Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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

?pbrfs

Refines the solution of a system of linear equations with a band symmetric (Hermitian) positive-definite coefficient matrix and estimates its error.

Syntax

lapack_int LAPACKE_spbrfs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const float*
ab
,
lapack_int
ldab
,
const float*
afb
,
lapack_int
ldafb
,
const float*
b
,
lapack_int
ldb
,
float*
x
,
lapack_int
ldx
,
float*
ferr
,
float*
berr
);
lapack_int LAPACKE_dpbrfs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const double*
ab
,
lapack_int
ldab
,
const double*
afb
,
lapack_int
ldafb
,
const double*
b
,
lapack_int
ldb
,
double*
x
,
lapack_int
ldx
,
double*
ferr
,
double*
berr
);
lapack_int LAPACKE_cpbrfs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const lapack_complex_float*
ab
,
lapack_int
ldab
,
const lapack_complex_float*
afb
,
lapack_int
ldafb
,
const lapack_complex_float*
b
,
lapack_int
ldb
,
lapack_complex_float*
x
,
lapack_int
ldx
,
float*
ferr
,
float*
berr
);
lapack_int LAPACKE_zpbrfs
(
int
matrix_layout
,
char
uplo
,
lapack_int
n
,
lapack_int
kd
,
lapack_int
nrhs
,
const lapack_complex_double*
ab
,
lapack_int
ldab
,
const lapack_complex_double*
afb
,
lapack_int
ldafb
,
const lapack_complex_double*
b
,
lapack_int
ldb
,
lapack_complex_double*
x
,
lapack_int
ldx
,
double*
ferr
,
double*
berr
);
Include Files
  • mkl.h
Description
The routine performs an iterative refinement of the solution to a system of linear equations
A*X
 =
B
 with a symmetric (Hermitian) positive definite band 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
)
.
Finally, the routine 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 factorization routine
    ?pbtrf
  • call the solver routine
    ?pbtrs
    .
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 upper triangle of
A
 is stored.
If
uplo
 =
'L'
, the lower triangle of
A
 is stored.
n
The order of the matrix
A
;
n
≥
 0.
kd
The number of superdiagonals or subdiagonals in the matrix
A
;
kd
≥
 0.
nrhs
The number of right-hand sides;
nrhs
≥
 0.
ab
Array
ab
 (size max(
ldab
*
n
)) contains the original band matrix
A
, as supplied to
?pbtrf
.
afb
Array
afb
 (size max(
ldafb
*
n
)) contains the factored band matrix
A
, as returned by
?pbtrf
.
b
Array
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
Array
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
.
ldab
The leading dimension of
ab
;
ldab
≥
kd
 + 1.
ldafb
The leading dimension of
afb
;
ldafb
≥
kd
 + 1.
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
;
ldx
≥
 max(1,
n
) for column major layout and
ldx
≥
nrhs
 for row major layout
.
Output Parameters
x
The refined solution matrix
X
.
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.
For each right-hand side, computation of the backward error involves a minimum of
8
n
*
kd
 floating-point operations (for real flavors) or
32
n
*
kd
 operations (for complex flavors). In addition, each step of iterative refinement involves
12
n
*
kd
 operations (for real flavors) or
48
n
*
kd
 operations (for complex flavors); the number of iterations may range from 1 to 5.
Estimating the forward error involves solving a number of systems of linear equations
A
*
x
 =
b
; the number is usually 4 or 5 and never more than 11. Each solution requires approximately
4
n
*
kd
 floating-point operations for real flavors or
16
n
*
kd
 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