Integer, Intent (In)	::	n
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (Out)	::	s(n), scond, amax
Complex (Kind=nag_wp), Intent (In)	::	ap(*)
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nag.h>

void	f07gtf_ (const char uplo, const Integer n, const Complex ap[], double s[], double scond, double amax, Integer *info, const Charlen length_uplo)

The routine may be called by the names f07gtf, nagf_lapacklin_zppequ or its LAPACK name zppequ.

3 Description

f07gtf 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: $uplo$ – Character(1) Input

On entry: indicates whether the upper or lower triangular part of

A

is stored in the array ap, as follows:

$uplo ='U'$: The upper triangle of $A$ is stored.
$uplo ='L'$: The lower triangle of $A$ is stored.

Constraint:

uplo ='U'

'L'

2: $n$ – Integer Input

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $ap (*)$ – Complex (Kind=nag_wp) array Input

Note: the dimension of the array ap must be at least

\max (1, n \times (n + 1) / 2)

On entry: the

n \times n

Hermitian matrix

A

, packed by columns.

More precisely,

if $uplo ='U'$ , the upper triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $uplo ='L'$ , the lower triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

Only the elements of ap corresponding to the diagonal elements

A

are referenced.

4: $s (n)$ – Real (Kind=nag_wp) array Output

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: $info$ – 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.

$info > 0$: 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

f07gtf is not threaded in any implementation.

9 Further Comments

The real analogue of this routine is f07gff.

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.

f07gt: FL CL CPP AD

NAG FL Interfacef07gtf (zppequ)

▸▿ 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
f07gtf (zppequ)