Contents

# ?hecon_3

Estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian matrix A.
Description
?hecon_3
estimates the reciprocal of the condition number (in the 1-norm) 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 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. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as
rcond
= 1 / (
anorm
* norm(inv(A))).
This routine uses BLAS3 solver
?hetrs_3
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
uplo
Specifies whether the details of the factorization are stored as an upper or lower triangular matrix: =
'U'
: Upper triangular, form is A = P*U*D*(U
H
)*(P
T
); =
'L'
: Lower triangular, form is A = P*L*D*(L
H
)*(P
T
).
n
The order of the matrix A.
n
≥ 0.
A
Array of size max(1,
lda
*
n
).
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 must be provided on entry in array
e
.
—and—
• If
uplo
=
'U'
, factor U in the superdiagonal part of A. If
uplo
=
'L'
, factor L in the subdiagonal part of A.
lda
The leading dimension of the array
A
.
e
Array of size
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
-
1]
is not referenced in both the
uplo
=
'U'
and
uplo
=
'L'
cases.
ipiv
Array of size
n
.
Details of the interchanges and the block structure of D as determined by
?hetrf_rk
.
anorm
The 1-norm of the original matrix A.
Output Parameters
rcond
The reciprocal of the condition number of the matrix A, computed as
rcond
= 1/(
anorm
*
AINVNM
), where
AINVNM
is an estimate of the 1-norm of inv(A) computed in this routine.
Return Values
This function returns a value
info
.
= 0: Successful exit.
< 0: If
info
=
-i
, the
i
th