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NAG Toolbox: nag_lapack_dsptri (f07pj)

Purpose

nag_lapack_dsptri (f07pj) computes the inverse of a real symmetric indefinite matrix AA, where AA has been factorized by nag_lapack_dsptrf (f07pd), using packed storage.

Syntax

[ap, info] = f07pj(uplo, ap, ipiv, 'n', n)
[ap, info] = nag_lapack_dsptri(uplo, ap, ipiv, 'n', n)

Description

nag_lapack_dsptri (f07pj) is used to compute the inverse of a real symmetric indefinite matrix AA, the function must be preceded by a call to nag_lapack_dsptrf (f07pd), which computes the Bunch–Kaufman factorization of AA, using packed storage.
If uplo = 'U'uplo='U', A = PUDUTPTA=PUDUTPT and A1A-1 is computed by solving UTPTXPU = D1UTPTXPU=D-1.
If uplo = 'L'uplo='L', A = PLDLTPTA=PLDLTPT and A1A-1 is computed by solving LTPTXPL = D1LTPTXPL=D-1.

References

Du Croz J J and Higham N J (1992) Stability of methods for matrix inversion IMA J. Numer. Anal. 12 1–19

Parameters

Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies how AA has been factorized.
uplo = 'U'uplo='U'
A = PUDUTPTA=PUDUTPT, where UU is upper triangular.
uplo = 'L'uplo='L'
A = PLDLTPTA=PLDLTPT, where LL is lower triangular.
Constraint: uplo = 'U'uplo='U' or 'L''L'.
2:     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 factorization of AA stored in packed form, as returned by nag_lapack_dsptrf (f07pd).
3:     ipiv( : :) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)max(1,n).
Details of the interchanges and the block structure of DD, as returned by nag_lapack_dsptrf (f07pd).

Optional Input Parameters

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

Input Parameters Omitted from the MATLAB Interface

work

Output Parameters

1:     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 factorization stores the nn by nn matrix A1A-1.
More precisely,
  • if uplo = 'U'uplo='U', the upper triangle of A1A-1 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 A1A-1 must be stored with element AijAij in ap(i + (2nj)(j1) / 2)api+(2n-j)(j-1)/2 for ijij.
2:     info – int64int32nag_int scalar
info = 0info=0 unless the function detects an error (see Section [Error Indicators and Warnings]).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_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: ipiv, 5: work, 6: 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.
W INFO > 0INFO>0
If info = iinfo=i, d(i,i)d(i,i) is exactly zero; DD is singular and the inverse of AA cannot be computed.

Accuracy

The computed inverse XX satisfies a bound of the form c(n)c(n) is a modest linear function of nn, and εε is the machine precision

Further Comments

The total number of floating point operations is approximately (2/3)n323n3.
The complex analogues of this function are nag_lapack_zhptri (f07pw) for Hermitian matrices and nag_lapack_zsptri (f07qw) for symmetric matrices.

Example

function nag_lapack_dsptri_example
uplo = 'L';
ap = [2.07;
     4.2;
     0.2230413840558341;
     0.6536583767489105;
     1.15;
     0.8115010321439103;
     -0.5959697237786296;
     -2.59067708640519;
     0.3030846795506181;
     0.4073851981348882];
ipiv = [int64(-3);-3;3;4];
[apOut, info] = nag_lapack_dsptri(uplo, ap, ipiv)
 

apOut =

    0.7485
    0.5221
   -1.0058
   -1.4386
   -0.1605
   -0.3131
   -0.7440
    1.3501
    2.0667
    2.4547


info =

                    0


function f07pj_example
uplo = 'L';
ap = [2.07;
     4.2;
     0.2230413840558341;
     0.6536583767489105;
     1.15;
     0.8115010321439103;
     -0.5959697237786296;
     -2.59067708640519;
     0.3030846795506181;
     0.4073851981348882];
ipiv = [int64(-3);-3;3;4];
[apOut, info] = f07pj(uplo, ap, ipiv)
 

apOut =

    0.7485
    0.5221
   -1.0058
   -1.4386
   -0.1605
   -0.3131
   -0.7440
    1.3501
    2.0667
    2.4547


info =

                    0



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

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