NAG CL Interface
d05byc (abel_​weak_​weights)

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1 Purpose

d05byc computes the fractional quadrature weights associated with the Backward Differentiation Formulae (BDF) of orders 4, 5 and 6. These weights can then be used in the solution of weakly singular equations of Abel type.

2 Specification

#include <nag.h>
void  d05byc (Integer iorder, Integer iq, double omega[], double sw[], NagError *fail)
The function may be called by the names: d05byc or nag_inteq_abel_weak_weights.

3 Description

d05byc computes the weights Wi,j and ωi for a family of quadrature rules related to a BDF method for approximating the integral:
1π0tϕ(s) t-s dshj=0 2p-2Wi,jϕ(j×h)+hj=2p-1iωi-jϕ(j×h),  0tT, (1)
with t=i×h(i0), for some given h. In (1), p is the order of the BDF method used and Wi,j, ωi are the fractional starting and the fractional convolution weights respectively. The algorithm for the generation of ωi is based on Newton's iteration. Fast Fourier transform (FFT) techniques are used for computing these weights and subsequently Wi,j (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: iorder Integer Input
On entry: p, the order of the BDF method to be used.
Constraint: 4iorder6.
2: iq Integer Input
On entry: determines the number of weights to be computed. By setting iq to a value, 2iq+1 fractional convolution weights are computed.
Constraint: iq0.
3: omega[dim] double Output
On exit: the first 2iq+1 elements of omega contains the fractional convolution weights ωi, for i=0,1,,2iq+1-1. The remainder of the array is used as workspace.
4: sw[N×(2×iorder-1)] double Output
Note: the (i,j)th element of the matrix is stored in sw[(j-1)×N+i-1].
On exit: sw[j×N+i-1] contains the fractional starting weights Wi,j, for i=1,2,,N and j=0,1,,2×iorder-2, where N=(2iq+1+2×iorder-1).
5: fail NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument value had an illegal value.
On entry, iorder=value.
Constraint: 4iorder6.
On entry, iq=value.
Constraint: iq0.
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 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

d05byc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
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.

9 Further Comments

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
y(t)=f(t)+1π0tK(t,s)y(s) t-s ds,  0tT, (2)
using d05byc. In (2), K(t,s) and f(t) are given and the solution y(t) is sought on a uniform mesh of size h such that T=N×h. Discretization of (2) yields
yi = f(i×h) + h j=0 2p-2 W i,j K (i×h,j×h) yj + h j=2p-1 i ωi-j K (i×h,j×h) yj , (3)
where yiy(i×h), for i=1,2,,N. We propose the following algorithm for computing yi from (3) after a call to d05byc:
  1. (a)Set N=2iq+1+2×iorder-2 and h=T/N.
  2. (b)Equation (3) requires 2×iorder-2 starting values, yj, for j=1,2,,2×iorder-2, with y0=f(0). These starting values can be computed by solving the system
    yi = f(i×h) + h j=0 2×iorder-2 sw[j×N+i] K (i×h,j×h) yj ,  i=1,2,,2×iorder-2 .  
  3. (c)Compute the inhomogeneous terms
    σi = f(i×h) + h j=0 2×iorder- 2 sw[j×N+i] K (i×h,j×h) yj ,   i = 2 × iorder-1 , 2×iorder , , N .  
  4. (d)Start the iteration for i=2×iorder-1,2×iorder,,N to compute yi from:
    (1- h omega[0]K(i×h,i×h)) yi = σi + h j=2×iorder-1 i-1 omega[i-j] K (i×h,j×h) yj .  
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 16 fractional convolution and 23 fractional starting weights generated by the fourth-order BDF method.

10.1 Program Text

Program Text (d05byce.c)

10.2 Program Data

Program Data (d05byce.d)

10.3 Program Results

Program Results (d05byce.r)