Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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

?tprfs

Estimates the error in the solution of a system of linear equations with a packed triangular coefficient matrix.

Syntax

lapack_int LAPACKE_stprfs
(
int
matrix_layout
,
char
uplo
,
char
trans
,
char
diag
,
lapack_int
n
,
lapack_int
nrhs
,
const float*
ap
,
const float*
b
,
lapack_int
ldb
,
const float*
x
,
lapack_int
ldx
,
float*
ferr
,
float*
berr
);
lapack_int LAPACKE_dtprfs
(
int
matrix_layout
,
char
uplo
,
char
trans
,
char
diag
,
lapack_int
n
,
lapack_int
nrhs
,
const double*
ap
,
const double*
b
,
lapack_int
ldb
,
const double*
x
,
lapack_int
ldx
,
double*
ferr
,
double*
berr
);
lapack_int LAPACKE_ctprfs
(
int
matrix_layout
,
char
uplo
,
char
trans
,
char
diag
,
lapack_int
n
,
lapack_int
nrhs
,
const lapack_complex_float*
ap
,
const lapack_complex_float*
b
,
lapack_int
ldb
,
const lapack_complex_float*
x
,
lapack_int
ldx
,
float*
ferr
,
float*
berr
);
lapack_int LAPACKE_ztprfs
(
int
matrix_layout
,
char
uplo
,
char
trans
,
char
diag
,
lapack_int
n
,
lapack_int
nrhs
,
const lapack_complex_double*
ap
,
const lapack_complex_double*
b
,
lapack_int
ldb
,
const lapack_complex_double*
x
,
lapack_int
ldx
,
double*
ferr
,
double*
berr
);
Include Files
  • mkl.h
Description
The routine estimates the errors in the solution to a system of linear equations
A*X
 =
B
 or
A
T
*X
 =
B
 or
A
H
*X
 =
B
 with a packed triangular matrix
A
, with multiple right-hand sides. For each computed solution vector
x
, the routine computes the component-wise backward error
β
. This error is the smallest relative perturbation in elements of
A
 and
b
 such that
x
 is the exact solution of the perturbed system:
|
δ
a
i
j
|
≤
β
|
a
i
j
|, |
δ
b
i
|
≤
β
|
b
i
|
 such that
(
A
 +
δ
A
)
x
 = (
b
 +
δ
b
)
.
The routine also estimates the component-wise forward error in the computed solution
||
x
 -
x
e
||
∞
/||
x
||
∞
 (here
x
e
 is the exact solution).
Before calling this routine, call the solver routine
?tptrs
.
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
A
 is upper or lower triangular:
If
uplo
 =
'U'
, then
A
 is upper triangular.
If
uplo
 =
'L'
, then
A
 is lower triangular.
trans
Must be
'N'
 or
'T'
 or
'C'
.
Indicates the form of the equations:
If
trans
 =
'N'
, the system has the form
A*X
 =
B
.
If
trans
 =
'T'
, the system has the form
A
T
*X
 =
B
.
If
trans
 =
'C'
, the system has the form
A
H
*X
 =
B
.
diag
Must be
'N'
 or
'U'
.
If
diag
 =
'N'
,
A
 is not a unit triangular matrix.
If
diag
 =
'U'
,
A
 is unit triangular: diagonal elements of
A
 are assumed to be 1 and not referenced in the array
ap
.
n
The order of the matrix
A
;
n
≥
 0.
nrhs
The number of right-hand sides;
nrhs
≥
 0.
ap
,
b
,
x
Arrays:
ap
max(1,
n
(
n
 + 1)/2)
 contains the upper or lower triangular matrix
A
, as specified by
uplo
.
b
of size max(1,
ldb
*
nrhs
) for column major layout and max(1,
ldb
*
n
) for row major layout
 contains the right-hand side matrix
B
.
x
of size max(1,
ldx
*
nrhs
) for column major layout and max(1,
ldx
*
n
) for row major layout
 contains the solution matrix
X
.
ldb
The leading dimension of
b
;
ldb
≥
 max(1,
n
) for column major layout and
ldb
≥
nrhs
 for row major layout
.
ldx
The leading dimension of
x
;
ldb
≥
 max(1,
n
) for column major layout and
ldb
≥
nrhs
 for row major layout
.
Output Parameters
ferr
,
berr
Arrays, size at least
max(1,
nrhs
)
. Contain the component-wise forward and backward errors, respectively, for each solution vector.
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
The bounds returned in
ferr
 are not rigorous, but in practice they almost always overestimate the actual error.
A call to this routine involves, for each right-hand side, solving a number of systems of linear equations
A
*
x
 =
b
; the number of systems is usually 4 or 5 and never more than 11. Each solution requires approximately
n
2
 floating-point operations for real flavors or
4
n
2
 for complex flavors.

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