Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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

?hptrs

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

Syntax

lapack_int
LAPACKE_chptrs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
nrhs
,
const
lapack_complex_float
*
ap
,
const
lapack_int
*
ipiv
,
lapack_complex_float
*
b
,
lapack_int
ldb
);
lapack_int
LAPACKE_zhptrs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
nrhs
,
const
lapack_complex_double
*
ap
,
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 packed 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 arrays
ap
 (containing
U
 or
L
)and
ipiv
 in the form returned by the factorization routine
?hptrf
.
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
ap
 stores the packed factor
U
 of the factorization
A
 =
U*D*U
H
. If
uplo
 =
'L'
, the array
ap
 stores the packed 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
?hptrf
.
ap
The dimension of array
ap
 must be at least max(1,
n
(
n
+1)/2). The array
ap
 contains the factor
U
 or
L
, as specified by
uplo
, in packed storage (see Matrix Storage Schemes).
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 max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout.
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
 for complex flavors.
To estimate the condition number
κ
∞
(
A
)
, call
?hpcon
.
To refine the solution and estimate the error, call
?hprfs
.

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