d05byc (abel_weak_weights) : NAG Library CL Interface, Mark 27

3 Description

d05byc 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

iorder

– Integer Input

On entry:

p

, the order of the BDF method to be used.

Constraint:

4 \leq iorder \leq 6

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

omega [\dim]

– double Output

On exit: the first

2^{iq + 1}

elements of omega 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.

sw [N \times (2 \times iorder - 1)]

– double Output

Note: the

(i, j)

th element of the matrix is stored in

sw [(j - 1) \times N + i - 1]

On exit:

sw [j \times N + i - 1]

contains the fractional starting weights

W_{i, 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)

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

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.

NE_BAD_PARAM

On entry, argument

〈value〉

had an illegal value.

NE_INT

On entry,

iorder = 〈value〉

.
Constraint:

4 \leq iorder \leq 6

On entry,

iq = 〈value〉

.
Constraint:

iq \geq 0

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 7.5 in the Introduction to the NAG Library CL Interface for further information.

NE_NO_LICENCE

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.

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) + \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 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 \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 d05byc:

(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
$y_{i} = f (i \times h) + \sqrt{h} \sum_{j = 0}^{2 \times iorder - 2} sw [j \times N + i] K (i \times h, j \times h) y_{j}, i = 1, 2, \dots, 2 \times iorder - 2 .$
(c)Compute the inhomogeneous terms
$σ_{i} = f (i \times h) + \sqrt{h} \sum_{j = 0}^{2 \times iorder - 2} sw [j \times N + i] K (i \times h, j \times h) y_{j}, i = 2 \times iorder - 1, 2 \times iorder, \dots, N .$
(d)Start the iteration for $i = 2 \times iorder - 1, 2 \times iorder, \dots, N$ to compute $y_{i}$ from:
$(1 - \sqrt{h} omega [0] K (i \times h, i \times h)) y_{i} = σ_{i} + \sqrt{h} \sum_{j = 2 \times iorder - 1}^{i - 1} omega [i - j] K (i \times h, j \times h) y_{j} .$

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).

NAG CL Interface
d05byc (abel_weak_weights)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interfaced05byc (abel_​weak_​weights)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
d05byc (abel_weak_weights)