d02cjc integrates a system of first order ordinary differential equations over a range with suitable initial conditions, using a variable-order, variable-step Adams' method until a user-specified function, if supplied, of the solution is zero, and returns the solution at specified points, if desired.

2 Specification

#include <nag.h>

void

d02cjc (Integer neq,

void	(fcn)(Integer neq, double x, const double y[], double f[], Nag_User comm),

double *x, double y[], double xend, double tol, Nag_ErrorControl err_c,

void	(output)(Integer neq, double xsol, const double y[], Nag_User *comm),

double

(*g)(Integer neq, double x, const double y[], Nag_User *comm),

Nag_User *comm, NagError *fail)

The function may be called by the names: d02cjc, nag_ode_ivp_adams_zero_simple or nag_ode_ivp_adams_gen.

3 Description

d02cjc advances the solution of a system of ordinary differential equations

y_{i}^{'} = f_{i} (x, y_{1}, y_{2}, \dots, y_{neq}), i = 1, 2, \dots, neq,

from

x = x

x = xend

using a variable-order, variable-step Adams' method. The system is defined by fcn, which evaluates

f_{i}

in terms of

x

and

y_{1}, y_{2}, \dots, y_{neq}

. The initial values of

y_{1}, y_{2}, \dots, y_{neq}

must be given at

x = x

The solution is returned via output at specified points, if desired: this solution is obtained by

C^{1}

interpolation on solution values produced by the method. As the integration proceeds a check can be made on the user-specified function

g (x, y)

to determine an interval where it changes sign. The position of this sign change is then determined accurately. It is assumed that

g (x, y)

is a continuous function of the variables, so that a solution of

g (x, y) = 0.0

can be determined by searching for a change in sign in

g (x, y)

. The accuracy of the integration, the interpolation and, indirectly, of the determination of the position where

g (x, y) = 0.0

, is controlled by the arguments tol and err_c.

For a description of Adams' methods and their practical implementation see Hall and Watt (1976).

4 References

Hall G and Watt J M (ed.) (1976) Modern Numerical Methods for Ordinary Differential Equations Clarendon Press, Oxford

5 Arguments

1: $neq$ – Integer Input

On entry: the number of differential equations.

Constraint:

neq \geq 1

2: $fcn$ – function, supplied by the user External Function

fcn must evaluate the first derivatives

y_{i}^{'}

(i.e., the functions

f_{i}

) for given values of their arguments

x, y_{1}, y_{2}, \dots, y_{neq}

The specification of fcn is:

void	fcn (Integer neq, double x, const double y[], double f[], Nag_User *comm)

1: $neq$ – Integer Input

On entry: the number of differential equations.

2: $x$ – double Input

On entry: the value of the independent variable

x

3: $y [neq]$ – const double Input

On entry:

y [i - 1]

holds the value of the variable

y_{i}

, for

i = 1, 2, \dots, neq

4: $f [neq]$ – double Output

On exit:

f [i - 1]

must contain the value of

f_{i}

, for

i = 1, 2, \dots, neq

5: $comm$ – Nag_User *

Pointer to a structure of type Nag_User with the following member:

p – Pointer: On entry/exit: the pointer $comm \to p$ should be cast to the required type, e.g., struct user *s = (struct user *)comm → p, to obtain the original object's address with appropriate type. (See the argument comm below.)

Note: fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02cjc. If your code inadvertently does return any NaNs or infinities, d02cjc is likely to produce unexpected results.

3: $x$ – double * Input/Output

On entry: the initial value of the independent variable

x

Constraint:

x \neq xend

On exit: if

g

is supplied, x contains the point where

g (x, y) = 0.0

, unless

g (x, y) \neq 0.0

anywhere on the range x to xend, in which case, x will contain xend. If

g

is not supplied x contains xend, unless an error has occurred, when it contains the value of

x

at the error.

4: $y [neq]$ – double Input/Output

On entry: the initial values of the solution

y_{1}, y_{2}, \dots, y_{neq}

x = x

On exit: the computed values of the solution at the final point

x = x

5: $xend$ – double Input

On entry: the final value of the independent variable.

$xend < x$: Integration proceeds in the negative direction.

Constraint:

xend \neq x

6: $tol$ – double Input

On entry: a positive tolerance for controlling the error in the integration. Hence tol affects the determination of the position where

g (x, y) = 0.0

, if

g

is supplied.

d02cjc has been designed so that, for most problems, a reduction in tol leads to an approximately proportional reduction in the error in the solution. However, the actual relation between tol and the accuracy achieved cannot be guaranteed. You are strongly recommended to call d02cjc with more than one value for tol and to compare the results obtained to estimate their accuracy. In the absence of any prior knowledge, you might compare the results obtained by calling d02cjc with

tol = {10.0}^{- p}

and

tol = {10.0}^{- p - 1}

where

p

correct decimal digits are required in the solution.

Constraint:

tol > 0.0

7: $err_c$ – Nag_ErrorControl Input

On entry: the type of error control. At each step in the numerical solution an estimate of the local error, est, is made. For the current step to be accepted the following condition must be satisfied:

est = \sqrt{\sum_{i = 1}^{neq} {(e_{i} / (τ_{r} \times | y_{i} | + τ_{a}))}^{2}} \leq 1.0

where

τ_{r}

and

τ_{a}

are defined by

err_c	$τ_{r}$	$τ_{a}$
$Nag_Relative$	tol	$ε$
$Nag_Absolute$	$0.0$	tol
$Nag_Mixed$	tol	tol

where

ε

is a small machine-dependent number and

e_{i}

is an estimate of the local error at

y_{i}

, computed internally. If the appropriate condition is not satisfied, the step size is reduced and the solution is recomputed on the current step. If you wish to measure the error in the computed solution in terms of the number of correct decimal places, then err_c should be set to

Nag_Absolute

on entry, whereas if the error requirement is in terms of the number of correct significant digits, then err_c should be set to

Nag_Relative

. If you prefer a mixed error test, then err_c should be set to

Nag_Mixed

. The recommended value for err_c is

Nag_Mixed

and this should be chosen unless there are good reasons for a different choice.

Constraint:

err_c = Nag_Relative

Nag_Absolute

Nag_Mixed

8: $output$ – function, supplied by the user External Function

output permits access to intermediate values of the computed solution (for example to print or plot them), at successive user-specified points. It is initially called by d02cjc with

xsol = x

(the initial value of

x

). You must reset xsol to the next point (between the current xsol and xend) where output is to be called, and so on at each call to output. If, after a call to output, the reset point xsol is beyond xend, d02cjc will integrate to xend with no further calls to output; if a call to output is required at the point

xsol = xend

, then xsol must be given precisely the value xend.

The specification of output is:

void	output (Integer neq, double xsol, const double y[], Nag_User comm)

1: $neq$ – Integer Input

On entry: the number of differential equations.

2: $xsol$ – double * Input/Output

On entry: the value of the independent variable

x

On exit: you must set xsol to the next value of

x

at which output is to be called.

3: $y [neq]$ – const double Input

On entry: the computed solution at the point xsol.

4: $comm$ – Nag_User *

Pointer to a structure of type Nag_User with the following member:

p – Pointer: On entry/exit: the pointer $comm \to p$ should be cast to the required type, e.g., struct user *s = (struct user *)comm → p, to obtain the original object's address with appropriate type. (See the argument comm below.)

Note: output should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02cjc. If your code inadvertently does return any NaNs or infinities, d02cjc is likely to produce unexpected results.

If you do not wish to access intermediate output, the actual argument output must be the NAG defined null function pointer NULLFN.

9: $g$ – function, supplied by the user External Function

g must evaluate

g (x, y)

for specified values

x, y

. It specifies the function

g

for which the first position

x

where

g (x, y) = 0

is to be found.

The specification of g is:

double

g (Integer neq, double x, const double y[], Nag_User *comm)

1: $neq$ – Integer Input

On entry: the number of differential equations.

2: $x$ – double Input

On entry: the value of the independent variable

x

3: $y [neq]$ – const double Input

On entry:

y [i - 1]

holds the value of the variable

y_{i}

, for

i = 1, 2, \dots, neq

4: $comm$ – Nag_User *

Pointer to a structure of type Nag_User with the following member:

p – Pointer: On entry/exit: the pointer $comm \to p$ should be cast to the required type, e.g., struct user *s = (struct user *)comm → p, to obtain the original object's address with appropriate type. (See the argument comm below.)

Note: g should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02cjc. If your code inadvertently does return any NaNs or infinities, d02cjc is likely to produce unexpected results.

If you do not require the root finding option, the actual argument g must be the NAG defined null double function pointer NULLDFN.

10: $comm$ – Nag_User *

Pointer to a structure of type Nag_User with the following member:

p – Pointer: On entry/exit: the pointer $comm \to p$ , of type Pointer, allows you to communicate information to and from fcn, output and g. An object of the required type should be declared, e.g., a structure, and its address assigned to the pointer $comm \to p$ by means of a cast to Pointer in the calling program. E.g. comm.p = (Pointer)&s. The type pointer will be void * with a C compiler that defines void * and char * otherwise.

11: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_REAL_ARG_EQ: On entry, $x = ⟨ value ⟩$ while $xend = ⟨ value ⟩$ . These arguments must satisfy $x \neq xend$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument err_c had an illegal value.
NE_INT_ARG_LT: On entry, $neq = ⟨ value ⟩$ .
Constraint: $neq \geq 1$ .
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.
NE_NO_SIGN_CHANGE: No change in sign of the function $g (x, y)$ was detected in the integration range.
NE_REAL_ARG_LE: On entry, tol must not be less than or equal to 0.0: $tol = ⟨ value ⟩$ .
NE_TOL_PROGRESS: The value of tol, $⟨ value ⟩$ , is too small for the function to make any further progress across the integration range. Current value of $x = ⟨ value ⟩$ .
NE_TOL_TOO_SMALL: The value of tol, $⟨ value ⟩$ , is too small for the function to take an initial step.
NE_XSOL_INCONSIST: On call $⟨ value ⟩$ to the supplied print function xsol was set to a value behind the previous value of xsol in the direction of integration.
Previous $xsol = ⟨ value ⟩$ , $xend = ⟨ value ⟩$ , new $xsol = ⟨ value ⟩$ .
NE_XSOL_NOT_RESET: On call $⟨ value ⟩$ to the supplied print function xsol was not reset.
NE_XSOL_SET_WRONG: xsol was set to a value behind x in the direction of integration by the first call to the supplied print function.
The integration range is $[⟨ value1 ⟩, ⟨ value2 ⟩]$ , $xsol = ⟨ value ⟩$ .

7 Accuracy

The accuracy of the computation of the solution vector y may be controlled by varying the local error tolerance tol. In general, a decrease in local error tolerance should lead to an increase in accuracy. You are advised to choose

err_c = Nag_Mixed

unless you have a good reason for a different choice.

If the problem is a root-finding one, then the accuracy of the root determined will depend on the properties of

g (x, y)

. You should try to code g without introducing any unnecessary cancellation errors.

8 Parallelism and Performance

d02cjc is not threaded in any implementation.

9 Further Comments

If more than one root is required then d02qfc should be used.

If the function fails with error exit of

fail . code = NE_TOL_TOO_SMALL

, then it can be called again with a larger value of tol if this has not already been tried. If the accuracy requested is really needed and cannot be obtained with this function, the system may be very stiff (see below) or so badly scaled that it cannot be solved to the required accuracy.

If the function fails with error exit of

fail . code = NE_TOL_PROGRESS

, it is probable that it has been called with a value of tol which is so small that a solution cannot be obtained on the range x to xend. This can happen for well-behaved systems and very small values of tol. You should, however, consider whether there is a more fundamental difficulty. For example:

(a)in the region of a singularity (infinite value) of the solution, the function will usually stop with error exit of $fail . code = NE_TOL_PROGRESS$ , unless overflow occurs first. Numerical integration cannot be continued through a singularity, and analytic treatment should be considered;
(b)for ‘stiff’ equations where the solution contains rapidly decaying components, the function will use very small steps in $x$ (internally to d02cjc) to preserve stability. This will exhibit itself by making the computing time excessively long, or occasionally by an exit with $fail . code = NE_TOL_PROGRESS$ . Adams' methods are not efficient in such cases.

10 Example

We illustrate the solution of four different problems. In each case the differential system (for a projectile) is

\begin{array}{l} y^{'} = \tan ϕ \\ v^{'} = \frac{- 0.032 \tan ϕ}{v} - \frac{0.02 v}{\cos ϕ} \\ ϕ^{'} = \frac{- 0.032}{v^{2}} \end{array}

over an interval

x = 0.0

xend = 10.0

starting with values

y = 0.5

v = 0.5

and

ϕ = π / 5

. We solve each of the following problems with local error tolerances

1.0e−4

and

1.0e−5

(i)To integrate to $x = 10.0$ producing output at intervals of 2.0 until a point is encountered where $y = 0.0$ .
(ii)As (i) but with no intermediate output.
(iii)As (i) but with no termination on a root-finding condition.
(iv)As (i) but with no intermediate output and no root-finding termination condition.