NAG Library Routine Document

F07FFF (DPOEQU)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

F07FFF (DPOEQU) computes a diagonal scaling matrix

S

intended to equilibrate a real

n

n

symmetric positive definite matrix

A

and reduce its condition number.

2 Specification

SUBROUTINE F07FFF (

N, A, LDA, S, SCOND, AMAX, INFO)

INTEGER	N, LDA, INFO
REAL (KIND=nag_wp)	A(LDA,*), S(N), SCOND, AMAX

The routine may be called by its LAPACK name dpoequ.

3 Description

F07FFF (DPOEQU) 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 Parameters

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).

7 Accuracy

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

8 Parallelism and Performance

Not applicable.

9 Further Comments

The complex analogue of this routine is F07FTF (ZPOEQU).

10 Example

This example equilibrates the symmetric positive definite matrix

A