# NAG Library Routine Document

## 1Purpose

d05byf 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

Fortran Interface
 Subroutine d05byf ( iq, wt, sw, ldsw, work, lwk,
 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)
#include <nagmk26.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)

## 3Description

d05byf 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:     $\mathbf{lenfw}$ – IntegerInput
On entry: the dimension of the array wt as declared in the (sub)program from which d05byf is called.
Constraint: ${\mathbf{lenfw}}\ge {2}^{{\mathbf{iq}}+2}$.
4:     $\mathbf{wt}\left({\mathbf{lenfw}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the first ${2}^{{\mathbf{iq}}+1}$ elements of wt 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.
5:     $\mathbf{sw}\left({\mathbf{ldsw}},2×{\mathbf{iorder}}-1\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{sw}}\left(\mathit{i},\mathit{j}+1\right)$ contains the fractional starting weights ${W}_{\mathit{i}-1,\mathit{j}}$ , for $\mathit{i}=1,2,\dots ,\mathit{N}$ and $\mathit{j}=0,1,\dots ,2×{\mathbf{iorder}}-2$, where $\mathit{N}=\left({2}^{{\mathbf{iq}}+1}+2×{\mathbf{iorder}}-1\right)$.
6:     $\mathbf{ldsw}$ – IntegerInput
On entry: the first dimension of the array sw as declared in the (sub)program from which d05byf is called.
Constraint: ${\mathbf{ldsw}}\ge {2}^{{\mathbf{iq}}+1}+2×{\mathbf{iorder}}-1$.
7:     $\mathbf{work}\left({\mathbf{lwk}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
8:     $\mathbf{lwk}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which d05byf is called.
Constraint: ${\mathbf{lwk}}\ge {2}^{{\mathbf{iq}}+3}$.
9:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  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  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{iorder}}=〈\mathit{\text{value}}〉$.
Constraint: $4\le {\mathbf{iorder}}\le 6$.
On entry, ${\mathbf{iq}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{iq}}\ge 0$.
On entry, ${\mathbf{ldsw}}=〈\mathit{\text{value}}〉$, ${\mathbf{iq}}=〈\mathit{\text{value}}〉$ and ${\mathbf{iorder}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldsw}}\ge {2}^{{\mathbf{iq}}+1}+2×{\mathbf{iorder}}-1$.
On entry, ${\mathbf{lenfw}}=〈\mathit{\text{value}}〉$ and ${\mathbf{iq}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lenfw}}\ge {2}^{{\mathbf{iq}}+2}$.
On entry, ${\mathbf{lwk}}=〈\mathit{\text{value}}〉$ and ${\mathbf{iq}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{lwk}}\ge {2}^{{\mathbf{iq}}+3}$
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

Not applicable.

## 8Parallelism 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.

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 d05byf. 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 d05byf:
(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
 $yi = fi×h + h ∑ j=0 2×iorder-2 swi+1j+1 K i×h,j×h yj , i=1,2,…,2×iorder-2 .$
(c) Compute the inhomogeneous terms
 $σi = fi×h + h ∑ j=0 2×iorder- 2 swi+1j+1 K i×h,j×h yj , i = 2 × iorder-1 , 2×iorder , … , N .$
(d) Start the iteration for $i=2×{\mathbf{iorder}}-1,2×{\mathbf{iorder}},\dots ,\mathit{N}$ to compute ${y}_{i}$ from:
 $1 - h wt1 K i×h,i×h yi = σi + h ∑ j=2×iorder-1 i-1 wti-j+1 K i×h,j×h yj .$
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 (d05byfe.f90)

None.

### 10.3Program Results

Program Results (d05byfe.r)