NAG Library Routine Document

F07AFF (DGEEQU)

F07AFF (DGEEQU) 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 Parameters

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, $*$ ) – 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.
4: LDA – INTEGERInput: On entry: the first dimension of the array A as declared in the (sub)program from which F07AFF (DGEEQU) 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

Errors or warnings detected by the routine:

$INFO < 0$: If $INFO = - i$ , the $i$ th argument had an illegal value. An explanatory message is output, and execution of the program is terminated.

$INFO > 0 and INFO \leq M$: If $INFO = i$ , the $i$ th row of $A$ is exactly zero.

$INFO > M$: If $INFO = i$ , the $(i - M)$ th column of $A$ is exactly zero.

7 Accuracy

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

8 Further Comments

The complex analogue of this routine is F07ATF (ZGEEQU).

9 Example

This example equilibrates the general matrix

A

given by

A = (\begin{array}{l} 1.80 \times 10^{10} & 2.88 \times 10^{10} & 2.05 & - 8.90 \times 10^{9} \\ 5.25 & - 2.95 & - 9.50 \times 10^{- 9} & - 3.80 \\ 1.58 & - 2.69 & - 2.90 \times 10^{- 10} & - 1.04 \\ - 1.11 & - 0.66 & - 5.90 \times 10^{- 11} & 0.80 \end{array}) .

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

NAG Library Routine DocumentF07AFF (DGEEQU)

+− Contents

1 Purpose

2 Specification

3 Description