NAG Library Routine Document

D05AAF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     LAMBDA – REAL (KIND=nag_wp)Input

On entry: the value of the parameter $\lambda$ of the integral equation.

2:     A – REAL (KIND=nag_wp)Input

On entry: $a$, the lower limit of integration.

3:     B – REAL (KIND=nag_wp)Input

On entry: $b$, the upper limit of integration.

Constraint: $\mathbf{B}>\mathbf{A}$.

4:     K1 – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure

K1 must evaluate the kernel $k\left(x,s\right)=k_1\left(x,s\right)$ of the integral equation for $a\le s\le x$.

The specification of K1 is:

FUNCTION K1 ( X, S) REAL (KIND=nag_wp) K1 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 $k_1\left(x,s\right)$ 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. Parameters denoted as Input must not be changed by this procedure.

5:     K2 – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure

K2 must evaluate the kernel $k\left(x,s\right)=k_2\left(x,s\right)$ of the integral equation for $x<s\le b$.

The specification of K2 is:

FUNCTION K2 ( X, S) REAL (KIND=nag_wp) K2 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 $k_2\left(x,s\right)$ 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. Parameters denoted as Input must not be changed by this procedure.

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:     G – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure

G must evaluate the function $g\left(x\right)$ for $a\le x\le b$.

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 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. Parameters denoted as Input must not be changed by this procedure.

7:     F(N) – 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:     C(N) – 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:     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:   IND – INTEGERInput

On entry: determines the forms of the kernel, $k\left(x,s\right)$, and the function $g\left(x\right)$.

$\mathbf{IND}=0$

$k\left(x,s\right)$ is not centro-symmetric (or no account is to be taken of centro-symmetry).

$\mathbf{IND}=1$

$k\left(x,s\right)$ is centro-symmetric and $g\left(x\right)$ is odd.

$\mathbf{IND}=2$

$k\left(x,s\right)$ is centro-symmetric and $g\left(x\right)$ is even.

$\mathbf{IND}=3$

$k\left(x,s\right)$ is centro-symmetric but $g\left(x\right)$ is neither odd nor even.

Constraint: $\mathbf{IND}=0$, $1$, $2$ or $3$.

11:   W1(LDW1,LDW2) – REAL (KIND=nag_wp) arrayWorkspace

12:   W2(LDW2,$4$) – REAL (KIND=nag_wp) arrayWorkspace

13:   WD(LDW2) – REAL (KIND=nag_wp) arrayWorkspace

14:   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:   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.

Constraint: $\mathbf{LDW2}\ge 2\times \mathbf{N}+2$.

16:   IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. 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 $-1\text{ or }1$ 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 $-\mathbf{1}\text{ or }\mathbf{1}$ 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).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

On entry,

$\mathbf{A}\ge \mathbf{B}$ or $\mathbf{N}<1$.

$\mathbf{IFAIL}=2$

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.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (d05aafe.f90)

9.2  Program Data

9.3  Program Results

Program Results (d05aafe.r)