NAG C Library Function Document

nag_inteq_fredholm2_smooth (d05abc)

1
Purpose

nag_inteq_fredholm2_smooth (d05abc) solves any linear nonsingular Fredholm integral equation of the second kind with a smooth kernel.

2
Specification

#include <nag.h>
#include <nagd05.h>
void  nag_inteq_fredholm2_smooth (double lambda, double a, double b, Integer n,
double (*k)(double x, double s, Nag_Comm *comm),
double (*g)(double x, Nag_Comm *comm),
Nag_Boolean odorev, Nag_Boolean ev, double f[], double c[], Nag_Comm *comm, NagError *fail)

3
Description

nag_inteq_fredholm2_smooth (d05abc) uses the method of El–Gendi (1969) to solve an integral equation of the form
fx-λabkx,sfsds=gx  
for the function fx in the range axb.
An approximation to the solution fx is found in the form of an n term Chebyshev series i=1nciTix, where  indicates that the first term is halved in the sum. The coefficients ci, for i=1,2,,n, of this series are determined directly from approximate values fi, for i=1,2,,n, of the function fx at the first n of a set of m+1 Chebyshev points
xi=12a+b+b-a×cosi-1×π/m,  i=1,2,,m+1.  
The values fi are obtained by solving a set of simultaneous linear algebraic equations formed by applying a quadrature formula (equivalent to the scheme of Clenshaw and Curtis (1960)) to the integral equation at each of the above points.
In general m=n-1. However, advantage may be taken of any prior knowledge of the symmetry of fx. Thus if fx is symmetric (i.e., even) about the mid-point of the range a,b, it may be approximated by an even Chebyshev series with m=2n-1. Similarly, if fx is anti-symmetric (i.e., odd) about the mid-point of the range of integration, it may be approximated by an odd Chebyshev series with m=2n.

4
References

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

5
Arguments

1:     lambda doubleInput
On entry: the value of the parameter λ of the integral equation.
2:     a doubleInput
On entry: a, the lower limit of integration.
3:     b doubleInput
On entry: b, the upper limit of integration.
Constraint: b>a.
4:     n IntegerInput
On entry: the number of terms in the Chebyshev series which approximates the solution fx.
Constraint: n1.
5:     k function, supplied by the userExternal Function
k must compute the value of the kernel kx,s of the integral equation over the square axb, asb.
The specification of k is:
double  k (double x, double s, Nag_Comm *comm)
1:     x doubleInput
2:     s doubleInput
On entry: the values of x and s at which kx,s is to be calculated.
3:     comm Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to k.
userdouble *
iuserInteger *
pPointer 
The type Pointer will be void *. Before calling nag_inteq_fredholm2_smooth (d05abc) you may allocate memory and initialize these pointers with various quantities for use by k when called from nag_inteq_fredholm2_smooth (d05abc) (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
Note: k should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by nag_inteq_fredholm2_smooth (d05abc). If your code inadvertently does return any NaNs or infinities, nag_inteq_fredholm2_smooth (d05abc) is likely to produce unexpected results.
6:     g function, supplied by the userExternal Function
g must compute the value of the function gx of the integral equation in the interval axb.
The specification of g is:
double  g (double x, Nag_Comm *comm)
1:     x doubleInput
On entry: the value of x at which gx is to be calculated.
2:     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_smooth (d05abc) you may allocate memory and initialize these pointers with various quantities for use by g when called from nag_inteq_fredholm2_smooth (d05abc) (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_smooth (d05abc). If your code inadvertently does return any NaNs or infinities, nag_inteq_fredholm2_smooth (d05abc) is likely to produce unexpected results.
7:     odorev Nag_BooleanInput
On entry: indicates whether it is known that the solution fx is odd or even about the mid-point of the range of integration. If odorev is Nag_TRUE then an odd or even solution is sought depending upon the value of ev.
8:     ev Nag_BooleanInput
On entry: is ignored if odorev is Nag_FALSE. Otherwise, if ev is Nag_TRUE, an even solution is sought, whilst if ev is Nag_FALSE, an odd solution is sought.
9:     f[n] doubleOutput
On exit: the approximate values fi, for i=1,2,,n, of the function fx at the first n of m+1 Chebyshev points (see Section 3), where
m=2n-1 if odorev=Nag_TRUE and ev=Nag_TRUE.
m=2n if odorev=Nag_TRUE and ev=Nag_FALSE.
m=n-1 if odorev=Nag_FALSE.
10:   c[n] doubleOutput
On exit: the coefficients ci, for i=1,2,,n, of the Chebyshev series approximation to fx. When odorev is Nag_TRUE, this series contains polynomials of even order only or of odd order only, according to ev being Nag_TRUE or Nag_FALSE respectively.
11:   comm Nag_Comm *
The NAG communication argument (see Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
12:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6
Error 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 value had an illegal value.
NE_EIGENVALUES
A failure has occurred due to proximity of an eigenvalue.
In general, if lambda is near an eigenvalue of the integral equation, the corresponding matrix will be nearly singular. In the special case, m=1, the matrix reduces to a zero-valued number.
NE_INT
On entry, n=value.
Constraint: n1.
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, a=value and b=value.
Constraint: b>a.

7
Accuracy

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

8
Parallelism and Performance

nag_inteq_fredholm2_smooth (d05abc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_inteq_fredholm2_smooth (d05abc) 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

The time taken by nag_inteq_fredholm2_smooth (d05abc) depends upon the value of n and upon the complexity of the kernel function kx,s.

10
Example

This example solves Love's equation:
fx+1π -11fs 1+ x-s 2 ds=1 .  
It will solve the slightly more general equation:
fx-λ ab kx,sfs ds=1  
where kx,s=α/α2+ x-s 2. The values λ=-1/π,a=-1,b=1,α=1 are used below.
It is evident from the symmetry of the given equation that fx is an even function. Advantage is taken of this fact both in the application of nag_inteq_fredholm2_smooth (d05abc), to obtain the fifxi and the ci, and in subsequent applications of nag_sum_cheby_series (c06dcc) to obtain fx at selected points.
The program runs for n=5 and n=10.

10.1
Program Text

Program Text (d05abce.c)

10.2
Program Data

None.

10.3
Program Results

Program Results (d05abce.r)