Integer, Intent (In)	::	n, lda
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Out)	::	s(n), scond, amax
Complex (Kind=nag_wp), Intent (In)	::	a(lda,*)

C Header Interface

#include nagmk26.h

void	f07ftf_ ( const Integer n, const Complex a[], const Integer lda, double s[], double scond, double amax, Integer *info)

The routine may be called by its LAPACK name zpoequ.

3

Description

f07ftf (zpoequ) computes a diagonal scaling matrix

S

chosen so that

s_{j} = 1 / \sqrt{a_{j j}} .

This means that the matrix

B

given by

B = S A S,

has diagonal elements equal to unity. This in turn means that the condition number of

B

κ_{2} (B)

, is within a factor

n

of the matrix of smallest possible condition number over all possible choices of diagonal scalings (see Corollary 7.6 of Higham (2002)).

4

References

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

5

Arguments

1: $n$ – IntegerInput: On entry: $n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $a (lda *)$ – 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. Only the diagonal elements of the array a are referenced.
3: $lda$ – IntegerInput: On entry: the first dimension of the array a as declared in the (sub)program from which f07ftf (zpoequ) is called.

Constraint: $lda \geq \max (1, n)$ .
4: $s (n)$ – Real (Kind=nag_wp) arrayOutput: On exit: if $info = 0$ , s contains the diagonal elements of the scaling matrix $S$ .
5: $scond$ – Real (Kind=nag_wp)Output: On exit: if $info = 0$ , scond contains the ratio of the smallest value of s to the largest value of s. If $scond \geq 0.1$ and amax is neither too large nor too small, it is not worth scaling by $S$ .
6: $amax$ – 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.
7: $info$ – 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.

$info > 0$: The $〈value〉$ th diagonal element of $A$ is not positive (and hence $A$ cannot be positive definite).

7

Accuracy

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

8

Parallelism and Performance

f07ftf (zpoequ) is not threaded in any implementation.

9

Further Comments

The real analogue of this routine is f07fff (dpoequ).

10

Example

This example equilibrates the Hermitian positive definite matrix

A

given by

A = (\begin{array}{l} 3.23 & 1.51 - 1.92 i & (1.90 + 0.84 i) \times 10^{5} & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 & (- 0.23 + 1.11 i) \times 10^{5} & - 1.18 + 1.37 i \\ (1.90 - 0.84 i) \times 10^{5} & (- 0.23 - 1.11 i) \times 10^{5} & 4.09 \times 10^{10} & (2.33 - 0.14 i) \times 10^{5} \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & (2.33 + 0.14 i) \times 10^{5} & 4.29 \end{array}) .

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

NAG Library Routine Document

f07ftf (zpoequ)

▸▿ 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