Contents

# ?sytrf_rk

Computes the factorization of a real or complex symmetric indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
Description
?sytrf_rk
computes the factorization of a real or complex symmetric matrix A using the bounded Bunch-Kaufman (rook) diagonal pivoting method: 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.
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 upper or lower triangular part of the symmetric matrix A is stored:
• =
'U'
: Upper triangular
• =
'L'
: Lower triangular
n
The order of the matrix A.
n
≥ 0.
A
Array of size max(1,
lda
*
n
).
On entry, the symmetric matrix A. If
uplo
=
'U'
n
-by-
n
upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If
uplo
=
'L'
n
-by-
n
lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.
lda
The leading dimension of the array
A
.
Output Parameters
A
On exit, contains:
• 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 are stored on exit in array
e
).
• If
uplo
=
'U'
, factor U in the superdiagonal part of A. If
uplo
=
'L'
, factor L in the subdiagonal part of A.
e
Array of size
n
.
On exit, 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 set to 0. If
uplo
=
'L'
, e(
i
) = D(
i
+1,
i
),
i
=1:N
-
1, and e(
n
) is set to 0.
For 1-by-1 diagonal block D(
k
), where 1 ≤
k
n
, the element
e
[
k
-
1] is set to 0 in both the
uplo
=
'U'
and
uplo
=
'L'
cases.
ipiv
Array of size
n
.
ipiv
describes the permutation matrix P in the factorization of matrix A as follows: The absolute value of ipiv(
k
) represents the index of the row and column that were interchanged with the
k
th
row and column. The value of
uplo
describes the order in which the interchanges were applied. Also, the sign of
ipiv
represents the block structure of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks, which correspond to 1 or 2 interchanges at each factorization step. If
uplo
=
'U'
(in factorization order,
k
decreases from
n
to 1):
1. A single positive entry ipiv(
k
) > 0 means that D(
k
,
k
) is a 1-by-1 diagonal block. If ipiv(
k
) !=
k
, rows and columns
k
and ipiv(
k
) were interchanged in the matrix A(1:N,1:N). If ipiv(
k
) =
k
, no interchange occurred.
2. A pair of consecutive negative entries ipiv(
k
) < 0 and ipiv(
k
-
1). < 0 means that D(
k
-
1:
k
,
k
-
1:
k
) is a 2-by-2 diagonal block. (Note that negative entries in
ipiv
appear
only
in pairs.)
• If
-
ipiv(
k
) !=
k
, rows and columns
k
and
-
ipiv(
k
) were interchanged in the matrix A(1:N,1:N). If
-
ipiv(
k
) =
k
, no interchange occurred.
• If
-
ipiv(
k
-
1) !=
k
-
1, rows and columns
k
-
1 and
-
ipiv(
k
-
1) were interchanged in the matrix A(1:N,1:N). If
-
ipiv(
k
-
1) =
k
-
1, no interchange occurred.
3. In both cases 1 and 2, always ABS( ipiv(
k
) ) ≤
k
.
Any entry ipiv(
k
) is always nonzero on output.
If
uplo
=
'L'
(in factorization order,
k
increases from 1 to
n
):
1. A single positive entry ipiv(
k
) > 0 means that D(
k
,
k
) is a 1-by-1 diagonal block. If ipiv(
k
) !=
k
, rows and columns
k
and ipiv(
k
) were interchanged in the matrix A(1:N,1:N). If ipiv(
k
) =
k
, no interchange occurred.
2. A pair of consecutive negative entries ipiv(
k
) < 0 and ipiv(
k
+1) < 0 means that D(
k
:
k
+1,
k
:
k
+1) is a 2-by-2 diagonal block. (Note that negative entries in
ipiv
appear
only
in pairs.)
• If
-
ipiv(
k
) !=
k
, rows and columns
k
and
-
ipiv(
k
) were interchanged in the matrix A(1:N,1:N). If
-
ipiv(
k
) =
k
, no interchange occurred.
• If
-
ipiv(
k
+1) !=
k
+1, rows and columns
k
-
1 and
-
ipiv(
k
-
1) were interchanged in the matrix A(1:N,1:N). If
-
ipiv(
k
+1) =
k
+1, no interchange occurred.
3. In both cases 1 and 2, always ABS( ipiv(
k
) ) ≥
k
.
Any entry ipiv(
k
) is always nonzero on output.
Return Values
This function returns a value
info
.
= 0: Successful exit.
< 0: If
info
=
-
k
, the
k
th
> 0: If
info
=
k
, the matrix A is singular. If
uplo
=
'U'
, column
k
in the upper triangular part of A contains all zeros. If
uplo
=
'L'
, column
k
in the lower triangular part of A contains all zeros. Therefore, D(
k
,
k
) is exactly zero, and superdiagonal elements of column
k
of U (or subdiagonal elements of column
k
of L) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.

#### Product and Performance Information

1

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