Contents

# ?sycon_3

Estimates the reciprocal of the condition number (in the 1-norm) of a real or complex symmetric matrix A using the factorization computed by
?sytrf_rk
.
Description
?sycon_3
estimates the reciprocal of the condition number (in the 1-norm) of a real or complex symmetric matrix A using the factorization computed by
?sytrf_rk
. 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.
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
?sytrs_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. The form is A = P*U*D*(U
T
)*(P
T
).
• =
'L'
: Lower triangular. The form is A = P*L*D*(L
T
)*(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 factors U or L as computed by
?sytrf_rk
:
• Only
diagonal elements of the symmetric 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
).
—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 symmetric 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
?sytrf_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