Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

  • 2021.1
  • 12/04/2020
  • 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

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