?gbrfs
?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 or or
A*X
= B
A
T
*X
= B
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 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).
- trans
- Must be'N'or'T'or'C'.Indicates the form of the equations:If, the system has the formtrans='N'.A*X=BIf, the system has the formtrans='T'AT*X=B.If, the system has the formtrans='C'AH*X=B.
- n
- The order of the matrixA;n≥0.
- kl
- The number of sub-diagonals within the band ofA;kl≥0.
- ku
- The number of super-diagonals within the band ofA;ku≥0.
- nrhs
- The number of right-hand sides;nrhs≥0.
- ab,afb,b,x
- Arrays:ab(size max(1,contains the original band matrixldab*n))A, as supplied to?gbtrf, but stored in rows from1 to.kl+ku+ 1 for column major layout, and columns from 1 tokl+ku+ 1 for row major layoutbof size max(1,contains the right-hand side matrixldb*nrhs) for column major layout and max(1,ldb*n) for row major layoutB.xof size max(1,contains the solution matrixldx*nrhs) for column major layout and max(1,ldx*n) for row major layoutX.
- ldab
- The leading dimension ofab,.ldab≥kl+ku+ 1
- ldafb
- The leading dimension ofafb,.ldafb≥2*kl+ku+ 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)
- ipiv
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 4; the number is usually 4 or 5 and never more than 11. Each solution requires approximately
n
(kl
+ ku
) floating-point operations (for real flavors) or 16n
(kl
+ ku
) operations (for complex flavors). In addition, each step of iterative refinement involves 2n
(4kl
+ 3ku
) operations (for real flavors) or 8n
(4kl
+ 3ku
) 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
2
floating-point operations for real flavors or n
2
8
for complex flavors.n
2