Developer Reference

  • 0.9
  • 09/09/2020
  • Public Content
Contents

?sytrf

Computes the Bunch-Kaufman factorization of a symmetric matrix.

Syntax

lapack_int
LAPACKE_ssytrf
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
float
*
a
,
lapack_int
lda
,
lapack_int
*
ipiv
);
lapack_int
LAPACKE_dsytrf
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
double
*
a
,
lapack_int
lda
,
lapack_int
*
ipiv
);
lapack_int
LAPACKE_csytrf
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_complex_float
*
a
,
lapack_int
lda
,
lapack_int
*
ipiv
);
lapack_int
LAPACKE_zsytrf
(
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 Bunch-Kaufman 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
.
This routine supports the Progress Routine feature. See Progress Routine for details.
Input Parameters
matrix_layout
Specifies whether matrix storage layout 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
max(1,
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
Array, size at least
max(1,
n
)
. Contains details of the interchanges and the block structure of
D
. If
ipiv
[
i
-1] =
k
>0
, then
d
i
i
is a 1-by-1 block, and the
i
-th row and column of
A
was interchanged with the
k
-th row and column.
If
uplo
=
'U'
and
ipiv
[
i
] =
ipiv
[
i
-1] = -
m
< 0, then
D
has a 2-by-2 block in rows/columns
i
and
i
+1
, and
i
-th row and column of
A
was interchanged with the
m
-th row and column.
If
uplo
=
'L'
and
ipiv
[
i
] =
ipiv
[
i
-1] = -
m
< 0, then
D
has a 2-by-2 block in rows/columns
i
and
i
+1, and (
i
+1)-th row and column of
A
was interchanged with the
m
-th row and column.
Return Values
This function returns a value
info
.
If
info
= 0
, the execution is successful.
If
info
=
-i
, parameter
i
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 2-by-2 unit diagonal blocks and the unit diagonal elements of
U
and
L
are not stored. The remaining elements of
U
and
L
are stored in the corresponding columns of the array
a
, but additional row interchanges are required to recover
U
or
L
explicitly (which is seldom necessary).
If
ipiv
[
i
-1] =
i
for all
i
=1...
n
, then all off-diagonal elements of
U
(
L
) are stored explicitly in the corresponding elements of the array
a
.
If
uplo
=
'U'
, the computed factors
U
and
D
are the exact factors of a perturbed matrix
A
+
E
, where
|E| ≤ c(n)ε P|U||D||U
T
|P
T
c
(
n
)
is a modest linear function of
n
, and
ε
is the machine precision. A similar estimate holds for the computed
L
and
D
when
uplo
=
'L'
.
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
to estimate the condition number of
A
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

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804