# hetri_3x - Man Page

{he,sy}tri_3x: inverse

## Synopsis

### Functions

subroutine chetri_3x (uplo, n, a, lda, e, ipiv, work, nb, info)
CHETRI_3X
subroutine csytri_3x (uplo, n, a, lda, e, ipiv, work, nb, info)
CSYTRI_3X
subroutine dsytri_3x (uplo, n, a, lda, e, ipiv, work, nb, info)
DSYTRI_3X
subroutine ssytri_3x (uplo, n, a, lda, e, ipiv, work, nb, info)
SSYTRI_3X
subroutine zhetri_3x (uplo, n, a, lda, e, ipiv, work, nb, info)
ZHETRI_3X
subroutine zsytri_3x (uplo, n, a, lda, e, ipiv, work, nb, info)
ZSYTRI_3X

## Function Documentation

### subroutine chetri_3x (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, complex, dimension( n+nb+1, * ) work, integer nb, integer info)

CHETRI_3X

Purpose:

``` CHETRI_3X computes the inverse of a complex Hermitian indefinite
matrix A using the factorization computed by CHETRF_RK or CHETRF_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.

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 array, dimension (LDA,N)
On entry, diagonal of the block diagonal matrix D and
factors U or L as computed by CHETRF_RK and CHETRF_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 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 CHETRF_RK or CHETRF_BK.```

WORK

`          WORK is COMPLEX array, dimension (N+NB+1,NB+3).`

NB

```          NB is INTEGER
Block size.```

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:

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

Definition at line 158 of file chetri_3x.f.

### subroutine csytri_3x (character uplo, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( * ) e, integer, dimension( * ) ipiv, complex, dimension( n+nb+1, * ) work, integer nb, integer info)

CSYTRI_3X

Purpose:

``` CSYTRI_3X computes the inverse of a complex symmetric indefinite
matrix A using the factorization computed by CSYTRF_RK or CSYTRF_BK:

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

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

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 array, dimension (LDA,N)
On entry, diagonal of the block diagonal matrix D and
factors U or L as computed by CSYTRF_RK and CSYTRF_BK:
a) ONLY diagonal elements of the symmetric 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 symmetric 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 array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric 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 CSYTRF_RK or CSYTRF_BK.```

WORK

`          WORK is COMPLEX array, dimension (N+NB+1,NB+3).`

NB

```          NB is INTEGER
Block size.```

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:

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

Definition at line 158 of file csytri_3x.f.

### subroutine dsytri_3x (character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) e, integer, dimension( * ) ipiv, double precision, dimension( n+nb+1, * ) work, integer nb, integer info)

DSYTRI_3X

Purpose:

``` DSYTRI_3X computes the inverse of a real symmetric indefinite
matrix A using the factorization computed by DSYTRF_RK or DSYTRF_BK:

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

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

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 DOUBLE PRECISION array, dimension (LDA,N)
On entry, diagonal of the block diagonal matrix D and
factors U or L as computed by DSYTRF_RK and DSYTRF_BK:
a) ONLY diagonal elements of the symmetric 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 symmetric 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 DOUBLE PRECISION array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric 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 DSYTRF_RK or DSYTRF_BK.```

WORK

`          WORK is DOUBLE PRECISION array, dimension (N+NB+1,NB+3).`

NB

```          NB is INTEGER
Block size.```

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:

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

Definition at line 158 of file dsytri_3x.f.

### subroutine ssytri_3x (character uplo, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) e, integer, dimension( * ) ipiv, real, dimension( n+nb+1, * ) work, integer nb, integer info)

SSYTRI_3X

Purpose:

``` SSYTRI_3X computes the inverse of a real symmetric indefinite
matrix A using the factorization computed by SSYTRF_RK or SSYTRF_BK:

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

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

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 REAL array, dimension (LDA,N)
On entry, diagonal of the block diagonal matrix D and
factors U or L as computed by SYTRF_RK and SSYTRF_BK:
a) ONLY diagonal elements of the symmetric 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 symmetric 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 REAL array, dimension (N)
On entry, contains the superdiagonal (or subdiagonal)
elements of the symmetric 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 SSYTRF_RK or SSYTRF_BK.```

WORK

`          WORK is REAL array, dimension (N+NB+1,NB+3).`

NB

```          NB is INTEGER
Block size.```

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:

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

Definition at line 158 of file ssytri_3x.f.

### subroutine zhetri_3x (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, complex*16, dimension( n+nb+1, * ) work, integer nb, integer info)

ZHETRI_3X

Purpose:

``` ZHETRI_3X 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.

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).`

NB

```          NB is INTEGER
Block size.```

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:

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

Definition at line 158 of file zhetri_3x.f.

### subroutine zsytri_3x (character uplo, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, complex*16, dimension( n+nb+1, * ) work, integer nb, integer info)

ZSYTRI_3X

Purpose:

``` ZSYTRI_3X computes the inverse of a complex symmetric indefinite
matrix A using the factorization computed by ZSYTRF_RK or ZSYTRF_BK:

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

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

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 ZSYTRF_RK and ZSYTRF_BK:
a) ONLY diagonal elements of the symmetric 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 symmetric 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 symmetric 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 ZSYTRF_RK or ZSYTRF_BK.```

WORK

`          WORK is COMPLEX*16 array, dimension (N+NB+1,NB+3).`

NB

```          NB is INTEGER
Block size.```

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:

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

Definition at line 158 of file zsytri_3x.f.

## Author

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## Info

Tue Nov 28 2023 12:08:43 Version 3.12.0 LAPACK