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_INT_3: On entry, $p = ⟨ value ⟩$ , $m = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $0 \leq p \leq n \leq m + p$ .
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR: The $(N - P) \times (N - P)$ 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.

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.

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

Background information to multithreading can be found in the Multithreading documentation.

f08zac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08zac 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

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)

e04ncc may also be used to solve LSE problems. It differs from f08zac 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}

f08za: FL CL CPP AD PY MB

NAG CL Interfacef08zac (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

NAG CL Interface
f08zac (dgglse)