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NAG Toolbox: nag_lapack_zhptri (f07pw)

Purpose

nag_lapack_zhptri (f07pw) computes the inverse of a complex Hermitian indefinite matrix AA, where AA has been factorized by nag_lapack_zhptrf (f07pr), using packed storage.

Syntax

[ap, info] = f07pw(uplo, ap, ipiv, 'n', n)
[ap, info] = nag_lapack_zhptri(uplo, ap, ipiv, 'n', n)

Description

nag_lapack_zhptri (f07pw) is used to compute the inverse of a complex Hermitian indefinite matrix AA, the function must be preceded by a call to nag_lapack_zhptrf (f07pr), which computes the Bunch–Kaufman factorization of AA, using packed storage.
If uplo = 'U'uplo='U', A = PUDUHPTA=PUDUHPT and A1A-1 is computed by solving UHPTXPU = D1UHPTXPU=D-1 for XX.
If uplo = 'L'uplo='L', A = PLDLHPTA=PLDLHPT and A1A-1 is computed by solving LHPTXPL = D1LHPTXPL=D-1 for XX.

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 = PUDUHPTA=PUDUHPT, where UU is upper triangular.
uplo = 'L'uplo='L'
A = PLDLHPTA=PLDLHPT, 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_zhptrf (f07pr).
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_zhptrf (f07pr).

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_zhptri_example
uplo = 'L';
ap = [-1.36;
      3.91 - 1.5i;
      0.3100287981271241 + 0.04333020743962702i;
      -0.1518120207240102 + 0.3742958425613706i;
      -1.84 + 0i;
      0.5637050486508776 + 0.2850349501519716i;
      0.339658279960361 + 0.03031451811355637i;
      -5.417624387291579 + 0i;
      0.2997244646075835 + 0.1578268372785777i;
      -7.102809895801842 + 0i];
ipiv = [int64(-4);-4;3;4];
[apOut, info] = nag_lapack_zhptri(uplo, ap, ipiv)
 

apOut =

   0.0826 + 0.0000i
  -0.0335 + 0.0440i
   0.0603 - 0.0105i
   0.2391 - 0.0926i
  -0.1408 + 0.0000i
   0.0422 - 0.0222i
   0.0304 + 0.0203i
  -0.2007 + 0.0000i
   0.0982 - 0.0635i
   0.0073 + 0.0000i


info =

                    0


function f07pw_example
uplo = 'L';
ap = [-1.36;
      3.91 - 1.5i;
      0.3100287981271241 + 0.04333020743962702i;
      -0.1518120207240102 + 0.3742958425613706i;
      -1.84 + 0i;
      0.5637050486508776 + 0.2850349501519716i;
      0.339658279960361 + 0.03031451811355637i;
      -5.417624387291579 + 0i;
      0.2997244646075835 + 0.1578268372785777i;
      -7.102809895801842 + 0i];
ipiv = [int64(-4);-4;3;4];
[apOut, info] = f07pw(uplo, ap, ipiv)
 

apOut =

   0.0826 + 0.0000i
  -0.0335 + 0.0440i
   0.0603 - 0.0105i
   0.2391 - 0.0926i
  -0.1408 + 0.0000i
   0.0422 - 0.0222i
   0.0304 + 0.0203i
  -0.2007 + 0.0000i
   0.0982 - 0.0635i
   0.0073 + 0.0000i


info =

                    0



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