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NAG Toolbox: nag_lapack_zhetri (f07mw)

Purpose

nag_lapack_zhetri (f07mw) computes the inverse of a complex Hermitian indefinite matrix AA, where AA has been factorized by nag_lapack_zhetrf (f07mr).

Syntax

[a, info] = f07mw(uplo, a, ipiv, 'n', n)
[a, info] = nag_lapack_zhetri(uplo, a, ipiv, 'n', n)

Description

nag_lapack_zhetri (f07mw) is used to compute the inverse of a complex Hermitian indefinite matrix AA, the function must be preceded by a call to nag_lapack_zhetrf (f07mr), which computes the Bunch–Kaufman factorization of AA.
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:     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)
Details of the factorization of AA, as returned by nag_lapack_zhetrf (f07mr).
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_zhetrf (f07mr).

Optional Input Parameters

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

Input Parameters Omitted from the MATLAB Interface

lda work

Output Parameters

1:     a(lda, : :) – complex array
The first dimension of the array a will be max (1,n)max(1,n)
The second dimension of the array will be max (1,n)max(1,n)
ldamax (1,n)ldamax(1,n).
The factorization stores the nn by nn Hermitian matrix A1A-1.
If uplo = 'U'uplo='U', the upper triangle of A1A-1 is stored in the upper triangular part of the array.
If uplo = 'L'uplo='L', the lower triangle of A1A-1 is stored in the lower triangular part of the array.
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: a, 4: lda, 5: ipiv, 6: work, 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.
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_dsytri (f07mj).

Example

function nag_lapack_zhetri_example
uplo = 'L';
a = [complex(-1.36),  0 + 0i,  0 + 0i,  0 + 0i;
      1.58 - 0.9i,  -8.87 + 0i,  0 + 0i,  0 + 0i;
      2.21 + 0.21i,  -1.84 + 0.03i,  -4.63 + 0i,  0 + 0i;
      3.91 - 1.5i,  -1.78 - 1.18i,  0.11 - 0.11i,  -1.84 + 0i];
[a, ipiv, info] = nag_lapack_zhetrf(uplo, a);
[aOut, info] = nag_lapack_zhetri(uplo, a, ipiv)
 

aOut =

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


info =

                    0


function f07mw_example
uplo = 'L';
a = [complex(-1.36),  0 + 0i,  0 + 0i,  0 + 0i;
      1.58 - 0.9i,  -8.87 + 0i,  0 + 0i,  0 + 0i;
      2.21 + 0.21i,  -1.84 + 0.03i,  -4.63 + 0i,  0 + 0i;
      3.91 - 1.5i,  -1.78 - 1.18i,  0.11 - 0.11i,  -1.84 + 0i];
[a, ipiv, info] = f07mr(uplo, a);
[aOut, info] = f07mw(uplo, a, ipiv)
 

aOut =

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


info =

                    0



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