NAG Library Function Document

nag_inteq_fredholm2_smooth (d05abc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

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

f (x) - λ \int_{a}^{b} k (x, s) f (s) d s = g (x)

for the function

f (x)

in the range

a \leq x \leq b

An approximation to the solution

f (x)

is found in the form of an

n

term Chebyshev series

{\sum^{'}}_{i = 1}^{n} c_{i} T_{i} (x)

, where

^{'}

indicates that the first term is halved in the sum. The coefficients

c_{i}

, for

i = 1, 2, \dots, n

, of this series are determined directly from approximate values

f_{i}

, for

i = 1, 2, \dots, n

, of the function

f (x)

at the first

n

of a set of

m + 1

Chebyshev points

x_{i} = \frac{1}{2} (a + b + (b - a) \times \cos [(i - 1) \times π / m]), i = 1, 2, \dots, m + 1 .

The values

f_{i}

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

f (x)

. Thus if

f (x)

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 = 2 n - 1

. Similarly, if

f (x)

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 = 2 n

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

f (x)

Constraint:

n \geq 1

5: k – function, supplied by the userExternal Function

k must compute the value of the kernel

k (x, s)

of the integral equation over the square

a \leq x \leq b

a \leq s \leq b

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

k (x, s)

is to be calculated.

3: comm – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to k.

user – double *
iuser – Integer *
p – Pointer: 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 3.2.1.1 in the Essential Introduction).

6: g – function, supplied by the userExternal Function

g must compute the value of the function

g (x)

of the integral equation in the interval

a \leq x \leq b

The specification of g is:

double

g (double x, Nag_Comm *comm)

1: x – doubleInput

On entry: the value of

x

at which

g (x)

is to be calculated.

2: comm – Nag_Comm *

Pointer to structure of type Nag_Comm; the following members are relevant to g.

user – double *
iuser – Integer *
p – Pointer: 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 3.2.1.1 in the Essential Introduction).

7: odorev – Nag_BooleanInput

On entry: indicates whether it is known that the solution

f (x)

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

f_{i}

, for

i = 1, 2, \dots, n

, of the function

f (x)

at the first n of

m + 1

Chebyshev points (see Section 3), where

$m = 2 n - 1$	if $odorev = Nag_TRUE$ and $ev = Nag_TRUE$ .
$m = 2 n$	if $odorev = Nag_TRUE$ and $ev = Nag_FALSE$ .
$m = n - 1$	if $odorev = Nag_FALSE$ .

10: c[n] – doubleOutput

On exit: the coefficients

c_{i}

, for

i = 1, 2, \dots, n

, of the Chebyshev series approximation to

f (x)

. 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 *Communication Structure

The NAG communication argument (see Section 3.2.1.1 in the Essential Introduction).

12: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_EIGENVALUES: A failure has occurred due to proximity of an eigenvalue.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
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.
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 $c_{i}$ , or
(ii)	by comparing the coefficients $c_{i}$ or the function values $f_{i}$ 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 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

k (x, s)

10 Example

This example solves Love's equation:

f (x) + \frac{1}{π} \int_{- 1}^{1} \frac{f (s)}{1 + {(x - s)}^{2}} d s = 1 .

It will solve the slightly more general equation:

f (x) - λ \int_{a}^{b} k (x, s) f (s) d s = 1

where

k (x, s) = α / (α^{2} + {(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

f (x)

is an even function. Advantage is taken of this fact both in the application of nag_inteq_fredholm2_smooth (d05abc), to obtain the

f_{i} ≃ f (x_{i})

and the

c_{i}

, and in subsequent applications of nag_sum_cheby_series (c06dcc) to obtain

f (x)

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)

nag_inteq_fredholm2_smooth (d05abc) (PDF version)

d05 Chapter Contents

d05 Chapter Introduction

NAG Library Manual

NAG Library Function Documentnag_inteq_fredholm2_smooth (d05abc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_inteq_fredholm2_smooth (d05abc)