## Developer Reference

• 2020.2
• 07/15/2020
• Public Content
Contents

# ?hetri_3

Computes the inverse of a complex Hermitian matrix using the factorization computed by
?hetrf_rk
.
Description
?hetri_3
computes the inverse of a complex Hermitian matrix A using the factorization computed by
?hetrf_rk
: A = P*U*D*(U
H
)*(P
T
) or A = P*L*D*(L
H
)*(P
T
), where U (or L) is a 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 a Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.
?hetri_3
sets the leading dimension of the workspace before calling
?hetri_3x
, which actually computes the inverse.
This is the blocked version of the algorithm, calling Level-3 BLAS.
Input Parameters
uplo
CHARACTER*1
Specifies whether the details of the factorization are stored as an upper or lower triangular matrix.
• =
'U'
: The upper triangle of A is stored.
• =
'L'
: The lower triangle of A is stored.
n
INTEGER
The order of the matrix A.
n
≥ 0.
A
COMPLEX
for
chetri_3
COMPLEX*16
for
zhetri_3
Array, dimension (
lda
,
n
).
On entry, diagonal of the block diagonal matrix D and factor U or L as computed by
?hetrf_rk
:
• Only
diagonal elements of the Hermitian block diagonal matrix D on the diagonal of A; that is, D(
k
,
k
) = A(
k
,
k
). Superdiagonal (or subdiagonal) elements of D should be provided on entry in array
e
.
• If
uplo
=
'U'
, factor U in the superdiagonal part of A. If
uplo
=
'L'
, factor L is the subdiagonal part of A.
lda
INTEGER
The leading dimension of the array
A
.
lda
≥ max(1,
n
).
e
COMPLEX
for
chetri_3
COMPLEX*16
for
zhetri_3
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. If
uplo
=
'U'
, e(
i
) = D(
i
-
1,
i
),
i
=2:N, and e(1) is not referenced. If
uplo
=
'L'
, e(
i
) = D(
i
+1,
i
),
i
=1:N
-
1, and e(
n
) is not referenced.
For 1-by-1 diagonal block D(
k
), where 1 ≤
k
n
, the element
e
(
k
) is not referenced in both the
uplo
=
'U'
and
uplo
=
'L'
cases.
ipiv
INTEGER
Array, dimension (
n
).
Details of the interchanges and the block structure of D as determined by
?hetrf_rk
.
lwork
INTEGER
The length of the array
work
.
If
LDWORK
=
-
1, a workspace query is assumed; the routine calculates only the optimal size of the
work
array and returns this value as the first entry of the
work
array, and no error message related to
lwork
is issued by XERBLA.
Output Parameters
A
COMPLEX
for
chetri_3
COMPLEX*16
for
zhetri_3
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.
work
COMPLEX
for
chetri_3
COMPLEX*16
for
zhetri_3
Array, dimension (
n
+
NB
+1)*(
NB
+3). On exit, if
info
= 0,
work
(1) returns the optimal
lwork
.
info
INTEGER
• = 0: Successful exit.