NAG Library Function Document

1Purpose

nag_inteq_fredholm2_split (d05aac) solves a linear, nonsingular Fredholm equation of the second kind with a split kernel.

2Specification

 #include #include
void  nag_inteq_fredholm2_split (double lambda, double a, double b, Integer n,
 double (*k1)(double x, double s, Nag_Comm *comm),
 double (*k2)(double x, double s, Nag_Comm *comm),
 double (*g)(double x, Nag_Comm *comm),
Nag_KernelForm kform, double f[], double c[], Nag_Comm *comm, NagError *fail)

3Description

nag_inteq_fredholm2_split (d05aac) solves an integral equation of the form
 $fx-λ∫abkx,sfsds=gx$
for $a\le x\le b$, when the kernel $k$ is defined in two parts: $k={k}_{1}$ for $a\le s\le x$ and $k={k}_{2}$ for $x. The method used is that of El–Gendi (1969) for which, it is important to note, each of the functions ${k}_{1}$ and ${k}_{2}$ must be defined, smooth and nonsingular, for all $x$ and $s$ in the interval $\left[a,b\right]$.
An approximation to the solution $f\left(x\right)$ is found in the form of an $n$ term Chebyshev series $\underset{i=1}{\overset{n}{{\sum }^{\prime }}}{c}_{i}{T}_{i}\left(x\right)$, where ${}^{\prime }$ indicates that the first term is halved in the sum. The coefficients ${c}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, of this series are determined directly from approximate values ${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, of the function $f\left(x\right)$ at the first $n$ of a set of $m+1$ Chebyshev points:
 $xi=12a+b+b-acosi-1π/m, i=1,2,…,m+1.$
The values ${f}_{i}$ are obtained by solving simultaneous linear algebraic equations formed by applying a quadrature formula (equivalent to the scheme of Clenshaw and Curtis (1960)) to the integral equation at the above points.
In general $m=n-1$. However, if the kernel $k$ is centro-symmetric in the interval $\left[a,b\right]$, i.e., if $k\left(x,s\right)=k\left(a+b-x,a+b-s\right)$, then the function is designed to take advantage of this fact in the formation and solution of the algebraic equations. In this case, symmetry in the function $g\left(x\right)$ implies symmetry in the function $f\left(x\right)$. In particular, if $g\left(x\right)$ is even about the mid-point of the range of integration, then so also is $f\left(x\right)$, which may be approximated by an even Chebyshev series with $m=2n-1$. Similarly, if $g\left(x\right)$ is odd about the mid-point then $f\left(x\right)$ may be approximated by an odd series with $m=2n$.

4References

Clenshaw C W and Curtis A R (1960) A method for numerical integration on an automatic computer Numer. Math. 2 197–205
El–Gendi S E (1969) Chebyshev solution of differential, integral and integro-differential equations Comput. J. 12 282–287

5Arguments

1:    $\mathbf{lambda}$doubleInput
On entry: the value of the parameter $\lambda$ of the integral equation.
2:    $\mathbf{a}$doubleInput
On entry: $a$, the lower limit of integration.
3:    $\mathbf{b}$doubleInput
On entry: $b$, the upper limit of integration.
Constraint: ${\mathbf{b}}>{\mathbf{a}}$.
4:    $\mathbf{n}$IntegerInput
On entry: the number of terms in the Chebyshev series required to approximate $f\left(x\right)$.
Constraint: ${\mathbf{n}}\ge 1$.
5:    $\mathbf{k1}$function, supplied by the userExternal Function
k1 must evaluate the kernel $k\left(x,s\right)={k}_{1}\left(x,s\right)$ of the integral equation for $a\le s\le x$.
The specification of k1 is:
 double k1 (double x, double s, Nag_Comm *comm)
1:    $\mathbf{x}$doubleInput
2:    $\mathbf{s}$doubleInput
On entry: the values of $x$ and $s$ at which ${k}_{1}\left(x,s\right)$ is to be evaluated.
3:    $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to k1.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_inteq_fredholm2_split (d05aac) you may allocate memory and initialize these pointers with various quantities for use by k1 when called from nag_inteq_fredholm2_split (d05aac) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: k1 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_inteq_fredholm2_split (d05aac). If your code inadvertently does return any NaNs or infinities, nag_inteq_fredholm2_split (d05aac) is likely to produce unexpected results.
6:    $\mathbf{k2}$function, supplied by the userExternal Function
k2 must evaluate the kernel $k\left(x,s\right)={k}_{2}\left(x,s\right)$ of the integral equation for $x.
The specification of k2 is:
 double k2 (double x, double s, Nag_Comm *comm)
1:    $\mathbf{x}$doubleInput
2:    $\mathbf{s}$doubleInput
On entry: the values of $x$ and $s$ at which ${k}_{2}\left(x,s\right)$ is to be evaluated.
3:    $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to k2.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_inteq_fredholm2_split (d05aac) you may allocate memory and initialize these pointers with various quantities for use by k2 when called from nag_inteq_fredholm2_split (d05aac) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: k2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_inteq_fredholm2_split (d05aac). If your code inadvertently does return any NaNs or infinities, nag_inteq_fredholm2_split (d05aac) is likely to produce unexpected results.
Note that the functions ${k}_{1}$ and ${k}_{2}$ must be defined, smooth and nonsingular for all $x$ and $s$ in the interval [$a,b$].
7:    $\mathbf{g}$function, supplied by the userExternal Function
g must evaluate the function $g\left(x\right)$ for $a\le x\le b$.
The specification of g is:
 double g (double x, Nag_Comm *comm)
1:    $\mathbf{x}$doubleInput
On entry: the values of $x$ at which $g\left(x\right)$ is to be evaluated.
2:    $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to g.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling nag_inteq_fredholm2_split (d05aac) you may allocate memory and initialize these pointers with various quantities for use by g when called from nag_inteq_fredholm2_split (d05aac) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: g should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_inteq_fredholm2_split (d05aac). If your code inadvertently does return any NaNs or infinities, nag_inteq_fredholm2_split (d05aac) is likely to produce unexpected results.
8:    $\mathbf{kform}$Nag_KernelFormInput
On entry: determines the forms of the kernel, $k\left(x,s\right)$, and the function $g\left(x\right)$.
${\mathbf{kform}}=\mathrm{Nag_NoCentroSymm}$
$k\left(x,s\right)$ is not centro-symmetric (or no account is to be taken of centro-symmetry).
${\mathbf{kform}}=\mathrm{Nag_CentroSymmOdd}$
$k\left(x,s\right)$ is centro-symmetric and $g\left(x\right)$ is odd.
${\mathbf{kform}}=\mathrm{Nag_CentroSymmEven}$
$k\left(x,s\right)$ is centro-symmetric and $g\left(x\right)$ is even.
${\mathbf{kform}}=\mathrm{Nag_CentroSymmNeither}$
$k\left(x,s\right)$ is centro-symmetric but $g\left(x\right)$ is neither odd nor even.
Constraint: ${\mathbf{kform}}=\mathrm{Nag_NoCentroSymm}$, $\mathrm{Nag_CentroSymmOdd}$, $\mathrm{Nag_CentroSymmEven}$ or $\mathrm{Nag_CentroSymmNeither}$.
9:    $\mathbf{f}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the approximate values ${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$, of $f\left(x\right)$ evaluated at the first n of $m+1$ Chebyshev points ${x}_{i}$, (see Section 3).
If ${\mathbf{kform}}=\mathrm{Nag_NoCentroSymm}$ or $\mathrm{Nag_CentroSymmNeither}$, $m={\mathbf{n}}-1$.
If ${\mathbf{kform}}=\mathrm{Nag_CentroSymmOdd}$, $m=2×{\mathbf{n}}$.
If ${\mathbf{kform}}=\mathrm{Nag_CentroSymmEven}$, $m=2×{\mathbf{n}}-1$.
10:  $\mathbf{c}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the coefficients ${c}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$, of the Chebyshev series approximation to $f\left(x\right)$.
If ${\mathbf{kform}}=\mathrm{Nag_CentroSymmOdd}$ this series contains polynomials of odd order only and if ${\mathbf{kform}}=\mathrm{Nag_CentroSymmEven}$ the series contains even order polynomials only.
11:  $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
12:  $\mathbf{fail}$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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_EIGENVALUES
A failure has occurred due to proximity of an eigenvalue.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
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.
NE_REAL_2
On entry, ${\mathbf{a}}=〈\mathit{\text{value}}〉$ and ${\mathbf{b}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{b}}>{\mathbf{a}}$.

7Accuracy

No explicit error estimate is provided by the function but it is usually possible to obtain a good indication of the accuracy of the solution either
 (i) by examining the size of the later Chebyshev coefficients ${c}_{i}$, or (ii) by comparing the coefficients ${c}_{i}$ or the function values ${f}_{i}$ for two or more values of n.

8Parallelism and Performance

nag_inteq_fredholm2_split (d05aac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_inteq_fredholm2_split (d05aac) 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.

The time taken by nag_inteq_fredholm2_split (d05aac) increases with n.
This function may be used to solve an equation with a continuous kernel by defining k1 and k2 to be identical.
This function may also be used to solve a Volterra equation by defining k2 (or k1) to be identically zero.

10Example

This example solves the equation
 $fx - ∫01 kx,s fs ds = 1 - 1 π2 sinπx$
where
 $kx,s = s1-x for ​ 0≤s≤x , x1-s for ​ x
Five terms of the Chebyshev series are sought, taking advantage of the centro-symmetry of the $k\left(x,s\right)$ and even nature of $g\left(x\right)$ about the mid-point of the range $\left[0,1\right]$.
The approximate solution at the point $x=0.1$ is calculated by calling nag_sum_cheby_series (c06dcc).

10.1Program Text

Program Text (d05aace.c)

None.

10.3Program Results

Program Results (d05aace.r)

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