NAG Library Function Document

nag_ode_ivp_rk_range (d02pcc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_ode_ivp_rk_range (d02pcc) is a function for solving the initial value problem for a first order system of ordinary differential equations using Runge–Kutta methods.

2 Specification

#include <nag.h>

#include <nagd02.h>

void

nag_ode_ivp_rk_range (Integer neq,

void	(f)(Integer neq, double t, const double y[], double yp[], Nag_User comm),

double twant, double *tgot, double ygot[], double ypgot[], double ymax[], Nag_ODE_RK *opt, Nag_User *comm, NagError *fail)

3 Description

nag_ode_ivp_rk_range (d02pcc) and its associated functions (nag_ode_ivp_rk_setup (d02pvc), nag_ode_ivp_rk_errass (d02pzc) ) solve the initial value problem for a first order system of ordinary differential equations. The functions, based on Runge–Kutta methods and derived from RKSUITE (Brankin et al. (1991)) integrate

y^{'} = f (t, y) given y (t_{0}) = y_{0}

where

y

is the vector of neq solution components and

t

is the independent variable.

This function is designed for the usual task, namely to compute an approximate solution at a sequence of points. You must first call nag_ode_ivp_rk_setup (d02pvc) to specify the problem and how it is to be solved. Thereafter you call nag_ode_ivp_rk_range (d02pcc) repeatedly with successive values of twant, the points at which you require the solution, in the range from tstart to tend (as specified in nag_ode_ivp_rk_setup (d02pvc)). In this manner nag_ode_ivp_rk_range (d02pcc) returns the point at which it has computed a solution tgot (usually twant), the solution there ygot and its derivative ypgot. If nag_ode_ivp_rk_range (d02pcc) encounters some difficulty in taking a step toward twant, then it returns the point of difficulty tgot and the solution and derivative computed there ygot and ypgot.

In the call to nag_ode_ivp_rk_setup (d02pvc) you can specify the first step size for nag_ode_ivp_rk_range (d02pcc) to attempt or that it compute automatically an appropriate value. Thereafter nag_ode_ivp_rk_range (d02pcc) estimates an appropriate step size for its next step. This value and other details of the integration can be obtained after any call to nag_ode_ivp_rk_range (d02pcc) by examining the contents of the structure opt, see Section 5. The local error is controlled at every step as specified in nag_ode_ivp_rk_setup (d02pvc). If you wish to assess the true error, you must set

errass = Nag_ErrorAssess_on

in the call to nag_ode_ivp_rk_setup (d02pvc). This assessment can be obtained after any call to nag_ode_ivp_rk_range (d02pcc) by a call to the function nag_ode_ivp_rk_errass (d02pzc).

For more complicated tasks, you are referred to functions nag_ode_ivp_rk_onestep (d02pdc), nag_ode_ivp_rk_interp (d02pxc) and nag_ode_ivp_rk_reset_tend (d02pwc).

4 References

Brankin R W, Gladwell I and Shampine L F (1991) RKSUITE: A suite of Runge–Kutta codes for the initial value problems for ODEs SoftReport 91-S1 Southern Methodist University

5 Arguments

1: neq – IntegerInput

On entry: the number of ordinary differential equations in the system to be solved.

Constraint:

neq \geq 1

2: f – function, supplied by the userExternal Function

f must evaluate the first derivatives

y_{i}^{'}

(that is the functions

f_{i}

) for given values of the arguments

t, y_{i}

The specification of f is:

void	f (Integer neq, double t, const double y[], double yp[], Nag_User *comm)

1: neq – IntegerInput

On entry: the number of differential equations.

2: t – doubleInput

On entry: the current value of the independent variable,

t

3: y[neq] – const doubleInput

On entry: the current values of the dependent variables,

y_{i}

, for

i = 1, 2, \dots, neq

4: yp[neq] – doubleOutput

On exit: the values 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.)

3: twant – doubleInput

On entry: the next value of the independent variable,

t

, where a solution is desired.

Constraint: twant must be closer to tend than the previous of tgot (or tstart on the first call to nag_ode_ivp_rk_range (d02pcc)); see nag_ode_ivp_rk_setup (d02pvc) for a description of tstart and tend. twant must not lie beyond tend in the direction of integration.

4: tgot – double *Output

On exit: the value of the independent variable

t

at which a solution has been computed. On successful exit with

fail . code = NE_NOERROR

, tgot will equal twant. For non-trivial values of fail (i.e., those not related to an invalid call of nag_ode_ivp_rk_range (d02pcc)) a solution has still been computed at the value of tgot but in general tgot will not equal twant.

5: ygot[neq] – doubleInput/Output

On entry: on the first call to nag_ode_ivp_rk_range (d02pcc), ygot need not be set. On all subsequent calls ygot must remain unchanged.

On exit: an approximation to the true solution at the value of tgot. At each step of the integration to tgot, the local error has been controlled as specified in nag_ode_ivp_rk_setup (d02pvc). The local error has still been controlled even when

tgot \neq twant

, that is after a return with a non-trivial error.

6: ypgot[neq] – doubleOutput

On exit: an approximation to the first derivative of the true solution at tgot.

7: ymax[neq] – doubleInput/Output

On entry: on the first call to nag_ode_ivp_rk_range (d02pcc), ymax need not be set. On all subsequent calls ymax must remain unchanged.

On exit:

ymax [i - 1]

contains the largest value of

|y_{i}|

computed at any step in the integration so far.

8: opt – Nag_ODE_RK *

Pointer to a structure of type Nag_ODE_RK as initialized by the setup function nag_ode_ivp_rk_setup (d02pvc) with the following members:

totfcn – IntegerOutput: On exit: the total number of evaluations of $f$ used in the primary integration so far; this does not include evaluations of $f$ for the secondary integration specified by a prior call to nag_ode_ivp_rk_setup (d02pvc) with $errass = Nag_ErrorAssess_on$ .

stpcst – IntegerOutput: On exit: the cost in terms of number of evaluations of $f$ of a typical step with the method being used for the integration. The method is specified by the argument method in a prior call to nag_ode_ivp_rk_setup (d02pvc).

waste – doubleOutput: On exit: the number of attempted steps that failed to meet the local error requirement divided by the total number of steps attempted so far in the integration. A ‘large’ fraction indicates that the integrator is having trouble with the problem being solved. This can happen when the problem is ‘stiff’ and also when the solution has discontinuities in a low order derivative.

stpsok – IntegerOutput: On exit: the number of accepted steps.

hnext – doubleOutput: On exit: the step size the integrator plans to use for the next step.

9: 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 f. 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.

10: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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_MEMORY_FREED: Internally allocated memory has been freed by a call to nag_ode_ivp_rk_free (d02ppc) without a subsequent call to the setup function nag_ode_ivp_rk_setup (d02pvc).
NE_NEQ: The value of neq supplied is not the same as that given to the setup function nag_ode_ivp_rk_setup (d02pvc).
$neq = ⟨value⟩$ but the value given to nag_ode_ivp_rk_setup (d02pvc) was $⟨value⟩$ .
NE_NO_SETUP: The setup function nag_ode_ivp_rk_setup (d02pvc) has not been called.
NE_PREV_CALL: The previous call to a function had resulted in a severe error. You must call nag_ode_ivp_rk_setup (d02pvc) to start another problem.
NE_PREV_CALL_INI: The previous call to the function nag_ode_ivp_rk_range (d02pcc) had resulted in a severe error. You must call nag_ode_ivp_rk_setup (d02pvc) to start another problem.
NE_RK_INVALID_CALL: The function to be called as specified in the setup function nag_ode_ivp_rk_setup (d02pvc) was nag_ode_ivp_rk_onestep (d02pdc). However the actual call was made to nag_ode_ivp_rk_range (d02pcc). This is not permitted.
NE_RK_PCC_METHOD: The efficiency of the integration has been degraded. Consider calling the setup function nag_ode_ivp_rk_setup (d02pvc) to re-initialize the integration at the current point with $method = Nag_RK_4_5$ . Alternatively nag_ode_ivp_rk_range (d02pcc) can be called again to resume at the current point.
NE_RK_PDC_GLOBAL_ERROR_S: The global error assessment algorithm failed at the start of the integration.
NE_RK_PDC_GLOBAL_ERROR_T: The global error assessment may not be reliable for t past tgot. $tgot = ⟨value⟩$ .
NE_RK_PDC_POINTS: More than 100 output points have been obtained by integrating to tend. They have been sufficiently close to one another that the efficiency of the integration has been degraded. It would probably be (much) more efficient to obtain output by interpolating with nag_ode_ivp_rk_interp (d02pxc) (after changing to $method = Nag_RK_4_5$ if you are using $method = Nag_RK_7_8$ ).
NE_RK_PDC_STEP: In order to satisfy the error requirements nag_ode_ivp_rk_range (d02pcc) would have to use a step size of $⟨value⟩$ at current $t = ⟨value⟩$ . This is too small for the machine precision.
NE_RK_PDC_TEND: $tend = ⟨value⟩$ has been reached already. To integrate further with same problem the function nag_ode_ivp_rk_reset_tend (d02pwc) must be called with a new value of tend.
NE_RK_TGOT_EQ_TEND: The call to nag_ode_ivp_rk_range (d02pcc) has been made after reaching tend. The previous call to nag_ode_ivp_rk_range (d02pcc) resulted in tgot (tstart on the first call) $= tend$ . You must call nag_ode_ivp_rk_setup (d02pvc) to start another problem.
NE_RK_TGOT_RANGE_TEND: The call to nag_ode_ivp_rk_range (d02pcc) has been made with a twant that does not lie between the previous value of tgot (tstart on the first call) and tend. This is not permitted.
NE_RK_TGOT_RANGE_TEND_CLOSE: The call to nag_ode_ivp_rk_range (d02pcc) has been made with a twant that does not lie between the previous value of tgot (tstart on the first call) and tend. This is not permitted. However twant is very close to tend, so you may have meant it to be tend exactly. Check your program.
NE_RK_TWANT_CLOSE_TGOT: The call to nag_ode_ivp_rk_range (d02pcc) has been made with a twant that is not sufficiently different from the last value of tgot (tstart on the first call). When using $method = Nag_RK_7_8$ , it must differ by at least $⟨value⟩$ .
NE_STIFF_PROBLEM: The problem appears to be stiff.
NW_RK_TOO_MANY: Approximately $⟨value⟩$ function evaluations have been used to compute the solution since the integration started or since this message was last printed.

7 Accuracy

The accuracy of integration is determined by the arguments tol and thres in a prior call to nag_ode_ivp_rk_setup (d02pvc). Note that only the local error at each step is controlled by these arguments. The error estimates obtained are not strict bounds but are usually reliable over one step. Over a number of steps the overall error may accumulate in various ways, depending on the properties of the differential system.

8 Parallelism and Performance

Not applicable.

9 Further Comments

If nag_ode_ivp_rk_range (d02pcc) returns with

fail . code = NE_RK_PDC_STEP

and the accuracy specified by tol and thres is really required then you should consider whether there is a more fundamental difficulty. For example, the solution may contain a singularity. In such a region the solution components will usually be of a large magnitude. Successive output values of ygot and ymax should be monitored (or the function nag_ode_ivp_rk_onestep (d02pdc) should be used since this takes one integration step at a time) with the aim of trapping the solution before the singularity. In any case numerical solution cannot be continued through a singularity, and analytical treatment may be necessary.

Performance statistics are available after any return from nag_ode_ivp_rk_range (d02pcc) by examining the structure opt, see Section 5. If

errass = Nag_ErrorAssess_on

in the call to nag_ode_ivp_rk_setup (d02pvc), global error assessment is available after any return from nag_ode_ivp_rk_range (d02pcc) (except when the error is due to incorrect input arguments or incorrect set up) by a call to the function nag_ode_ivp_rk_errass (d02pzc). The approximate extra number of evaluations of

f

used is given by

2 \times opt \to stpsok \times opt \to stpcst

for

method = Nag_RK_4_5

Nag_RK_7_8

and

3 \times opt \to stpsok \times opt \to stpcst

for

method = Nag_RK_2_3

After a failure with

fail . code = NE_RK_PDC_STEP

, NE_RK_PDC_GLOBAL_ERROR_T or NE_RK_PDC_GLOBAL_ERROR_S the diagnostic function nag_ode_ivp_rk_errass (d02pzc) may be called only once.

If nag_ode_ivp_rk_range (d02pcc) returns with

fail . code = NE_STIFF_PROBLEM

then it is advisable to change to another code more suited to the solution of stiff problems. nag_ode_ivp_rk_range (d02pcc) will not return with

fail . code = NE_STIFF_PROBLEM

if the problem is actually stiff but it is estimated that integration can be completed using less function evaluations than already computed.

10 Example

We solve the equation

y^{''} = - y, y (0) = 0, y^{'} (0) = 1

reposed as

y_{1}^{'} = y_{2} y_{2}^{'} = - y_{1}

over the range

[0, 2 π]

with initial conditions

y_{1} = 0.0

and

y_{2} = 1.0

. We use relative error control with threshold values of

1.0e−8

for each solution component and compute the solution at intervals of length

π / 4

across the range. We use a low order Runge–Kutta method (

method = Nag_RK_2_3

) with tolerances

tol = 1.0e−3

and

tol = 1.0e−4

in turn so that we may compare the solutions. The value of

π

is obtained by using nag_pi (X01AAC).

See also Section 10 in nag_ode_ivp_rk_errass (d02pzc).

10.1 Program Text

Program Text (d02pcce.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (d02pcce.r)

nag_ode_ivp_rk_range (d02pcc) (PDF version)

d02 Chapter Contents

d02 Chapter Introduction

NAG Library Manual

NAG Library Function Documentnag_ode_ivp_rk_range (d02pcc)