Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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

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