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

C Header Interface

#include <nag.h>

void	f08zbf_ (const Integer m, const Integer n, const Integer p, double a[], const Integer lda, double b[], const Integer ldb, double d[], double x[], double y[], double work[], const Integer lwork, Integer *info)

The routine may be called by the names f08zbf, nagf_lapackeig_dggglm or its LAPACK name dggglm.

3 Description

f08zbf solves the real general Gauss–Markov linear model (GLM) problem

\underset{x}{minimize} {‖ y ‖}_{2} subject to d = A x + B y

where

A

is an

m \times n

matrix,

B

is an

m \times p

matrix and

d

is an

m

element vector. It is assumed that

n \leq m \leq n + p

rank (A) = n

and

rank (E) = m

, where

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

. Under these assumptions, the problem has a unique solution

x

and a minimal

2

-norm solution

y

, which is obtained using a generalized

Q R

factorization of the matrices

A

and

B

In particular, if the matrix

B

is square and nonsingular, then the GLM problem is equivalent to the weighted linear least squares problem

\underset{x}{minimize} {‖ B^{- 1} (d - A x) ‖}_{2} .

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

5 Arguments

1: $m$ – Integer Input: On entry: $m$ , the number of rows of the matrices $A$ and $B$ .

Constraint: $m \geq 0$ .
2: $n$ – Integer Input: On entry: $n$ , the number of columns of the matrix $A$ .

Constraint: $0 \leq n \leq m$ .
3: $p$ – Integer Input: On entry: $p$ , the number of columns of the matrix $B$ .

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

On entry: the $m \times n$ matrix $A$ .

On exit: a is overwritten.
5: $lda$ – Integer Input: On entry: the first dimension of the array a as declared in the (sub)program from which f08zbf is called.

Constraint: $lda \geq \max (1, m)$ .
6: $b (ldb, *)$ – Real (Kind=nag_wp) array Input/Output: Note: the second dimension of the array b must be at least $\max (1, p)$ .

On entry: the $m \times p$ matrix $B$ .

On exit: b is overwritten.
7: $ldb$ – Integer Input: On entry: the first dimension of the array b as declared in the (sub)program from which f08zbf is called.

Constraint: $ldb \geq \max (1, m)$ .
8: $d (m)$ – Real (Kind=nag_wp) array Input/Output: On entry: the left-hand side vector $d$ of the GLM equation.

On exit: d is overwritten.
9: $x (n)$ – Real (Kind=nag_wp) array Output: On exit: the solution vector $x$ of the GLM problem.
10: $y (p)$ – Real (Kind=nag_wp) array Output: On exit: the solution vector $y$ of the GLM problem.
11: $work (\max (1, lwork))$ – Real (Kind=nag_wp) array Workspace: On exit: if $info = 0$ , $work (1)$ contains the minimum value of lwork required for optimal performance.
12: $lwork$ – Integer Input: On entry: the dimension of the array work as declared in the (sub)program from which f08zbf 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, $lwork \geq n + \min (m, p) + \max (m, p) \times nb$ , where $nb$ is the optimal block size.

Constraint: $lwork \geq \max (1, m + n + p)$ or $lwork = −1$ .
13: $info$ – Integer Output: 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 $A$ in the generalized $R Q$ factorization of the pair $(A, B)$ is singular, so that $rank (A) < n$ ; the least squares solution could not be computed.

$info = 2$: The bottom $(m - n) \times (m - n)$ part of the upper trapezoidal factor $T$ associated with $B$ in the generalized $Q R$ factorization of the pair $(A, B)$ is singular, so that $rank (\begin{matrix} A & B \end{matrix}) < n$ ; the least squares solutions could not be computed.

7 Accuracy

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

8 Parallelism and Performance

f08zbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f08zbf 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

p = m \geq n

, the total number of floating-point operations is approximately

\frac{2}{3} (2 m^{3} - n^{3}) + 4 n m^{2}

; when

p = m = n

, the total number of floating-point operations is approximately

\frac{14}{3} m^{3}

10 Example

This example solves the weighted least squares problem

\underset{x}{minimize} {‖ B^{- 1} (d - A x) ‖}_{2},

where

B = (\begin{matrix} 0.5 & 0.0 & 0.0 & 0.0 \\ 0.0 & 1.0 & 0.0 & 0.0 \\ 0.0 & 0.0 & 2.0 & 0.0 \\ 0.0 & 0.0 & 0.0 & 5.0 \end{matrix}), d = (\begin{matrix} 1.32 \\ - 4.00 \\ 5.52 \\ 3.24 \end{matrix}) and A = (\begin{array}{r} - 0.57 & - 1.28 & - 0.39 \\ - 1.93 & 1.08 & - 0.31 \\ 2.30 & 0.24 & - 0.40 \\ - 0.02 & 1.03 & - 1.43 \end{array}) .

f08zb: FL CL CPP AD

NAG FL Interfacef08zbf (dggglm)

▸▿ 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 FL Interface
f08zbf (dggglm)