C Header Interface

#include <nag.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 the names f07ftf, nagf_lapacklin_zpoequ or its LAPACK name zpoequ.

3 Description

f07ftf 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

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

Constraint: $n \geq 0$ .
2: $a (lda, *)$ – Complex (Kind=nag_wp) array Input: 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: $lda$ – Integer Input: On entry: the first dimension of the array a as declared in the (sub)program from which f07ftf is called.

Constraint: $lda \geq \max (1, n)$ .
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).

$info < 0$: 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).

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

f07ftf is not threaded in any implementation.

The real analogue of this routine is f07fff.

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.

f07ft: FL CL CPP AD