Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

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

?gbrfs

Refines the solution of a system of linear equations with a general band coefficient matrix and estimates its error.

Syntax

lapack_int LAPACKE_sgbrfs
(
int
matrix_layout
,
char
trans
,
lapack_int
n
,
lapack_int
kl
,
lapack_int
ku
,
lapack_int
nrhs
,
const float*
ab
,
lapack_int
ldab
,
const float*
afb
,
lapack_int
ldafb
,
const lapack_int*
ipiv
,
const float*
b
,
lapack_int
ldb
,
float*
x
,
lapack_int
ldx
,
float*
ferr
,
float*
berr
);
lapack_int LAPACKE_dgbrfs
(
int
matrix_layout
,
char
trans
,
lapack_int
n
,
lapack_int
kl
,
lapack_int
ku
,
lapack_int
nrhs
,
const double*
ab
,
lapack_int
ldab
,
const double*
afb
,
lapack_int
ldafb
,
const lapack_int*
ipiv
,
const double*
b
,
lapack_int
ldb
,
double*
x
,
lapack_int
ldx
,
double*
ferr
,
double*
berr
);
lapack_int LAPACKE_cgbrfs
(
int
matrix_layout
,
char
trans
,
lapack_int
n
,
lapack_int
kl
,
lapack_int
ku
,
lapack_int
nrhs
,
const lapack_complex_float*
ab
,
lapack_int
ldab
,
const lapack_complex_float*
afb
,
lapack_int
ldafb
,
const lapack_int*
ipiv
,
const lapack_complex_float*
b
,
lapack_int
ldb
,
lapack_complex_float*
x
,
lapack_int
ldx
,
float*
ferr
,
float*
berr
);
lapack_int LAPACKE_zgbrfs
(
int
matrix_layout
,
char
trans
,
lapack_int
n
,
lapack_int
kl
,
lapack_int
ku
,
lapack_int
nrhs
,
const lapack_complex_double*
ab
,
lapack_int
ldab
,
const lapack_complex_double*
afb
,
lapack_int
ldafb
,
const lapack_int*
ipiv
,
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
 or
A
T
*X
 =
B
 or
A
H
*X
 =
B
 with a 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
    ?gbtrf
  • call the solver routine
    ?gbtrs
    .
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
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
.
n
The order of the matrix
A
;
n
≥
 0.
kl
The number of sub-diagonals within the band of
A
;
kl
≥
 0.
ku
The number of super-diagonals within the band of
A
;
ku
≥
 0.
nrhs
The number of right-hand sides;
nrhs
≥
 0.
ab
,
afb
,
b
,
x
Arrays:
ab
(size max(1,
ldab
*
n
))
 contains the original band matrix
A
, as supplied to
?gbtrf
, but stored in rows from
1 to
kl
 +
ku
 + 1 for column major layout, and columns from 1 to
kl
 +
ku
 + 1 for row major layout
.
afb
(size max(1,
ldafb
*
n
))
 contains the factored band matrix
A
, as returned by
?gbtrf
.
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
.
ldab
The leading dimension of
ab
,
ldab
≥
kl
 +
ku
 + 1
.
ldafb
The leading dimension of
afb
,
ldafb
≥
 2*
kl
 +
ku
 + 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
)
.
ipiv
Array, size at least
max(1,
n
)
. The
ipiv
 array, as returned by
?gbtrf
.
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 4
n
(
kl
 +
ku
) floating-point operations (for real flavors) or 16
n
(
kl
 +
ku
) operations (for complex flavors). In addition, each step of iterative refinement involves 2
n
(4
kl
 + 3
ku
) operations (for real flavors) or 8
n
(4
kl
 + 3
ku
) 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
2
n
2
 floating-point operations for real flavors or
8
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