NAG FL Interface
f07gtf (zppequ)

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1 Purpose

f07gtf computes a diagonal scaling matrix S intended to equilibrate a complex n × n Hermitian positive definite matrix A , stored in packed format, and reduce its condition number.

2 Specification

Fortran Interface
Subroutine f07gtf ( uplo, n, ap, s, scond, amax, info)
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
sj=1 / ajj .  
This means that the matrix B given by
B=SAS ,  
has diagonal elements equal to unity. This in turn means that the condition number of B , κ2(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:
The upper triangle of A is stored.
The lower triangle of A is stored.
Constraint: uplo='U' or 'L'.
2: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n0.
3: ap(*) Complex (Kind=nag_wp) array Input
Note: the dimension of the array ap must be at least max(1,n×(n+1)/2).
On entry: the n×n Hermitian matrix A, packed by columns.
More precisely,
  • if uplo='U', the upper triangle of A must be stored with element Aij in ap(i+j(j-1)/2) for ij;
  • if uplo='L', the lower triangle of A must be stored with element Aij in ap(i+(2n-j)(j-1)/2) for ij.
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 scond0.1 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|aij|. 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

If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.
The valueth 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

Background information to multithreading can be found in the Multithreading documentation.
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 = ( (3.23 ((1.51-1.92i (1.90+0.84i)×105 ((0.42+2.50i (1.51+1.92i ((3.58 (-0.23+1.11i)×105 -1.18+1.37i (1.90-0.84i)×105 (-0.23-1.11i)×105 4.09×1010 ((2.33-0.14i)×105 (0.42-2.50i (-1.18-1.37i (2.33+0.14i)×105 ((4.29 ) .  
Details of the scaling factors and the scaled matrix are output.

10.1 Program Text

Program Text (f07gtfe.f90)

10.2 Program Data

Program Data (f07gtfe.d)

10.3 Program Results

Program Results (f07gtfe.r)