Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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

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