NAG Library Function Document
nag_complex_band_lin_solve (f04cbc)
1
Purpose
nag_complex_band_lin_solve (f04cbc) computes the solution to a complex system of linear equations , where is an by band matrix, with subdiagonals and superdiagonals, and and are by matrices. An estimate of the condition number of and an error bound for the computed solution are also returned.
2
Specification
#include <nag.h> |
#include <nagf04.h> |
void |
nag_complex_band_lin_solve (Nag_OrderType order,
Integer n,
Integer kl,
Integer ku,
Integer nrhs,
Complex ab[],
Integer pdab,
Integer ipiv[],
Complex b[],
Integer pdb,
double *rcond,
double *errbnd,
NagError *fail) |
|
3
Description
The decomposition with partial pivoting and row interchanges is used to factor as , where is a permutation matrix, is the product of permutation matrices and unit lower triangular matrices with subdiagonals, and is upper triangular with superdiagonals. The factored form of is then used to solve the system of equations .
4
References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
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:
– IntegerInput
-
On entry: the number of linear equations , i.e., the order of the matrix .
Constraint:
.
- 3:
– IntegerInput
-
On entry: the number of subdiagonals , within the band of .
Constraint:
.
- 4:
– IntegerInput
-
On entry: the number of superdiagonals , within the band of .
Constraint:
.
- 5:
– IntegerInput
-
On entry: the number of right-hand sides , i.e., the number of columns of the matrix .
Constraint:
.
- 6:
– ComplexInput/Output
-
Note: the dimension,
dim, of the array
ab
must be at least
.
On entry: the
by
matrix
.
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 for further details.
On exit:
ab is overwritten by details of the factorization.
The elements, , of the upper triangular band factor with super-diagonals, and the multipliers, , used to form the lower triangular factor are stored. The elements , for and , and , for and , are stored where is stored on entry.
- 7:
– IntegerInput
On entry: the stride separating row or column elements (depending on the value of
order) of the matrix
in the array
ab.
Constraint:
.
- 8:
– IntegerOutput
-
On exit: if NE_NOERROR, the pivot indices that define the permutation matrix ; at the th step row of the matrix was interchanged with row . indicates a row interchange was not required.
- 9:
– ComplexInput/Output
-
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 matrix of right-hand sides .
On exit: if
NE_NOERROR or
NE_RCOND, the
by
solution matrix
.
- 10:
– IntegerInput
-
On entry: the stride separating row or column elements (depending on the value of
order) in the array
b.
Constraints:
- if ,
;
- if , .
- 11:
– double *Output
-
On exit: if NE_NOERROR, an estimate of the reciprocal of the condition number of the matrix , computed as .
- 12:
– double *Output
-
On exit: if
NE_NOERROR or
NE_RCOND, an estimate of the forward error bound for a computed solution
, such that
, where
is a column of the computed solution returned in the array
b and
is the corresponding column of the exact solution
. If
rcond is less than
machine precision,
errbnd is returned as unity.
- 13:
– 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.
The double allocatable memory required is
n, and the Complex allocatable memory required is
. In this case the factorization and the solution
have been computed, but
rcond and
errbnd have not been computed.
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: .
- NE_INT_2
-
On entry, , and .
Constraint: .
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.
- NE_RCOND
-
A solution has been computed, but
rcond is less than
machine precision so that the matrix
is numerically singular.
- NE_SINGULAR
-
Diagonal element of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.
7
Accuracy
The computed solution for a single right-hand side,
, satisfies an equation of the form
where
and
is the
machine precision. An approximate error bound for the computed solution is given by
where
, the condition number of
with respect to the solution of the linear equations.
nag_complex_band_lin_solve (f04cbc) uses the approximation
to estimate
errbnd. See Section 4.4 of
Anderson et al. (1999)
for further details.
8
Parallelism and Performance
nag_complex_band_lin_solve (f04cbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_complex_band_lin_solve (f04cbc) 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.
The band storage scheme for the array
ab
is illustrated by the following example, when
,
, and
.
Storage of the band matrix
in the array
ab:
Band matrix |
Band storage in array ab |
|
|
|
|
|
Array elements marked
need not be set and are not referenced by the function. Array elements marked + need not be set, but are defined on exit from the function and contain the elements
,
,
,
and
. In this example when
the first referenced element of
ab is
; while for
the first referenced element is
.
In general, elements
are stored as follows:
- if , are stored in
- if , are stored in
where
.
The total number of floating-point operations required to solve the equations depends upon the pivoting required, but if then it is approximately bounded by for the factorization and for the solution following the factorization. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of
Higham (2002) for further details.
The real analogue of
nag_complex_band_lin_solve (f04cbc) is
nag_real_band_lin_solve (f04bbc).
10
Example
This example solves the equations
where
is the band matrix
and
An estimate of the condition number of and an approximate error bound for the computed solutions are also printed.
10.1
Program Text
Program Text (f04cbce.c)
10.2
Program Data
Program Data (f04cbce.d)
10.3
Program Results
Program Results (f04cbce.r)