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NAG Toolbox

NAG Toolbox: nag_lapack_zsytri (f07nw)

Purpose

nag_lapack_zsytri (f07nw) computes the inverse of a complex symmetric matrix AA, where AA has been factorized by nag_lapack_zsytrf (f07nr).

Syntax

[a, info] = f07nw(uplo, a, ipiv, 'n', n)
[a, info] = nag_lapack_zsytri(uplo, a, ipiv, 'n', n)

Description

nag_lapack_zsytri (f07nw) is used to compute the inverse of a complex symmetric matrix AA, the function must be preceded by a call to nag_lapack_zsytrf (f07nr), which computes the Bunch–Kaufman factorization of AA.
If uplo = 'U'uplo='U', A = PUDUTPTA=PUDUTPT and A1A-1 is computed by solving UTPTXPU = D1UTPTXPU=D-1 for XX.
If uplo = 'L'uplo='L', A = PLDLTPTA=PLDLTPT and A1A-1 is computed by solving LTPTXPL = D1LTPTXPL=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 = 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:     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_zsytrf (f07nr).
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_zsytrf (f07nr).

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 symmetric 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_zsytri_example
uplo = 'L';
a = [ -0.39 - 0.71i,  0 + 0i,  0 + 0i,  0 + 0i;
      5.14 - 0.64i,  8.86 + 1.81i,  0 + 0i,  0 + 0i;
      -7.86 - 2.96i,  -3.52 + 0.58i,  -2.83 - 0.03i,  0 + 0i;
      3.8 + 0.92i,  5.32 - 1.59i,  -1.54 - 2.86i,  -0.56 + 0.12i];
[a, ipiv, info] = nag_lapack_zsytrf(uplo, a);
[aOut, info] = nag_lapack_zsytri(uplo, a, ipiv)
 

aOut =

  -0.1562 - 0.1014i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0400 + 0.1527i   0.0946 - 0.1475i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0550 + 0.0845i  -0.0326 - 0.1370i  -0.1320 - 0.0102i   0.0000 + 0.0000i
   0.2162 - 0.0742i  -0.0995 - 0.0461i  -0.1793 + 0.1183i  -0.2269 + 0.2383i


info =

                    0


function f07nw_example
uplo = 'L';
a = [ -0.39 - 0.71i,  0 + 0i,  0 + 0i,  0 + 0i;
      5.14 - 0.64i,  8.86 + 1.81i,  0 + 0i,  0 + 0i;
      -7.86 - 2.96i,  -3.52 + 0.58i,  -2.83 - 0.03i,  0 + 0i;
      3.8 + 0.92i,  5.32 - 1.59i,  -1.54 - 2.86i,  -0.56 + 0.12i];
[a, ipiv, info] = f07nr(uplo, a);
[aOut, info] = f07nw(uplo, a, ipiv)
 

aOut =

  -0.1562 - 0.1014i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0400 + 0.1527i   0.0946 - 0.1475i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0550 + 0.0845i  -0.0326 - 0.1370i  -0.1320 - 0.0102i   0.0000 + 0.0000i
   0.2162 - 0.0742i  -0.0995 - 0.0461i  -0.1793 + 0.1183i  -0.2269 + 0.2383i


info =

                    0



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