Developer Reference

  • 2021.1
  • 12/04/2020
  • Public Content
Contents

?sytrf_rook

Computes the bounded Bunch-Kaufman factorization of a symmetric matrix.

Syntax

lapack_int
LAPACKE_ssytrf_rook
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
float
*
a
,
lapack_int
lda
,
lapack_int
*
ipiv
);
lapack_int
LAPACKE_dsytrf_rook
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
double
*
a
,
lapack_int
lda
,
lapack_int
*
ipiv
);
lapack_int
LAPACKE_csytrf_rook
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_complex_float
*
a
,
lapack_int
lda
,
lapack_int
*
ipiv
);
lapack_int
LAPACKE_zsytrf_rook
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_complex_double
*
a
,
lapack_int
lda
,
lapack_int
*
ipiv
);
Include Files
  • mkl.h
Description
The routine computes the factorization of a real/complex symmetric matrix
A
using the bounded Bunch-Kaufman ("rook") diagonal pivoting method. The form of the factorization is:
  • if
    uplo
    =
    'U'
    ,
    A
    =
    U*D*U
    T
  • if
    uplo
    =
    'L'
    ,
    A
    =
    L*D*L
    T
    ,
where
A
is the input matrix,
U
and
L
are products of permutation and triangular matrices with unit diagonal (upper triangular for
U
and lower triangular for
L
), and
D
is a symmetric block-diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks.
U
and
L
have 2-by-2 unit diagonal blocks corresponding to the 2-by-2 blocks of
D
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout for array
b
is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
uplo
Must be
'U'
or
'L'
.
Indicates whether the upper or lower triangular part of
A
is stored and how
A
is factored:
If
uplo
=
'U'
, the array
a
stores the upper triangular part of the matrix
A
, and
A
is factored as
U*D*U
T
.
If
uplo
=
'L'
, the array
a
stores the lower triangular part of the matrix
A
, and
A
is factored as
L*D*L
T
.
n
The order of matrix
A
;
n
0.
a
Array, size
lda
*
n
. The array
a
contains either the upper or the lower triangular part of the matrix
A
(see
uplo
).
lda
The leading dimension of
a
; at least
max(1,
n
)
.
Output Parameters
a
The upper or lower triangular part of
a
is overwritten by details of the block-diagonal matrix
D
and the multipliers used to obtain the factor
U
(or
L
).
ipiv
If
ipiv
(
k
) > 0
, then rows and columns
k
and
ipiv
(
k
) were interchanged and
D
k
,
k
is a 1-by-1 diagonal block.
If
uplo
=
'U'
and
ipiv
(
k
) < 0
and
ipiv
(
k
- 1) < 0
, then rows and columns
k
and -
ipiv
(
k
) were interchanged, rows and columns
k
- 1 and -
ipiv
(
k
- 1) were interchanged, and
D
k
-1:
k
,
k
-1:
k
is a 2-by-2 diagonal block.
If
uplo
=
'L'
and
ipiv
(
k
) < 0
and
ipiv
(
k
+ 1) < 0
, then rows and columns
k
and
-ipiv
(
k
) were interchanged, rows and columns
k
+ 1 and
-ipiv
(
k
+ 1) were interchanged, and
D
k
:
k
+1,
k
:
k
+1
is a 2-by-2 diagonal block.
Return Values
This function returns a value
info
.
If
info
= 0
, the execution is successful.
If
info
=
-i
, the
i-
th parameter had an illegal value.
If
info
=
i
,
D
i
i
is 0. The factorization has been completed, but
D
is exactly singular. Division by 0 will occur if you use
D
for solving a system of linear equations.
Application Notes
The total number of floating-point operations is approximately
(1/3)
n
3
for real flavors or
(4/3)
n
3
for complex flavors.
After calling this routine, you can call the following routines:
to solve
A*X
=
B
?sycon_rook
(Fortran only)
to estimate the condition number of
A
?sytri_rook
(Fortran only)
to compute the inverse of
A
.
 
If
uplo
=
'U'
, then
A
=
U
*
D
*
U'
, where
U = P(n)*U(n)* ... *P(k)*U(k)*...,
that is,
U
is a product of terms
P
(
k
)*
U
(
k
), where
  • k
    decreases from
    n
    to 1 in steps of 1 and 2.
  • D
    is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks
    D
    (
    k
    ).
  • P
    (
    k
    ) is a permutation matrix as defined by
    ipiv
    [
    k
    -1]
    .
  • U
    (
    k
    ) is a unit upper triangular matrix, such that if the diagonal block
    D
    (
    k
    ) is of order
    s
    (
    s
    = 1 or 2), then
    Equation
If
s
= 1,
D
(
k
) overwrites
A
(
k
,
k
), and
v
overwrites
A
(1:
k
-1,
k
).
If
s
= 2, the upper triangle of
D
(
k
) overwrites
A
(
k
-1,
k
-1),
A
(
k
-1,
k
) and
A
(
k
,
k
), and
v
overwrites
A
(1:
k
-2,
k
-1:
k
).
 
If
uplo
=
'L'
, then
A
=
L
*
D
*
L'
, where
L = P(1)*L(1)* ... *P(k)*L(k)*...,
that is,
L
is a product of terms
P
(
k
)*
L
(
k
), where
  • k
    increases from 1 to
    n
    in steps of 1 and 2.
  • D
    is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks
    D
    (
    k
    ).
  • P
    (
    k
    ) is a permutation matrix as defined by
    ipiv
    (
    k
    ).
  • L
    (
    k
    ) is a unit lower triangular matrix, such that if the diagonal block
    D
    (
    k
    ) is of order
    s
    (
    s
    = 1 or 2), then
    Equation
If
s
= 1,
D
(
k
) overwrites
A
(
k
,
k
), and
v
overwrites
A
(
k
+1:
n
,
k
).
If
s
= 2, the lower triangle of
D
(
k
) overwrites
A
(
k
,
k
),
A
(
k
+1,
k
), and
A
(
k
+1,
k
+1), and
v
overwrites
A
(
k
+2:
n
,
k
:
k
+1).

Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.