Contents

# ?sysv_rk

Computes the solution to system of linear equations A * X = B for
SY
matrices.
Description
?sysv_rk
computes the solution to a real or complex system of linear equations A * X = B, where A is an
n
-by-
n
symmetric matrix and X and B are
n
-by-
nrhs
matrices.
The bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as A= P*U*D*(U
T
)*(P
T
), if
uplo
=
'U'
, or A= P*L*D*(L
T
)*(P
T
), if
uplo
=
'L'
, 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.
?sytrf_rk
is called to compute the factorization of a real or complex symmetric matrix. The factored form of A is then used to solve the system of equations A * X = B by calling BLAS3 routine
?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 upper or lower triangular part of the symmetric matrix A is stored:
• =
'U'
: The upper triangle of A is stored.
• =
'L'
: The lower triangle of A is stored.
n
The number of linear equations; that is, the order of the matrix A.
n
≥ 0.
nrhs
The number of right-hand sides; that is, the number of columns of the matrix B.
nrhs
≥ 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
.
B
Array of size max(1,
ldb
*
nrhs
).
On entry, the
n
-by-
nrhs
right-hand side matrix B.
ldb
The leading dimension of the array
B
.
ldb
≥ max(1,
n
) for column-major layout and
ldb
nrhs
for row-major layout.
Output Parameters
A
On exit, if
info
= 0, the 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 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. For more information, see the description of the
?sytrf_rk
routine.
e
Array of size
n
.
On exit, contains the output computed by the factorization routine
?sytrf_rk
; that is, 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
=1:N
-
1, 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
) is set to 0 in both the
uplo
=
'U'
and
uplo
=
'L'
?sytrf_rk
routine.
ipiv
Array of size
n
.
Details of the interchanges and the block structure of D, as determined by
?sytrf_rk
?sytrf_rk
routine.
B
On exit, if
info
= 0, the
n
-by-
nrhs
solution matrix X.
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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