Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

Contents

?hetrs

Solves a system of linear equations with a UDU
T
- or LDL
T
-factored Hermitian coefficient matrix.

Syntax

lapack_int
LAPACKE_chetrs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
nrhs
,
const
lapack_complex_float
*
a
,
lapack_int
lda
,
const
lapack_int
*
ipiv
,
lapack_complex_float
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_zhetrs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
nrhs
,
const
lapack_complex_double
*
a
,
lapack_int
lda
,
const
lapack_int
*
ipiv
,
lapack_complex_double
*
b
,
lapack_int
ldb
);
Include Files
  • mkl.h
Description
The routine solves for
X
 the system of linear equations
A*X
 =
B
 with a Hermitian matrix
A
, given the Bunch-Kaufman factorization of
A
:
if
uplo
=
'U'
,
A
 =
U*D*U
H
if
uplo
=
'L'
,
A
 =
L*D*L
H
,
where
U
 and
L
 are upper and lower triangular matrices with unit diagonal and
D
 is a symmetric block-diagonal matrix. The system is solved with multiple right-hand sides stored in the columns of the matrix
B
. You must supply to this routine the factor
U
 (or
L
) and the array
ipiv
 returned by the factorization routine
?hetrf
.
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 how the input matrix
A
 has been factored:
If
uplo
 =
'U'
, the array
a
 stores the upper triangular factor
U
 of the factorization
A
 =
U*D*U
H
.
If
uplo
 =
'L'
, the array
a
 stores the lower triangular factor
L
 of the factorization
A
 =
L*D*L
H
.
n
The order of matrix
A
;
n
≥
 0.
nrhs
The number of right-hand sides;
nrhs
≥
 0.
ipiv
Array, size at least
max(1,
n
)
.
The
ipiv
 array, as returned by
?hetrf
.
a
The array
a
of size max(1,
lda
*
n
)
contains the factor
U
 or
L
 (see
uplo
).
b
The array
b
 contains the matrix
B
 whose columns are the right-hand sides for the system of equations.
The size of
b
 is at least max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout.
lda
The leading dimension of
a
;
lda
≥
 max(1,
n
)
.
ldb
The leading dimension of
b
;
ldb
≥
 max(1,
n
) for column major layout and
ldb
≥
nrhs
 for row major layout
.
Output Parameters
b
Overwritten by the solution matrix
X
.
Return Values
This function returns a value
info
.
If
info
=0
, the execution is successful.
If
info
 =
-i
, parameter
i
 had an illegal value.
Application Notes
For each right-hand side
b
, the computed solution is the exact solution of a perturbed system of equations
(
A
 +
E
)
x
 =
b
, where 
|E| ≤ c(n)ε P|U||D||U
H
|P
T
 or |E| ≤ c(n)ε P|L||D||L
H
|P
T
c
(
n
)
 is a modest linear function of
n
, and
ε
 is the machine precision.
If
x
0
 is the true solution, the computed solution
x
 satisfies this error bound:
Equation
where
cond(
A
,
x
)
= || |
A
-1
||
A
| |
x
| ||
∞
 / ||
x
||
∞
≤
 ||
A
-1
||
∞
 ||
A
||
∞
 =
κ
∞
(
A
).
Note that
cond(
A
,
x
)
 can be much smaller than
κ
∞
(
A
)
.
The total number of floating-point operations for one right-hand side vector is approximately
8
n
2
.
To estimate the condition number
κ
∞
(
A
)
, call
?hecon
.
To refine the solution and estimate the error, call
?herfs
.

Product and Performance Information

1

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