NAG Library Function Document
nag_inteq_abel_weak_weights (d05byc)
1
Purpose
nag_inteq_abel_weak_weights (d05byc) computes the fractional quadrature weights associated with the Backward Differentiation Formulae (BDF) of orders , and . These weights can then be used in the solution of weakly singular equations of Abel type.
2
Specification
#include <nag.h> |
#include <nagd05.h> |
void |
nag_inteq_abel_weak_weights (Integer iorder,
Integer iq,
double omega[],
double sw[],
NagError *fail) |
|
3
Description
nag_inteq_abel_weak_weights (d05byc) computes the weights
and
for a family of quadrature rules related to a BDF method for approximating the integral:
with
, for some given
. In
(1),
is the order of the BDF method used and
,
are the fractional starting and the fractional convolution weights respectively. The algorithm for the generation of
is based on Newton's iteration. Fast Fourier transform (FFT) techniques are used for computing these weights and subsequently
(see
Baker and Derakhshan (1987) and
Henrici (1979) for practical details and
Lubich (1986) for theoretical details). Some special functions can be represented as the fractional integrals of simpler functions and fractional quadratures can be employed for their computation (see
Lubich (1986)). A description of how these weights can be used in the solution of weakly singular equations of Abel type is given in
Section 9.
4
References
Baker C T H and Derakhshan M S (1987) Computational approximations to some power series Approximation Theory (eds L Collatz, G Meinardus and G Nürnberger) 81 11–20
Henrici P (1979) Fast Fourier methods in computational complex analysis SIAM Rev. 21 481–529
Lubich Ch (1986) Discretized fractional calculus SIAM J. Math. Anal. 17 704–719
5
Arguments
- 1:
– IntegerInput
-
On entry: , the order of the BDF method to be used.
Constraint:
.
- 2:
– IntegerInput
-
On entry: determines the number of weights to be computed. By setting
iq to a value,
fractional convolution weights are computed.
Constraint:
.
- 3:
– doubleOutput
-
On exit: the first
elements of
omega contains the fractional convolution weights
, for
. The remainder of the array is used as workspace.
- 4:
– doubleOutput
-
Note: the th element of the matrix is stored in .
On exit: contains the fractional starting weights , for and , where .
- 5:
– 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: .
- 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
Not applicable.
8
Parallelism and Performance
nag_inteq_abel_weak_weights (d05byc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_inteq_abel_weak_weights (d05byc) 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.
Fractional quadrature weights can be used for solving weakly singular integral equations of Abel type. In this section, we propose the following algorithm which you may find useful in solving a linear weakly singular integral equation of the form
using
nag_inteq_abel_weak_weights (d05byc). In
(2),
and
are given and the solution
is sought on a uniform mesh of size
such that
. Discretization of
(2) yields
where
, for
. We propose the following algorithm for computing
from
(3) after a call to
nag_inteq_abel_weak_weights (d05byc):
(a) |
Set and . |
(b) |
Equation (3) requires starting values, , for , with . These starting values can be computed by solving the system
|
(c) |
Compute the inhomogeneous terms
|
(d) |
Start the iteration for to compute from:
|
Note that for nonlinear weakly singular equations, the solution of a nonlinear algebraic system is required at step
(b) and a single nonlinear equation at step
(d).
10
Example
The following example generates the first fractional convolution and fractional starting weights generated by the fourth-order BDF method.
10.1
Program Text
Program Text (d05byce.c)
10.2
Program Data
None.
10.3
Program Results
Program Results (d05byce.r)