Integer, Intent (In)	::	iorder, iq, lenfw, ldsw, lwk
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), Intent (Inout)	::	sw(ldsw,2*iorder-1)
Real (Kind=nag_wp), Intent (Out)	::	wt(lenfw), work(lwk)

C Header Interface

#include <nag.h>

void	d05byf_ (const Integer iorder, const Integer iq, const Integer lenfw, double wt[], double sw[], const Integer ldsw, double work[], const Integer lwk, Integer ifail)

The routine may be called by the names d05byf or nagf_inteq_abel_weak_weights.

3 Description

d05byf computes the weights

W_{i, j}

and

ω_{i}

for a family of quadrature rules related to a BDF method for approximating the integral:

\frac{1}{\sqrt{π}} \int_{0}^{t} \frac{ϕ (s)}{\sqrt{t - s}} d s ≃ \sqrt{h} \sum_{j = 0}^{2 p - 2} W_{i, j} ϕ (j \times h) + \sqrt{h} \sum_{j = 2 p - 1}^{i} ω_{i - j} ϕ (j \times h), 0 \leq t \leq T,

(1)

with

t = i \times h (i \geq 0)

, for some given

h

. In (1),

p

is the order of the BDF method used and

W_{i, j}

ω_{i}

are the fractional starting and the fractional convolution weights respectively. The algorithm for the generation of

ω_{i}

is based on Newton's iteration. Fast Fourier transform (FFT) techniques are used for computing these weights and subsequently

W_{i, 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: $4 \leq iorder \leq 6$ .
2: $iq$ – Integer Input: On entry: determines the number of weights to be computed. By setting iq to a value, $2^{iq + 1}$ fractional convolution weights are computed.

Constraint: $iq \geq 0$ .
3: $lenfw$ – Integer Input: On entry: the dimension of the array wt as declared in the (sub)program from which d05byf is called.

Constraint: $lenfw \geq 2^{iq + 2}$ .
4: $wt (lenfw)$ – Real (Kind=nag_wp) array Output: On exit: the first $2^{iq + 1}$ elements of wt contains the fractional convolution weights $ω_{i}$ , for $i = 0, 1, \dots, 2^{iq + 1} - 1$ . The remainder of the array is used as workspace.
5: $sw (ldsw, 2 \times iorder - 1)$ – Real (Kind=nag_wp) array Output: On exit: $sw (i, j + 1)$ contains the fractional starting weights $W_{i - 1, j}$ , for $i = 1, 2, \dots, N$ and $j = 0, 1, \dots, 2 \times iorder - 2$ , where $N = (2^{iq + 1} + 2 \times iorder - 1)$ .
6: $ldsw$ – Integer Input: On entry: the first dimension of the array sw as declared in the (sub)program from which d05byf is called.

Constraint: $ldsw \geq 2^{iq + 1} + 2 \times iorder - 1$ .
7: $work (lwk)$ – Real (Kind=nag_wp) array Workspace
8: $lwk$ – Integer Input: On entry: the dimension of the array work as declared in the (sub)program from which d05byf is called.

Constraint: $lwk \geq 2^{iq + 3}$ .
9: $ifail$ – Integer Input/Output: On entry: ifail must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this argument you should refer to Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit: $ifail = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $iorder = 〈value〉$ .
Constraint: $4 \leq iorder \leq 6$ .

On entry, $iq = 〈value〉$ .
Constraint: $iq \geq 0$ .

On entry, $ldsw = 〈value〉$ , $iq = 〈value〉$ and $iorder = 〈value〉$ .
Constraint: $ldsw \geq 2^{iq + 1} + 2 \times iorder - 1$ .

On entry, $lenfw = 〈value〉$ and $iq = 〈value〉$ .
Constraint: $lenfw \geq 2^{iq + 2}$ .

On entry, $lwk = 〈value〉$ and $iq = 〈value〉$ .
Constraint: $lwk \geq 2^{iq + 3}$

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

d05byf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

d05byf 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 routine. 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) + \frac{1}{\sqrt{π}} \int_{0}^{t} \frac{K (t, s) y (s)}{\sqrt{t - s}} d s, 0 \leq t \leq T,

(2)

using d05byf. 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 \times h

. Discretization of (2) yields

y_{i} = f (i \times h) + \sqrt{h} \sum_{j = 0}^{2 p - 2} W_{i, j} K (i \times h, j \times h) y_{j} + \sqrt{h} \sum_{j = 2 p - 1}^{i} ω_{i - j} K (i \times h, j \times h) y_{j},

(3)

where

y_{i} ≃ y (i \times h)

, for

i = 1, 2, \dots, N

. We propose the following algorithm for computing

y_{i}

from (3) after a call to d05byf:

(a)Set $N = 2^{iq + 1} + 2 \times iorder - 2$ and $h = T / N$ .
(b)Equation (3) requires $2 \times iorder - 2$ starting values, $y_{j}$ , for $j = 1, 2, \dots, 2 \times iorder - 2$ , with $y_{0} = f (0)$ . These starting values can be computed by solving the system
$y_{i} = f (i \times h) + \sqrt{h} \sum_{j = 0}^{2 \times iorder - 2} sw (i + 1, j + 1) K (i \times h, j \times h) y_{j}, i = 1, 2, \dots, 2 \times iorder - 2 .$
(c)Compute the inhomogeneous terms
$σ_{i} = f (i \times h) + \sqrt{h} \sum_{j = 0}^{2 \times iorder - 2} sw (i + 1, j + 1) K (i \times h, j \times h) y_{j}, i = 2 \times iorder - 1, 2 \times iorder, \dots, N .$
(d)Start the iteration for $i = 2 \times iorder - 1, 2 \times iorder, \dots, N$ to compute $y_{i}$ from:
$(1 - \sqrt{h} wt (1) K (i \times h, i \times h)) y_{i} = σ_{i} + \sqrt{h} \sum_{j = 2 \times iorder - 1}^{i - 1} wt (i - j + 1) K (i \times h, j \times h) y_{j} .$

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).