# zhetri_3.f - Man Page

SRC/zhetri_3.f

## Synopsis

### Functions/Subroutines

subroutine zhetri_3 (uplo, n, a, lda, e, ipiv, work, lwork, info)
ZHETRI_3

## Function/Subroutine Documentation

### subroutine zhetri_3 (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, complex*16, dimension( * ) work, integer lwork, integer info)

ZHETRI_3

Purpose:

``` ZHETRI_3 computes the inverse of a complex Hermitian indefinite
matrix A using the factorization computed by ZHETRF_RK or ZHETRF_BK:

A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),

where U (or L) is unit upper (or lower) triangular matrix,
U**H (or L**H) is the conjugate of U (or L), P is a permutation
matrix, P**T is the transpose of P, and D is Hermitian and block
diagonal with 1-by-1 and 2-by-2 diagonal blocks.

ZHETRI_3 sets the leading dimension of the workspace  before calling
ZHETRI_3X that actually computes the inverse.  This is the blocked
version of the algorithm, calling Level 3 BLAS.```
Parameters

UPLO

```          UPLO is CHARACTER*1
Specifies whether the details of the factorization are
stored as an upper or lower triangular matrix.
= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.```

N

```          N is INTEGER
The order of the matrix A.  N >= 0.```

A

```          A is COMPLEX*16 array, dimension (LDA,N)
On entry, diagonal of the block diagonal matrix D and
factors U or L as computed by ZHETRF_RK and ZHETRF_BK:
a) ONLY diagonal elements of the Hermitian block diagonal
matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
(superdiagonal (or subdiagonal) elements of D
should be provided on entry in array E), and
b) If UPLO = 'U': factor U in the superdiagonal part of A.
If UPLO = 'L': factor L in the subdiagonal part of A.

On exit, if INFO = 0, the Hermitian inverse of the original
matrix.
If UPLO = 'U': the upper triangular part of the inverse
is formed and the part of A below the diagonal is not
referenced;
If UPLO = 'L': the lower triangular part of the inverse
is formed and the part of A above the diagonal is not
referenced.```

LDA

```          LDA is INTEGER
The leading dimension of the array A.  LDA >= max(1,N).```

E

```          E is COMPLEX*16 array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the Hermitian block diagonal matrix D
with 1-by-1 or 2-by-2 diagonal blocks, where
If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.

NOTE: For 1-by-1 diagonal block D(k), where
1 <= k <= N, the element E(k) is not referenced in both
UPLO = 'U' or UPLO = 'L' cases.```

IPIV

```          IPIV is INTEGER array, dimension (N)
Details of the interchanges and the block structure of D
as determined by ZHETRF_RK or ZHETRF_BK.```

WORK

```          WORK is COMPLEX*16 array, dimension (N+NB+1)*(NB+3).
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.```

LWORK

```          LWORK is INTEGER
The length of WORK. LWORK >= (N+NB+1)*(NB+3).

If LDWORK = -1, then a workspace query is assumed;
the routine only calculates the optimal size of the optimal
size of the WORK array, returns this value as the first
entry of the WORK array, and no error message related to
LWORK is issued by XERBLA.```

INFO

```          INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Contributors:

```  November 2017,  Igor Kozachenko,
Computer Science Division,
University of California, Berkeley```

Definition at line 168 of file zhetri_3.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page zhetri_3(3) is an alias of zhetri_3.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK