NAG Library Routine Document

d05bdf  (abel2_weak)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{ck}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure

ck must evaluate the kernel $k\left(t\right)$ of the integral equation (1).

The specification of ck is:

Fortran Interface

Function ck ( t) Real (Kind=nag_wp) :: ck Real (Kind=nag_wp), Intent (In) :: t

C Header Interface

#include nagmk26.h double  ck ( const double *t)

1:     $\mathbf{t}$ – Real (Kind=nag_wp)Input

On entry: $t$, the value of the independent variable.

ck must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d05bdf is called. Arguments denoted as Input must not be changed by this procedure.

Note: ck should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d05bdf. If your code inadvertently does return any NaNs or infinities, d05bdf is likely to produce unexpected results.

2:     $\mathbf{cf}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure

cf must evaluate the function $f\left(t\right)$ in (1).

The specification of cf is:

Fortran Interface

Function cf ( t) Real (Kind=nag_wp) :: cf Real (Kind=nag_wp), Intent (In) :: t

C Header Interface

#include nagmk26.h double  cf ( const double *t)

1:     $\mathbf{t}$ – Real (Kind=nag_wp)Input

On entry: $t$, the value of the independent variable.

cf must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d05bdf is called. Arguments denoted as Input must not be changed by this procedure.

Note: cf should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d05bdf. If your code inadvertently does return any NaNs or infinities, d05bdf is likely to produce unexpected results.

3:     $\mathbf{cg}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure

cg must evaluate the function $g\left(s,y\left(s\right)\right)$ in (1).

The specification of cg is:

Fortran Interface

Function cg ( s, y) Real (Kind=nag_wp) :: cg Real (Kind=nag_wp), Intent (In) :: s, y

C Header Interface

#include nagmk26.h double  cg ( const double *s, const double *y)

1:     $\mathbf{s}$ – Real (Kind=nag_wp)Input

On entry: $s$, the value of the independent variable.

2:     $\mathbf{y}$ – Real (Kind=nag_wp)Input

On entry: the value of the solution $y$ at the point s.

cg must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d05bdf is called. Arguments denoted as Input must not be changed by this procedure.

Note: cg should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d05bdf. If your code inadvertently does return any NaNs or infinities, d05bdf is likely to produce unexpected results.

4:     $\mathbf{initwt}$ – Character(1)Input

On entry: if the fractional weights required by the method need to be calculated by the routine then set $\mathbf{initwt}=\text{'I'}$ (Initial call).

If $\mathbf{initwt}=\text{'S'}$ (Subsequent call), the routine assumes the fractional weights have been computed on a previous call and are stored in work.

Constraint: $\mathbf{initwt}=\text{'I'}$ or $\text{'S'}$.

Note: when d05bdf is re-entered with the value of $\mathbf{initwt}=\text{'S'}$, the values of nmesh, iorder and the contents of work must not be changed.

5:     $\mathbf{iorder}$ – IntegerInput

On entry: $p$, the order of the BDF method to be used.

Suggested value: $\mathbf{iorder}=4$.

Constraint: $4\le \mathbf{iorder}\le 6$.

6:     $\mathbf{tlim}$ – Real (Kind=nag_wp)Input

On entry: the final point of the integration interval, $T$.

Constraint: $\mathbf{tlim}>10\times \mathit{machine precision}$.

7:     $\mathbf{tolnl}$ – Real (Kind=nag_wp)Input

On entry: the accuracy required for the computation of the starting value and the solution of the nonlinear equation at each step of the computation (see Section 9).

Suggested value: $\mathbf{tolnl}=\sqrt{\epsilon }$ where $\epsilon$ is the machine precision.

Constraint: $\mathbf{tolnl}>10\times \mathit{machine precision}$.

8:     $\mathbf{nmesh}$ – IntegerInput

On entry: $N$, the number of equispaced points at which the solution is sought.

Constraint: $\mathbf{nmesh}=2^m+2\times \mathbf{iorder}-1$, where $m\ge 1$.

9:     $\mathbf{yn}\left(\mathbf{nmesh}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{yn}\left(\mathit{i}\right)$ contains the approximate value of the true solution $y\left(t\right)$ at the point $t=\left(\mathit{i}-1\right)\times h$, for $\mathit{i}=1,2,\dots ,\mathbf{nmesh}$, where $h=\mathbf{tlim}/\left(\mathbf{nmesh}-1\right)$.

10:   $\mathbf{work}\left(\mathbf{lwk}\right)$ – Real (Kind=nag_wp) arrayCommunication Array

On entry: if $\mathbf{initwt}=\text{'S'}$, work must contain fractional weights computed by a previous call of d05bdf (see description of initwt).

On exit: contains fractional weights which may be used by a subsequent call of d05bdf.

11:   $\mathbf{lwk}$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which d05bdf is called.

Constraint: $\mathbf{lwk}\ge \left(2\times \mathbf{iorder}+6\right)\times \mathbf{nmesh}+8\times {\mathbf{iorder}}^2-16\times \mathbf{iorder}+1$.

12:   $\mathbf{nct}\left(\mathbf{nmesh}/32+1\right)$ – Integer arrayWorkspace

13:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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,	$\mathbf{iorder}<4$ or $\mathbf{iorder}>6$,
or	$\mathbf{tlim}\le 10\times \mathit{machine precision}$,
or	$\mathbf{initwt}\ne \text{'I'}$ or $\text{'S'}$,
or	$\mathbf{initwt}=\text{'S'}$ on the first call to d05bdf,
or	$\mathbf{tolnl}\le 10\times \mathit{machine precision}$,
or	$\mathbf{nmesh}\ne 2^m+2\times \mathbf{iorder}-1,m\ge 1$,
or	$\mathbf{lwk}<\left(2\times \mathbf{iorder}+6\right)\times \mathbf{nmesh}+8\times {\mathbf{iorder}}^2-16\times \mathbf{iorder}+1$.

$\mathbf{ifail}=2$

The routine cannot compute the $2p-1$ starting values due to an error solving the system of nonlinear equations. Relaxing the value of tolnl and/or increasing the value of nmesh may overcome this problem (see Section 9 for further details).

$\mathbf{ifail}=3$

The routine cannot compute the solution at a specific step due to an error in the solution of the nonlinear equation (2). Relaxing the value of tolnl and/or increasing the value of nmesh may overcome this problem (see Section 9 for further details).

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (d05bdfe.f90)

10.2

Program Data

10.3

Program Results

Program Results (d05bdfe.r)