Integer, Intent (In)	::	n, ldcm, nt2p1
Integer, Intent (Inout)	::	ifail
Real (Kind=nag_wp), External	::	k, g
Real (Kind=nag_wp), Intent (In)	::	lambda, a, b
Real (Kind=nag_wp), Intent (Inout)	::	cm(ldcm,ldcm), f1(ldcm,1)
Real (Kind=nag_wp), Intent (Out)	::	wk(2,nt2p1), f(n), c(n)
Logical, Intent (In)	::	odorev, ev

C Header Interface

#include <nag.h>

void

d05abf_ (
double (NAG_CALL *k)(const double *x, const double *s),
double (NAG_CALL *g)(const double *x),
const double *lambda, const double *a, const double *b, const logical *odorev, const logical *ev, const Integer *n, double cm[], double f1[], double wk[], const Integer *ldcm, const Integer *nt2p1, double f[], double c[], Integer *ifail)

The routine may be called by the names d05abf or nagf_inteq_fredholm2_smooth.

3 Description

d05abf 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: $k$ – real (Kind=nag_wp) Function, supplied by the user. External Procedure

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:

Fortran Interface

Function k (

x, s)

Real (Kind=nag_wp)	::	k
Real (Kind=nag_wp), Intent (In)	::	x, s

C Header Interface

double

k (const double *x, const double *s)

1: $x$ – Real (Kind=nag_wp) Input
2: $s$ – Real (Kind=nag_wp) Input: On entry: the values of $x$ and $s$ at which $k (x, s)$ is to be calculated.

k must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d05abf is called. Arguments denoted as Input must not be changed by this procedure.

Note: k should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d05abf. If your code inadvertently does return any NaNs or infinities, d05abf is likely to produce unexpected results.

2: $g$ – real (Kind=nag_wp) Function, supplied by the user. External Procedure

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:

Fortran Interface

Function g (

Real (Kind=nag_wp)	::	g
Real (Kind=nag_wp), Intent (In)	::	x

C Header Interface

double

g (const double *x)

1: $x$ – Real (Kind=nag_wp) Input: On entry: the value of $x$ at which $g (x)$ is to be calculated.

g must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d05abf is called. Arguments denoted as Input must not be changed by this procedure.

Note: g should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d05abf. If your code inadvertently does return any NaNs or infinities, d05abf is likely to produce unexpected results.

3: $lambda$ – Real (Kind=nag_wp) Input

On entry: the value of the parameter

λ

of the integral equation.

4: $a$ – Real (Kind=nag_wp) Input

On entry:

a

, the lower limit of integration.

5: $b$ – Real (Kind=nag_wp) Input

On entry:

b

, the upper limit of integration.

Constraint:

b > a

6: $odorev$ – Logical Input

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 .TRUE. then an odd or even solution is sought depending upon the value of ev.

7: $ev$ – Logical Input

On entry: is ignored if odorev is .FALSE.. Otherwise, if ev is .TRUE., an even solution is sought, whilst if ev is .FALSE., an odd solution is sought.

8: $n$ – Integer Input

On entry: the number of terms in the Chebyshev series which approximates the solution

f (x)

Constraint:

n \geq 1

9: $cm (ldcm, ldcm)$ – Real (Kind=nag_wp) array Workspace

10: $f1 (ldcm, 1)$ – Real (Kind=nag_wp) array Workspace

11: $wk (2, nt2p1)$ – Real (Kind=nag_wp) array Workspace

12: $ldcm$ – Integer Input

On entry: the first dimension of the arrays cm and f1 and the second dimension of the array cm as declared in the (sub)program from which d05abf is called.

Constraint:

ldcm \geq n

13: $nt2p1$ – Integer Input

On entry: the second dimension of the array wk as declared in the (sub)program from which d05abf is called. The value

2 \times n + 1

14: $f (n)$ – Real (Kind=nag_wp) array Output

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 $odorev = .TRUE.$ and $ev = .TRUE.$ .
$m = 2 n$	if $odorev = .TRUE.$ and $ev = .FALSE.$ .
$m = n - 1$	if $odorev = .FALSE.$ .

15: $c (n)$ – Real (Kind=nag_wp) array Output

On exit: the coefficients

c_{i}

, for

i = 1, 2, \dots, n

, of the Chebyshev series approximation to

f (x)

. When odorev is .TRUE., this series contains polynomials of even order only or of odd order only, according to ev being .TRUE. or .FALSE. respectively.

16: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

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

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $a = ⟨ value ⟩$ and $b = ⟨ value ⟩$ .
Constraint: $b > a$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

$ifail = 2$: 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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

No explicit error estimate is provided by the routine 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

d05abf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

d05abf 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.

9 Further Comments

The time taken by d05abf 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 d05abf, to obtain the

f_{i} ≃ f (x_{i})

and the

c_{i}

, and in subsequent applications of c06dcf to obtain

f (x)

at selected points.

The program runs for

n = 5

and

n = 10

10.1 Program Text

Program Text (d05abfe.f90)

10.2 Program Data

None.

10.3 Program Results

Program Results (d05abfe.r)

NAG Library Manual, Mark 27.2

Interfaces: FL CL CPP AD

NAG FL Interface Introduction

D05 (Inteq) Chapter Contents

D05 (Inteq) Chapter Introduction

d05ab: FL CL CPP AD

NAG FL Interfaced05abf (fredholm2_​smooth)

▸▿ 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 FL Interface
d05abf (fredholm2_smooth)