NAG Library Routine Document

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

Constraint: $n \geq 0$ .
2: $a (lda *)$ – Real (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 f07fff (dpoequ) 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).

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

f07fff (dpoequ) is not threaded in any implementation.

The complex analogue of this routine is f07ftf (zpoequ).

This example equilibrates the symmetric positive definite matrix

A

given by

A = (\begin{array}{l} 4.16 & - 3.12 \times 10^{5} & 0.56 & - 0.10 \\ - 3.12 \times 10^{5} & 5.03 \times 10^{10} & - 0.83 \times 10^{5} & 1.18 \times 10^{5} \\ 0.56 & - 0.83 \times 10^{5} & 0.76 & 0.34 \\ - 0.10 & 1.18 \times 10^{5} & 0.34 & 1.18 \end{array}) .

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