D05ABF (PDF version)
D05 Chapter Contents
D05 Chapter Introduction
NAG Library Manual

NAG Library Routine Document

D05ABF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

 Contents

    1  Purpose
    7  Accuracy

1  Purpose

D05ABF solves any linear nonsingular Fredholm integral equation of the second kind with a smooth kernel.

2  Specification

SUBROUTINE D05ABF ( K, G, LAMBDA, A, B, ODOREV, EV, N, CM, F1, WK, LDCM, NT2P1, F, C, IFAIL)
INTEGER  N, LDCM, NT2P1, IFAIL
REAL (KIND=nag_wp)  K, G, LAMBDA, A, B, CM(LDCM,LDCM), F1(LDCM,1), WK(2,NT2P1), F(N), C(N)
LOGICAL  ODOREV, EV
EXTERNAL  K, G

3  Description

D05ABF 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  Parameters

1:     K – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure
K must compute the value of the kernel kx,s of the integral equation over the square axb, asb.
The specification of K is:
FUNCTION K ( X, S)
REAL (KIND=nag_wp) K
REAL (KIND=nag_wp)  X, 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 kx,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. Parameters denoted as Input must not be changed by this procedure.
2:     G – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure
G must compute the value of the function gx of the integral equation in the interval axb.
The specification of G is:
FUNCTION G ( X)
REAL (KIND=nag_wp) G
REAL (KIND=nag_wp)  X
1:     X – REAL (KIND=nag_wp)Input
On entry: the value of x at which gx 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. Parameters denoted as Input must not be changed by this procedure.
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 – LOGICALInput
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 .TRUE. then an odd or even solution is sought depending upon the value of EV.
7:     EV – LOGICALInput
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 – INTEGERInput
On entry: the number of terms in the Chebyshev series which approximates the solution fx.
Constraint: N1.
9:     CMLDCMLDCM – REAL (KIND=nag_wp) arrayWorkspace
10:   F1LDCM1 – REAL (KIND=nag_wp) arrayWorkspace
11:   WK2NT2P1 – REAL (KIND=nag_wp) arrayWorkspace
12:   LDCM – INTEGERInput
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: LDCMN.
13:   NT2P1 – INTEGERInput
On entry: the second dimension of the array WK as declared in the (sub)program from which D05ABF is called. The value 2×N+1.
14:   FN – REAL (KIND=nag_wp) arrayOutput
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=.TRUE. and EV=.TRUE..
m=2N if ODOREV=.TRUE. and EV=.FALSE..
m=N-1 if ODOREV=.FALSE..
15:   CN – REAL (KIND=nag_wp) arrayOutput
On exit: the coefficients ci, for i=1,2,,N, of the Chebyshev series approximation to fx. 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 – INTEGERInput/Output
On entry: IFAIL must be set to 0, -1​ or ​1. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value -1​ or ​1 is recommended. If the output of error messages is undesirable, then the value 1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is 0. 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 or -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,AB or N<1.
IFAIL=2
A failure has occurred due to proximity to 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 3.8 in the Essential Introduction for further information.
IFAIL=-399
Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.
IFAIL=-999
Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction 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 ci, or
(ii) by comparing the coefficients ci or the function values fi 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 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 D05ABF, to obtain the fifxi and the ci, and in subsequent applications of C06DCF to obtain fx 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)


D05ABF (PDF version)
D05 Chapter Contents
D05 Chapter Introduction
NAG Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2015