# NAG FL Interfaced05byf (abel_​weak_​weights)

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## 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 <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.

## 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:
 $1π∫0tϕ(s) t-s ds≃h∑j=0 2p-2Wi,jϕ(j×h)+h∑j=2p-1iωi-jϕ(j×h), 0≤t≤T,$ (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.
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: $\mathbf{iorder}$Integer Input
On entry: $p$, the order of the BDF method to be used.
Constraint: $4\le {\mathbf{iorder}}\le 6$.
2: $\mathbf{iq}$Integer Input
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: $\mathbf{lenfw}$Integer Input
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) array Output
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) array Output
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: $\mathbf{ldsw}$Integer Input
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) array Workspace
8: $\mathbf{lwk}$Integer Input
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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ 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: ${\mathbf{lwk}}\ge {2}^{{\mathbf{iq}}+3}$
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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
 $y(t)=f(t)+1π∫0tK(t,s)y(s) t-s ds, 0≤t≤T,$ (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
 $yi = f(i×h) + h ∑ j=0 2p-2 W i,j K (i×h,j×h) yj + h ∑ j=2p-1 i ωi-j K (i×h,j×h) yj ,$ (3)
where ${y}_{\mathit{i}}\simeq y\left(\mathit{i}×h\right)$, for $\mathit{i}=1,2,\dots ,\mathit{N}$. We propose the following algorithm for computing ${y}_{i}$ from (3) after a call to d05byf:
1. (a)Set $\mathit{N}={2}^{{\mathbf{iq}}+1}+2×{\mathbf{iorder}}-2$ and $h=T/\mathit{N}$.
2. (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
 $yi = f(i×h) + h ∑ j=0 2×iorder-2 sw(i+1,j+1) K (i×h,j×h) yj , i=1,2,…,2×iorder-2 .$
3. (c)Compute the inhomogeneous terms
 $σi = f(i×h) + h ∑ j=0 2×iorder- 2 sw(i+1,j+1) K (i×h,j×h) yj , i = 2 × iorder-1 , 2×iorder , … , N .$
4. (d)Start the iteration for $i=2×{\mathbf{iorder}}-1,2×{\mathbf{iorder}},\dots ,\mathit{N}$ to compute ${y}_{i}$ from:
 $(1- h wt(1)K(i×h,i×h)) yi = σi + h ∑ j=2×iorder-1 i-1 wt(i-j+1) K (i×h,j×h) yj .$
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)