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NAG Toolbox: nag_lapack_zsptri (f07qw)

Purpose

nag_lapack_zsptri (f07qw) computes the inverse of a complex symmetric matrix AA, where AA has been factorized by nag_lapack_zsptrf (f07qr), using packed storage.

Syntax

[ap, info] = f07qw(uplo, ap, ipiv, 'n', n)
[ap, info] = nag_lapack_zsptri(uplo, ap, ipiv, 'n', n)

Description

nag_lapack_zsptri (f07qw) is used to compute the inverse of a complex symmetric matrix AA, the function must be preceded by a call to nag_lapack_zsptrf (f07qr), 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( : :) – complex 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_zsptrf (f07qr).
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_zsptrf (f07qr).

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( : :) – complex 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 real floating point operations is approximately (8/3)n383n3.
The real analogue of this function is nag_lapack_dsptri (f07pj).

Example

function nag_lapack_zsptri_example
uplo = 'L';
ap = [ -0.39 - 0.71i;
      -7.86 - 2.96i;
      0.5278724801640799 - 0.3714660014825906i;
      0.442558238872675 + 0.1936483698297402i;
      -2.83 - 0.03i;
      -0.6078391056683192 + 0.281079647893122i;
      -0.4822822975185383 + 0.01498936219105284i;
      4.407906236731014 + 5.399120676796941i;
      -0.1070821880092683 - 0.3156780862488454i;
      -2.095414887840057 - 2.201139281440786i];
ipiv = [int64(-3);-3;3;4];
[apOut, info] = nag_lapack_zsptri(uplo, ap, ipiv)
 

apOut =

  -0.1562 - 0.1014i
   0.0400 + 0.1527i
   0.0550 + 0.0845i
   0.2162 - 0.0742i
   0.0946 - 0.1475i
  -0.0326 - 0.1370i
  -0.0995 - 0.0461i
  -0.1320 - 0.0102i
  -0.1793 + 0.1183i
  -0.2269 + 0.2383i


info =

                    0


function f07qw_example
uplo = 'L';
ap = [ -0.39 - 0.71i;
      -7.86 - 2.96i;
      0.5278724801640799 - 0.3714660014825906i;
      0.442558238872675 + 0.1936483698297402i;
      -2.83 - 0.03i;
      -0.6078391056683192 + 0.281079647893122i;
      -0.4822822975185383 + 0.01498936219105284i;
      4.407906236731014 + 5.399120676796941i;
      -0.1070821880092683 - 0.3156780862488454i;
      -2.095414887840057 - 2.201139281440786i];
ipiv = [int64(-3);-3;3;4];
[apOut, info] = f07qw(uplo, ap, ipiv)
 

apOut =

  -0.1562 - 0.1014i
   0.0400 + 0.1527i
   0.0550 + 0.0845i
   0.2162 - 0.0742i
   0.0946 - 0.1475i
  -0.0326 - 0.1370i
  -0.0995 - 0.0461i
  -0.1320 - 0.0102i
  -0.1793 + 0.1183i
  -0.2269 + 0.2383i


info =

                    0



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Chapter Introduction
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