# NAG Library Function Document

## 1Purpose

nag_inteq_abel_weak_weights (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.

## 2Specification

 #include #include
 void nag_inteq_abel_weak_weights (Integer iorder, Integer iq, double omega[], double sw[], NagError *fail)

## 3Description

nag_inteq_abel_weak_weights (d05byc) computes the weights ${W}_{i,j}$ and ${\omega }_{i}$ for a family of quadrature rules related to a BDF method for approximating the integral:
 $1π∫0tϕs t-s ds≃h∑j=0 2p-2Wi,jϕj×h+h∑j=2p-1iωi-jϕj×h, 0≤t≤T,$ (1)
with $t=i×h\left(i\ge 0\right)$, for some given $h$. In (1), $p$ is the order of the BDF method used and ${W}_{i,j}$, ${\omega }_{i}$ are the fractional starting and the fractional convolution weights respectively. The algorithm for the generation of ${\omega }_{i}$ 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.

## 4References

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

## 5Arguments

1:    $\mathbf{iorder}$IntegerInput
On entry: $p$, the order of the BDF method to be used.
Constraint: $4\le {\mathbf{iorder}}\le 6$.
2:    $\mathbf{iq}$IntegerInput
On entry: determines the number of weights to be computed. By setting iq to a value, ${2}^{{\mathbf{iq}}+1}$ fractional convolution weights are computed.
Constraint: ${\mathbf{iq}}\ge 0$.
3:    $\mathbf{omega}\left[{2}^{{\mathbf{iq}}+2}\right]$doubleOutput
On exit: the first ${2}^{{\mathbf{iq}}+1}$ elements of omega contains the fractional convolution weights ${\omega }_{i}$, for $\mathit{i}=0,1,\dots ,{2}^{{\mathbf{iq}}+1}-1$. The remainder of the array is used as workspace.
4:    $\mathbf{sw}\left[\mathit{N}×\left(2×{\mathbf{iorder}}-1\right)\right]$doubleOutput
Note: the $\left(i,j\right)$th element of the matrix is stored in ${\mathbf{sw}}\left[\left(j-1\right)×\mathit{N}+i-1\right]$.
On exit: ${\mathbf{sw}}\left[\mathit{j}×\mathit{N}+\mathit{i}-1\right]$ contains the fractional starting weights ${W}_{\mathit{i},\mathit{j}}$, for $\mathit{i}=1,2,\dots ,\mathit{N}$ and $\mathit{j}=0,1,\dots ,2×{\mathbf{iorder}}-2$, where $\mathit{N}=\left({2}^{{\mathbf{iq}}+1}+2×{\mathbf{iorder}}-1\right)$.
5:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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 $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{iorder}}=〈\mathit{\text{value}}〉$.
Constraint: $4\le {\mathbf{iorder}}\le 6$.
On entry, ${\mathbf{iq}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{iq}}\ge 0$.
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 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.

Not applicable.

## 8Parallelism 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.

## 9Further 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
 $yt=ft+1π∫0tKt,sys t-s ds, 0≤t≤T,$ (2)
using nag_inteq_abel_weak_weights (d05byc). In (2), $K\left(t,s\right)$ and $f\left(t\right)$ are given and the solution $y\left(t\right)$ is sought on a uniform mesh of size $h$ such that $T=\mathit{N}×h$. Discretization of (2) yields
 $yi = fi×h + h ∑ j=0 2p-2 W i,j K i×h,j×h yj + h ∑ j=2p-1 i ωi-j K i×h,j×h yj ,$ (3)
where ${y}_{\mathit{i}}\simeq y\left(\mathit{i}×h\right)$, for $\mathit{i}=1,2,\dots ,\mathit{N}$. We propose the following algorithm for computing ${y}_{i}$ from (3) after a call to nag_inteq_abel_weak_weights (d05byc):
(a) Set $\mathit{N}={2}^{{\mathbf{iq}}+1}+2×{\mathbf{iorder}}-2$ and $h=T/\mathit{N}$.
(b) Equation (3) requires $2×{\mathbf{iorder}}-2$ starting values, ${y}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,2×{\mathbf{iorder}}-2$, with ${y}_{0}=f\left(0\right)$. These starting values can be computed by solving the system
 $yi = fi×h + h ∑ j=0 2×iorder-2 sw[j×N+i] K i×h,j×h yj , i=1,2,…,2×iorder-2 .$
(c) Compute the inhomogeneous terms
 $σi = fi×h + h ∑ j=0 2×iorder- 2 sw[j×N+i] K i×h,j×h yj , i = 2 × iorder-1 , 2×iorder , … , N .$
(d) Start the iteration for $i=2×{\mathbf{iorder}}-1,2×{\mathbf{iorder}},\dots ,\mathit{N}$ to compute ${y}_{i}$ 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).

## 10Example

The following example generates the first $16$ fractional convolution and $23$ fractional starting weights generated by the fourth-order BDF method.

### 10.1Program Text

Program Text (d05byce.c)

None.

### 10.3Program Results

Program Results (d05byce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017