NAG Library Function Document
nag_inteq_fredholm2_split (d05aac)
1
Purpose
nag_inteq_fredholm2_split (d05aac) solves a linear, nonsingular Fredholm equation of the second kind with a split kernel.
2
Specification
#include <nag.h> |
#include <nagd05.h> |
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) |
|
3
Description
nag_inteq_fredholm2_split (d05aac) solves an integral equation of the form
for
, when the kernel
is defined in two parts:
for
and
for
. The method used is that of
El–Gendi (1969) for which, it is important to note, each of the functions
and
must be defined, smooth and nonsingular, for all
and
in the interval
.
An approximation to the solution
is found in the form of an
term Chebyshev series
, where
indicates that the first term is halved in the sum. The coefficients
, for
, of this series are determined directly from approximate values
, for
, of the function
at the first
of a set of
Chebyshev points:
The values
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 . However, if the kernel is centro-symmetric in the interval , i.e., if , 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 implies symmetry in the function . In particular, if is even about the mid-point of the range of integration, then so also is , which may be approximated by an even Chebyshev series with . Similarly, if is odd about the mid-point then may be approximated by an odd series with .
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:
– doubleInput
-
On entry: the value of the parameter of the integral equation.
- 2:
– doubleInput
-
On entry: , the lower limit of integration.
- 3:
– doubleInput
-
On entry: , the upper limit of integration.
Constraint:
.
- 4:
– IntegerInput
-
On entry: the number of terms in the Chebyshev series required to approximate .
Constraint:
.
- 5:
– function, supplied by the userExternal Function
-
k1 must evaluate the kernel
of the integral equation for
.
The specification of
k1 is:
double |
k1 (double x,
double s,
Nag_Comm *comm)
|
|
- 1:
– doubleInput
- 2:
– doubleInput
-
On entry: the values of and at which is to be evaluated.
- 3:
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
k1.
- user – double *
- iuser – Integer *
- p – Pointer
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:
– function, supplied by the userExternal Function
-
k2 must evaluate the kernel
of the integral equation for
.
The specification of
k2 is:
double |
k2 (double x,
double s,
Nag_Comm *comm)
|
|
- 1:
– doubleInput
- 2:
– doubleInput
-
On entry: the values of and at which is to be evaluated.
- 3:
– Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to
k2.
- user – double *
- iuser – Integer *
- p – Pointer
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 and must be defined, smooth and nonsingular for all and in the interval [].
- 7:
– function, supplied by the userExternal Function
-
g must evaluate the function
for
.
The specification of
g is:
double |
g (double x,
Nag_Comm *comm)
|
|
- 1:
– doubleInput
-
On entry: the values of at which is to be evaluated.
- 2:
– 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_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:
– Nag_KernelFormInput
-
On entry: determines the forms of the kernel,
, and the function
.
- is not centro-symmetric (or no account is to be taken of centro-symmetry).
- is centro-symmetric and is odd.
- is centro-symmetric and is even.
- is centro-symmetric but is neither odd nor even.
Constraint:
, , or .
- 9:
– doubleOutput
-
On exit: the approximate values
, for
, of
evaluated at the first
n of
Chebyshev points
, (see
Section 3).
If or , .
If , .
If , .
- 10:
– doubleOutput
-
On exit: the coefficients
, for
, of the Chebyshev series approximation to
.
If this series contains polynomials of odd order only and if the series contains even order polynomials only.
- 11:
– Nag_Comm *
-
The NAG communication argument (see
Section 3.3.1.1 in How to Use the NAG Library and its Documentation).
- 12:
– 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 had an illegal value.
- NE_EIGENVALUES
-
A failure has occurred due to proximity of an eigenvalue.
- NE_INT
-
On entry, .
Constraint: .
- 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, and .
Constraint: .
7
Accuracy
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 , or |
(ii) |
by comparing the coefficients or the function values for two or more values of n. |
8
Parallelism 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.
10
Example
This example solves the equation
where
Five terms of the Chebyshev series are sought, taking advantage of the centro-symmetry of the
and even nature of
about the mid-point of the range
.
The approximate solution at the point
is calculated by calling
nag_sum_cheby_series (c06dcc).
10.1
Program Text
Program Text (d05aace.c)
10.2
Program Data
None.
10.3
Program Results
Program Results (d05aace.r)