Integer, Intent (In)	::	m, n, lda
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Out)	::	r(m), c(n), rowcnd, colcnd, amax
Complex (Kind=nag_wp), Intent (In)	::	a(lda,*)

C Header Interface

#include nagmk26.h

void	f07atf_ ( const Integer m, const Integer n, const Complex a[], const Integer lda, double r[], double c[], double rowcnd, double colcnd, double amax, Integer *info)

The routine may be called by its LAPACK name zgeequ.

3

Description

f07atf (zgeequ) computes the diagonal scaling matrices. The diagonal scaling matrices are chosen to try to make the elements of largest absolute value in each row and column of the matrix

B

given by

B = D_{R} A D_{C}

have absolute value

1

. The diagonal elements of

D_{R}

and

D_{C}

are restricted to lie in the safe range

(δ, 1 / δ)

, where

δ

is the value returned by routine x02amf. Use of these scaling factors is not guaranteed to reduce the condition number of

A

but works well in practice.

4

References

None.

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 matrix $A$ .

Constraint: $n \geq 0$ .
3: $a (lda *)$ – Complex (Kind=nag_wp) arrayInput: Note: the second dimension of the array a must be at least $\max (1, n)$ .

On entry: the matrix $A$ whose scaling factors are to be computed.
4: $lda$ – IntegerInput: On entry: the first dimension of the array a as declared in the (sub)program from which f07atf (zgeequ) is called.

Constraint: $lda \geq \max (1, m)$ .
5: $r (m)$ – Real (Kind=nag_wp) arrayOutput: On exit: if $info = 0$ or $info > m$ , r contains the row scale factors, the diagonal elements of $D_{R}$ . The elements of r will be positive.
6: $c (n)$ – Real (Kind=nag_wp) arrayOutput: On exit: if $info = 0$ , c contains the column scale factors, the diagonal elements of $D_{C}$ . The elements of c will be positive.
7: $rowcnd$ – Real (Kind=nag_wp)Output: On exit: if $info = 0$ or $info > m$ , rowcnd contains the ratio of the smallest value of $r (i)$ to the largest value of $r (i)$ . If $rowcnd \geq 0.1$ and amax is neither too large nor too small, it is not worth scaling by $D_{R}$ .
8: $colcnd$ – Real (Kind=nag_wp)Output: On exit: if $info = 0$ , colcnd contains the ratio of the smallest value of $c (i)$ to the largest value of $c (i)$ .
If $colcnd \geq 0.1$ , it is not worth scaling by $D_{C}$ .
9: $amax$ – Real (Kind=nag_wp)Output: On exit: $\max |a_{i j}|$ . If amax is very close to overflow or underflow, the matrix $A$ should be scaled.
10: $info$ – 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.

$info > 0 and info \leq m$: Row $〈value〉$ of $A$ is exactly zero.

$info > m$: Column $〈value〉$ of $A$ is exactly zero.

7

Accuracy

The computed scale factors will be close to the exact scale factors.

8

Parallelism and Performance

f07atf (zgeequ) 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

The real analogue of this routine is f07aff (dgeequ).

10

Example

This example equilibrates the general matrix

A

given by

A = (\begin{array}{l} - 1.34 + 2.55 i & (0.28 + 3.17 i) \times 10^{10} & - 6.39 - 2.20 i \\ - 1.70 - 1.41 i & (3.31 - 0.15 i) \times 10^{10} & - 0.15 + 1.34 i \\ (2.41 + 0.39 i) \times 10^{- 10} & - 0.56 + 1.47 i & (- 0.83 - 0.69 i) \times 10^{- 10} \end{array}) .

Details of the scaling factors, and the scaled matrix are output.

NAG Library Routine Document

f07atf (zgeequ)

▸▿ 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