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NAG Toolbox: nag_lapack_zpoequ (f07ft)

Purpose

nag_lapack_zpoequ (f07ft) computes a diagonal scaling matrix S S  intended to equilibrate a complex n n  by n n  Hermitian positive definite matrix A A  and reduce its condition number.

Syntax

[s, scond, amax, info] = f07ft(a, 'n', n)
[s, scond, amax, info] = nag_lapack_zpoequ(a, 'n', n)

Description

nag_lapack_zpoequ (f07ft) computes a diagonal scaling matrix S S  chosen so that
sj = 1 / sqrt(ajj) .
sj=1 / ajj .
This means that the matrix B B  given by
B = SAS ,
B=SAS ,
has diagonal elements equal to unity. This in turn means that the condition number of B B , κ2(B) κ2(B) , is within a factor n n  of the matrix of smallest possible condition number over all possible choices of diagonal scalings (see Corollary 7.6 of Higham (2002)).

References

Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

Parameters

Compulsory Input Parameters

1:     a(lda, : :) – complex array
The first dimension of the array a must be at least max (1,n)max(1,n)
The second dimension of the array must be at least max (1,n)max(1,n)
The matrix AA whose scaling factors are to be computed. Only the diagonal elements of the array a are referenced.

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array a The second dimension of the array a.
nn, the order of the matrix AA.
Constraint: n0n0.

Input Parameters Omitted from the MATLAB Interface

lda

Output Parameters

1:     s(n) – double array
If INFO = 0INFO=0, s contains the diagonal elements of the scaling matrix SS.
2:     scond – double scalar
If INFO = 0INFO=0, scond contains the ratio of the smallest value of s to the largest value of s. If scond0.1scond0.1 and amax is neither too large nor too small, it is not worth scaling by SS.
3:     amax – double scalar
max|aij|max|aij|. If amax is very close to overflow or underflow, the matrix AA should be scaled.
4:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

  info = iinfo=-i
If info = iinfo=-i, parameter ii had an illegal value on entry. The parameters are numbered as follows:
1: n, 2: a, 3: lda, 4: s, 5: scond, 6: amax, 7: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.
  INFO > 0INFO>0
If info = iinfo=i, the iith diagonal element of AA is not positive (and hence AA cannot be positive definite).

Accuracy

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

Further Comments

The real analogue of this function is nag_lapack_dpoequ (f07ff).

Example

function nag_lapack_zpoequ_example
a = [3.23,  1.51 - 1.92i,  190000 + 84000i,  0.42 + 2.5i;
      0 + 0i,  3.58 + 0i,  -23000 + 111000i,  -1.18 + 1.37i;
      0 + 0i,  0 + 0i,  40900000000 + 0i,  233000 - 14000i;
      0 + 0i,  0 + 0i,  0 + 0i,  4.29 + 0i];
[s, scond, amax, info] = nag_lapack_zpoequ(a)
 

s =

    0.5564
    0.5285
    0.0000
    0.4828


scond =

   8.8867e-06


amax =

   4.0900e+10


info =

                    0


function f07ft_example
a = [3.23,  1.51 - 1.92i,  190000 + 84000i,  0.42 + 2.5i;
      0 + 0i,  3.58 + 0i,  -23000 + 111000i,  -1.18 + 1.37i;
      0 + 0i,  0 + 0i,  40900000000 + 0i,  233000 - 14000i;
      0 + 0i,  0 + 0i,  0 + 0i,  4.29 + 0i];
[s, scond, amax, info] = f07ft(a)
 

s =

    0.5564
    0.5285
    0.0000
    0.4828


scond =

   8.8867e-06


amax =

   4.0900e+10


info =

                    0



PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
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