NAG Library Function Document
nag_zgbrfs (f07bvc)
1
Purpose
nag_zgbrfs (f07bvc) returns error bounds for the solution of a complex band system of linear equations with multiple right-hand sides, , or . It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.
2
Specification
#include <nag.h> |
#include <nagf07.h> |
void |
nag_zgbrfs (Nag_OrderType order,
Nag_TransType trans,
Integer n,
Integer kl,
Integer ku,
Integer nrhs,
const Complex ab[],
Integer pdab,
const Complex afb[],
Integer pdafb,
const Integer ipiv[],
const Complex b[],
Integer pdb,
Complex x[],
Integer pdx,
double ferr[],
double berr[],
NagError *fail) |
|
3
Description
nag_zgbrfs (f07bvc) returns the backward errors and estimated bounds on the forward errors for the solution of a complex band system of linear equations with multiple right-hand sides , or . The function handles each right-hand side vector (stored as a column of the matrix ) independently, so we describe the function of nag_zgbrfs (f07bvc) in terms of a single right-hand side and solution .
Given a computed solution
, the function computes the
component-wise
backward error
. This is the size of the smallest relative perturbation in each element of
and
such that
is the exact solution of a perturbed system
Then the function estimates a bound for the
component-wise forward
error in the computed solution, defined by:
where
is the true solution.
For details of the method, see the
f07 Chapter Introduction.
4
References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5
Arguments
- 1:
– Nag_OrderTypeInput
-
On entry: the
order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by
. See
Section 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_TransTypeInput
-
On entry: indicates the form of the linear equations for which
is the computed solution as follows:
- The linear equations are of the form .
- The linear equations are of the form .
- The linear equations are of the form .
Constraint:
, or .
- 3:
– IntegerInput
-
On entry: , the order of the matrix .
Constraint:
.
- 4:
– IntegerInput
-
On entry: , the number of subdiagonals within the band of the matrix .
Constraint:
.
- 5:
– IntegerInput
-
On entry: , the number of superdiagonals within the band of the matrix .
Constraint:
.
- 6:
– IntegerInput
-
On entry: , the number of right-hand sides.
Constraint:
.
- 7:
– const ComplexInput
-
Note: the dimension,
dim, of the array
ab
must be at least
.
On entry: the original
by
band matrix
as supplied to
nag_zgbtrf (f07brc) but with reduced requirements since the matrix is not factorized.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements
, for row
and column
, depends on the
order argument as follows:
- if , is stored as ;
- if , is stored as .
See
Section 9 in
nag_zgbsv (f07bnc) for further details.
- 8:
– IntegerInput
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
ab.
Constraint:
.
- 9:
– const ComplexInput
-
Note: the dimension,
dim, of the array
afb
must be at least
.
On entry: the
factorization of
, as returned by
nag_zgbtrf (f07brc).
- 10:
– IntegerInput
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix in the array
afb.
Constraint:
.
- 11:
– const IntegerInput
-
Note: the dimension,
dim, of the array
ipiv
must be at least
.
On entry: the pivot indices, as returned by
nag_zgbtrf (f07brc).
- 12:
– const ComplexInput
-
Note: the dimension,
dim, of the array
b
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the by right-hand side matrix .
- 13:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 14:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
x
must be at least
- when
;
- when
.
The
th element of the matrix
is stored in
- when ;
- when .
On entry: the
by
solution matrix
, as returned by
nag_zgbtrs (f07bsc).
On exit: the improved solution matrix .
- 15:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
x.
Constraints:
- if ,
;
- if , .
- 16:
– doubleOutput
-
On exit: contains an estimated error bound for the th solution vector, that is, the th column of , for .
- 17:
– doubleOutput
-
On exit: contains the component-wise backward error bound for the th solution vector, that is, the th column of , for .
- 18:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- NE_INT_3
-
On entry, , and .
Constraint: .
On entry, , and .
Constraint: .
- NE_INTERNAL_ERROR
-
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
The bounds returned in
ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.
8
Parallelism and Performance
nag_zgbrfs (f07bvc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zgbrfs (f07bvc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementation-specific information.
For each right-hand side, computation of the backward error involves a minimum of real floating-point operations. Each step of iterative refinement involves an additional real operations. This assumes and . At most five steps of iterative refinement are performed, but usually only one or two steps are required.
Estimating the forward error involves solving a number of systems of linear equations of the form or ; the number is usually and never more than . Each solution involves approximately real operations.
The real analogue of this function is
nag_dgbrfs (f07bhc).
10
Example
This example solves the system of equations
using iterative refinement and to compute the forward and backward error bounds, where
and
Here
is nonsymmetric and is treated as a band matrix, which must first be factorized by
nag_zgbtrf (f07brc).
10.1
Program Text
Program Text (f07bvce.c)
10.2
Program Data
Program Data (f07bvce.d)
10.3
Program Results
Program Results (f07bvce.r)