Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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

?sptrf

Computes the Bunch-Kaufman factorization of a symmetric matrix using packed storage.

Syntax

lapack_int
LAPACKE_ssptrf
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
float
*
ap
,
lapack_int
*
ipiv
);
lapack_int
LAPACKE_dsptrf
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
double
*
ap
,
lapack_int
*
ipiv
);
lapack_int
LAPACKE_csptrf
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_complex_float
*
ap
,
lapack_int
*
ipiv
);
lapack_int
LAPACKE_zsptrf
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_complex_double
*
ap
,
lapack_int
*
ipiv
);
Include Files
  • mkl.h
Description
The routine computes the factorization of a real/complex symmetric matrix
A
 stored in the packed format 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
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 Function 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 packed in the array
ap
 and how
A
 is factored:
If
uplo
 =
'U'
, the array
ap
 stores the upper triangular part of the matrix
A
, and
A
 is factored as
U*D*U
T
.
If
uplo
 =
'L'
, the array
ap
 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.
ap
Array, size at least max(1,
n
(
n
+1)/2). The array
ap
 contains the upper or the lower triangular part of the matrix
A
 (as specified by
uplo
) in packed storage.
Output Parameters
ap
The upper or lower triangle of
A
 (as specified by
uplo
) 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
 overwrite elements of the corresponding columns of the array
ap
, but additional row interchanges are required to recover
U
 or
L
 explicitly (which is seldom necessary).
If
ipiv
(
i
) =
i
 for all
i
 = 1...
n
, then all off-diagonal elements of
U
 (
L
) are stored explicitly in packed form.
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
.

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