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NAG Toolbox: nag_lapack_dppequ (f07gf)

Purpose

nag_lapack_dppequ (f07gf) computes a diagonal scaling matrix S S  intended to equilibrate a real n n  by n n  symmetric positive definite matrix A A , stored in packed format, and reduce its condition number.

Syntax

[s, scond, amax, info] = f07gf(uplo, n, ap)
[s, scond, amax, info] = nag_lapack_dppequ(uplo, n, ap)

Description

nag_lapack_dppequ (f07gf) 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:     uplo – string (length ≥ 1)
Indicates whether the upper or lower triangular part of AA is stored in the array ap, as follows:
uplo = 'U'uplo='U'
The upper triangle of AA is stored.
uplo = 'L'uplo='L'
The lower triangle of AA is stored.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     n – int64int32nag_int scalar
nn, the order of the matrix AA.
Constraint: n0n0.
3:     ap( : :) – double array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)max(1,n×(n+1)/2).
The nn by nn symmetric matrix AA, packed by columns.
More precisely,
  • if uplo = 'U'uplo='U', the upper triangle of AA must be stored with element AijAij in ap(i + j(j1) / 2)api+j(j-1)/2 for ijij;
  • if uplo = 'L'uplo='L', the lower triangle of AA must be stored with element AijAij in ap(i + (2nj)(j1) / 2)api+(2n-j)(j-1)/2 for ijij.
Only the elements of ap corresponding to the diagonal elements AA are referenced.

Optional Input Parameters

None.

Input Parameters Omitted from the MATLAB Interface

None.

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: uplo, 2: n, 3: ap, 4: s, 5: scond, 6: amax, 7: info.
  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 complex analogue of this function is nag_lapack_zppequ (f07gt).

Example

function nag_lapack_dppequ_example
uplo = 'U';
n = int64(4);
ap = [4.16;
     -312000;
     50300000000;
     0.56;
     -83000;
     0.76;
     -0.1;
     118000;
     0.34;
     1.18];
[s, scond, amax, info] = nag_lapack_dppequ(uplo, n, ap)
 

s =

    0.4903
    0.0000
    1.1471
    0.9206


scond =

   3.8871e-06


amax =

   5.0300e+10


info =

                    0


function f07gf_example
uplo = 'U';
n = int64(4);
ap = [4.16;
     -312000;
     50300000000;
     0.56;
     -83000;
     0.76;
     -0.1;
     118000;
     0.34;
     1.18];
[s, scond, amax, info] = f07gf(uplo, n, ap)
 

s =

    0.4903
    0.0000
    1.1471
    0.9206


scond =

   3.8871e-06


amax =

   5.0300e+10


info =

                    0



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Chapter Introduction
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