?pbrfs
?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 with a symmetric (Hermitian) positive definite band matrix
A*X
= B
A
, with multiple right-hand sides. For each computed solution vector x
, the routine computes the component-wise backward errorβ
. This error is the smallest relative perturbation in elements of A
and b
such that x
is the exact solution of the perturbed system:|| |, || |
such that δ
a
i
j
≤
β
|a
i
j
δ
b
i
≤
β
|b
i
(
.A
+ δ
A
)x
= (b
+ δ
b
)Finally, the routine estimates the component-wise forward error in the computed solution is the exact solution).
||||/||
(here x
- x
e
∞
x
||∞
x
e
Before calling this routine:
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 matrixAhas been factored:If, the upper triangle ofuplo='U'Ais stored.If, the lower triangle ofuplo='L'Ais stored.
- n
- The order of the matrixA;n≥0.
- kd
- The number of superdiagonals or subdiagonals in the matrixA;kd≥0.
- nrhs
- The number of right-hand sides;nrhs≥0.
- ab
- afb
- b
- Arraybof size max(1,ldb*nrhs) for column major layout and max(1,ldb*n) for row major layout contains the right-hand side matrixB.
- x
- Arrayxof size max(1,ldx*nrhs) for column major layout and max(1,ldx*n) for row major layout contains the solution matrixX.
- ldab
- The leading dimension ofab;ldab≥kd+ 1.
- ldafb
- The leading dimension ofafb;ldafb≥kd+ 1.
- ldb
- The leading dimension ofb;.ldb≥max(1,n) for column major layout andldb≥nrhsfor row major layout
- ldx
- The leading dimension ofx;.ldx≥max(1,n) for column major layout andldx≥nrhsfor row major layout
Output Parameters
- x
- The refined solution matrixX.
- ferr,berr
- Arrays, size at leastmax(1,. Contain the component-wise forward and backward errors, respectively, for each solution vector.nrhs)
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
floating-point operations (for real flavors) or n
*kd
32
operations (for complex flavors). In addition, each step of iterative refinement involves n
*kd
12
operations (for real flavors) or n
*kd
48
operations (for complex flavors); the number of iterations may range from 1 to 5. n
*kd
Estimating the forward error involves solving a number of systems of linear equations ; the number is usually 4 or 5 and never more than 11. Each solution requires approximately
A
*x
= b
4
floating-point operations for real flavors or n
*kd
16
for complex flavors.n
*kd