for $a\le x\le b$, when the kernel $k$ is defined in two parts: $k={k}_{1}$ for $a\le s\le x$ and $k={k}_{2}$ for $x<s\le b$. The method used is that of El–Gendi (1969) for which, it is important to note, each of the functions ${k}_{1}$ and ${k}_{2}$ must be defined, smooth and nonsingular, for all $x$ and $s$ in the interval $[a,b]$.
An approximation to the solution $f\left(x\right)$ is found in the form of an $n$ term Chebyshev series $\underset{i=1}{\overset{n}{{\sum}^{\prime}}}}{c}_{i}{T}_{i}\left(x\right)$, where ${}^{\prime}$ indicates that the first term is halved in the sum. The coefficients ${c}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, of this series are determined directly from approximate values ${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, of the function $f\left(x\right)$ at the first $n$ of a set of $m+1$ Chebyshev points:
The values ${f}_{i}$ 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 $m=n-1$. However, if the kernel $k$ is centro-symmetric in the interval $[a,b]$, i.e., if $k(x,s)=k(a+b-x,a+b-s)$, then the routine is designed to take advantage of this fact in the formation and solution of the algebraic equations. In this case, symmetry in the function $g\left(x\right)$ implies symmetry in the function $f\left(x\right)$. In particular, if $g\left(x\right)$ is even about the mid-point of the range of integration, then so also is $f\left(x\right)$, which may be approximated by an even Chebyshev series with $m=2n-1$. Similarly, if $g\left(x\right)$ is odd about the mid-point then $f\left(x\right)$ may be approximated by an odd series with $m=2n$.
4References
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
5Arguments
1: $\mathbf{lambda}$ – Real (Kind=nag_wp)Input
On entry: the value of the parameter $\lambda $ of the integral equation.
2: $\mathbf{a}$ – Real (Kind=nag_wp)Input
On entry: $a$, the lower limit of integration.
3: $\mathbf{b}$ – Real (Kind=nag_wp)Input
On entry: $b$, the upper limit of integration.
Constraint:
${\mathbf{b}}>{\mathbf{a}}$.
4: $\mathbf{k1}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure
k1 must evaluate the kernel $k(x,s)={k}_{1}(x,s)$ of the integral equation for $a\le s\le x$.
On entry: the values of $x$ and $s$ at which ${k}_{1}(x,s)$ is to be evaluated.
k1 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d05aaf is called. Arguments denoted as Input must not be changed by this procedure.
Note:k1 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d05aaf. If your code inadvertently does return any NaNs or infinities, d05aaf is likely to produce unexpected results.
5: $\mathbf{k2}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure
k2 must evaluate the kernel $k(x,s)={k}_{2}(x,s)$ of the integral equation for $x<s\le b$.
On entry: the values of $x$ and $s$ at which ${k}_{2}(x,s)$ is to be evaluated.
k2 must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d05aaf is called. Arguments denoted as Input must not be changed by this procedure.
Note:k2 should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d05aaf. If your code inadvertently does return any NaNs or infinities, d05aaf is likely to produce unexpected results.
Note that the functions ${k}_{1}$ and ${k}_{2}$ must be defined, smooth and nonsingular for all $x$ and $s$ in the interval [$a,b$].
6: $\mathbf{g}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure
g must evaluate the function $g\left(x\right)$ for $a\le x\le b$.
On entry: the values of $x$ at which $g\left(x\right)$ is to be evaluated.
g must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d05aaf 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 d05aaf. If your code inadvertently does return any NaNs or infinities, d05aaf is likely to produce unexpected results.
7: $\mathbf{f}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the approximate values
${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$, of $f\left(x\right)$ evaluated at the first n of $m+1$ Chebyshev points ${x}_{i}$, (see Section 3).
If ${\mathbf{ind}}=0$ or $3$, $m={\mathbf{n}}-1$.
If ${\mathbf{ind}}=1$, $m=2\times {\mathbf{n}}$.
If ${\mathbf{ind}}=2$, $m=2\times {\mathbf{n}}-1$.
8: $\mathbf{c}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the coefficients
${c}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$, of the Chebyshev series approximation to $f\left(x\right)$.
If ${\mathbf{ind}}=1$ this series contains polynomials of odd order only and if ${\mathbf{ind}}=2$ the series contains even order polynomials only.
9: $\mathbf{n}$ – IntegerInput
On entry: the number of terms in the Chebyshev series required to approximate $f\left(x\right)$.
Constraint:
${\mathbf{n}}\ge 1$.
10: $\mathbf{ind}$ – IntegerInput
On entry: determines the forms of the kernel, $k(x,s)$, and the function $g\left(x\right)$.
${\mathbf{ind}}=0$
$k(x,s)$ is not centro-symmetric (or no account is to be taken of centro-symmetry).
${\mathbf{ind}}=1$
$k(x,s)$ is centro-symmetric and $g\left(x\right)$ is odd.
${\mathbf{ind}}=2$
$k(x,s)$ is centro-symmetric and $g\left(x\right)$ is even.
${\mathbf{ind}}=3$
$k(x,s)$ is centro-symmetric but $g\left(x\right)$ is neither odd nor even.
Constraint:
${\mathbf{ind}}=0$, $1$, $2$ or $3$.
11: $\mathbf{w1}({\mathbf{ldw1}},{\mathbf{ldw2}})$ – Real (Kind=nag_wp) arrayWorkspace
12: $\mathbf{w2}({\mathbf{ldw2}},4)$ – Real (Kind=nag_wp) arrayWorkspace
13: $\mathbf{wd}\left({\mathbf{ldw2}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
14: $\mathbf{ldw1}$ – IntegerInput
On entry: the first dimension of the array w1 as declared in the (sub)program from which d05aaf is called.
Constraint:
${\mathbf{ldw1}}\ge {\mathbf{n}}$.
15: $\mathbf{ldw2}$ – IntegerInput
On entry: the second dimension of the array w1 and the first dimension of the array w2 and the dimension of the array wd as declared in the (sub)program from which d05aaf is called.
On entry: ifail must be set to $0$, $-1$ or $1$ 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 $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{a}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{b}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{b}}>{\mathbf{a}}$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=2$
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.
${\mathbf{ifail}}=-99$
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.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
No explicit error estimate is provided by the routine 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 ${c}_{i}$, or
(ii)by comparing the coefficients ${c}_{i}$ or the function values ${f}_{i}$ for two or more values of n.
8Parallelism and Performance
d05aaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d05aaf 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.
Five terms of the Chebyshev series are sought, taking advantage of the centro-symmetry of the $k(x,s)$ and even nature of $g\left(x\right)$ about the mid-point of the range $[0,1]$.
The approximate solution at the point $x=0.1$ is calculated by calling c06dcf.