void	nag_zggglm (Nag_OrderType order, Integer m, Integer n, Integer p, Complex a[], Integer pda, Complex b[], Integer pdb, Complex d[], Complex x[], Complex y[], NagError *fail)

3

Description

nag_zggglm (f08zpc) solves the complex general Gauss–Markov linear model (GLM) problem

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

where

A

is an

m

n

matrix,

B

is an

m

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 QR factorization and its applications Linear Algebra Appl. (Volume 162–164) 243–271

5

Arguments

1: $order$ – Nag_OrderTypeInput

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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: $m$ – IntegerInput

On entry:

m

, the number of rows of the matrices

A

and

B

Constraint:

m \geq 0

3: $n$ – IntegerInput

On entry:

n

, the number of columns of the matrix

A

Constraint:

0 \leq n \leq m

4: $p$ – IntegerInput

On entry:

p

, the number of columns of the matrix

B

Constraint:

p \geq m - n

5: $a [\dim]$ – ComplexInput/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

n

matrix

A

On exit: a is overwritten.

6: $pda$ – IntegerInput

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: $b [\dim]$ – ComplexInput/Output

Note: the dimension, dim, of the array b must be at least

$\max (1, pdb \times p)$ when $order = Nag_ColMajor$ ;
$\max (1, m \times pdb)$ 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

m

p

matrix

B

On exit: b is overwritten.

8: $pdb$ – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) in the array b.

Constraints:

if $order = Nag_ColMajor$ , $pdb \geq \max (1, m)$ ;
if $order = Nag_RowMajor$ , $pdb \geq \max (1, p)$ .

9: $d [m]$ – ComplexInput/Output

On entry: the left-hand side vector

d

of the GLM equation.

On exit: d is overwritten.

10: $x [n]$ – ComplexOutput

On exit: the solution vector

x

of the GLM problem.

11: $y [p]$ – ComplexOutput

On exit: the solution vector

y

of the GLM problem.

12: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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, $pda = 〈value〉$ .
Constraint: $pda > 0$ .

On entry, $pdb = 〈value〉$ .
Constraint: $pdb > 0$ .
NE_INT_2: On entry, $m = 〈value〉$ and $n = 〈value〉$ .
Constraint: $0 \leq n \leq m$ .

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 $m = 〈value〉$ .
Constraint: $pdb \geq \max (1, m)$ .

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: $p \geq m - n$ .
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_SINGULAR: The bottom $(N - M)$ by $(N - M)$ 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.

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 $rank (\begin{matrix} B & A \end{matrix}) < n$ ; 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

nag_zggglm (f08zpc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_zggglm (f08zpc) 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

p = m \geq n

, the total number of real floating-point operations is approximately

\frac{8}{3} (2 m^{3} - n^{3}) + 16 n m^{2}

; when

p = m = n

, the total number of real floating-point operations is approximately

\frac{56}{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{array}{r} 0.5 - 1.0 i \\ 1.0 - 2.0 i \\ 2.0 - 3.0 i \\ 5.0 - 4.0 i \end{array}),

d = (\begin{array}{r} 6.00 - 0.40 i \\ - 5.27 + 0.90 i \\ 2.72 - 2.13 i \\ - 1.30 - 2.80 i \end{array})

and

A = (\begin{array}{r} 0.96 - 0.81 i & - 0.03 + 0.96 i & - 0.91 + 2.06 i \\ - 0.98 + 1.98 i & - 1.20 + 0.19 i & - 0.66 + 0.42 i \\ 0.62 - 0.46 i & 1.01 + 0.02 i & 0.63 - 0.17 i \\ 1.08 - 0.28 i & 0.20 - 0.12 i & - 0.07 + 1.23 i \end{array}) .

NAG Library Function Document

nag_zggglm (f08zpc)

▸▿ 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

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