NAG Toolbox: nag_ode_ivp_2nd_rkn_interp (d02lz)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{t}$ – double scalar

$t$, the current value at which the solution and its derivative have been computed (as returned in argument t on output from nag_ode_ivp_2nd_rkn (d02la)).

2:     $\mathrm{y}\left(\mathbf{neq}\right)$ – double array

The $\mathit{i}$th component of the solution at $t$, for $\mathit{i}=1,2,\dots ,\mathbf{neq}$, as returned from nag_ode_ivp_2nd_rkn (d02la).

3:     $\mathrm{yp}\left(\mathbf{neq}\right)$ – double array

The $\mathit{i}$th component of the derivative at $t$, for $\mathit{i}=1,2,\dots ,\mathbf{neq}$, as returned from nag_ode_ivp_2nd_rkn (d02la).

4:     $\mathrm{nwant}$ – int64int32nag_int scalar

The number of components of the solution and derivative whose values at twant are required. The first nwant components are evaluated.

Constraint: $1\le \mathbf{nwant}\le \mathbf{neq}$.

5:     $\mathrm{twant}$ – double scalar

The point at which components of the solution and derivative are to be evaluated. twant should not normally be an extrapolation point, that is twant should satisfy

$$\mathit{told}\le \mathbf{twant}\le \mathbf{t}\text{,}$$

or if integration is proceeding in the negative direction

$$\mathit{told}\ge \mathbf{twant}\ge \mathbf{t}\text{,}$$

where told is the previous integration point which is held in an element of the array rwork and is, to within rounding, $\mathbf{t}-\mathbf{hused}$. (hused is given by nag_ode_ivp_2nd_rkn_diag (d02ly).) Extrapolation is permitted but not recommended, and $\mathbf{ifail}=\mathbf{2}$ is returned whenever extrapolation is attempted.

6:     $\mathrm{rwork}\left(\mathit{lrwork}\right)$ – double array

This must be the same argument rwork as supplied to nag_ode_ivp_2nd_rkn (d02la). It is used to pass information from nag_ode_ivp_2nd_rkn (d02la) to nag_ode_ivp_2nd_rkn_interp (d02lz) and therefore the contents of this array must not be changed before calling nag_ode_ivp_2nd_rkn_interp (d02lz).

Optional Input Parameters

1:     $\mathrm{neq}$ – int64int32nag_int scalar

Default: the dimension of the arrays y, yp. (An error is raised if these dimensions are not equal.)

The number of second-order ordinary differential equations being solved by nag_ode_ivp_2nd_rkn (d02la). It must contain the same value as the argument neq in a prior call to nag_ode_ivp_2nd_rkn (d02la).

Output Parameters

1:     $\mathrm{ywant}\left(\mathbf{nwant}\right)$ – double array

The calculated value of the $\mathit{i}$th component of the solution at $t=\mathbf{twant}$, for $\mathit{i}=1,2,\dots ,\mathbf{nwant}$.

2:     $\mathrm{ypwant}\left(\mathbf{nwant}\right)$ – double array

The calculated value of the $\mathit{i}$th component of the derivative at $t=\mathbf{twant}$, for $\mathit{i}=1,2,\dots ,\mathbf{nwant}$.

3:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

If nag_ode_ivp_2nd_rkn_interp (d02lz) is to be used for extrapolation at twant, ifail should be set to $1$ before entry. It is then essential to test the value of ifail on exit.

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

   $\mathbf{ifail}=1$

Illegal input detected, i.e., one of the following conditions:

–	nag_ode_ivp_2nd_rkn (d02la) has not been called;
–	one or both of the arguments neq and lrwork does not match the corresponding argument supplied to the setup function nag_ode_ivp_2nd_rkn_setup (d02lx);
–	no integration steps have been taken since the last call to nag_ode_ivp_2nd_rkn_setup (d02lx) with $\mathbf{start}=\mathit{true}$;
–	$\mathbf{nwant}<1$ or $\mathbf{nwant}>\mathbf{neq}$.

This error exit can be caused if elements of rwork have been overwritten.

W  $\mathbf{ifail}=2$

nag_ode_ivp_2nd_rkn_interp (d02lz) has been called for extrapolation. The values of the solution and its derivative at twant have been calculated and placed in ywant and ypwant before returning with this error number (see Accuracy).

   $\mathbf{ifail}=3$

nag_ode_ivp_2nd_rkn (d02la) last used the high order method to integrate the system of differential equations. Interpolation is not permitted with this method.

   $\mathbf{ifail}=-99$

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

   $\mathbf{ifail}=-399$

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

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: d02lz_example

function d02lz_example


fprintf('d02lz example results\n\n');

% Variables and arrays for integration setup
neq    = 2;
h      = 0;
tol    = 1e-4;
thres  = zeros(neq, 1);
thresp = zeros(neq, 1);
maxstp = int64(1000);
start  = true;
onestp = true;
high   = false;
rwork  = zeros(16+20*neq,1);

% Integration setup
[startOut, rwork, ifail] = d02lx...
   (h, tol, thres, thresp, maxstp, start, onestp, high, rwork);

% Integration variables
t      = 0;
tend   = 20;
y      = [0.5; 0];
yp     = [0;   sqrt(3)];
ydp    = zeros(neq, 1);
tnext  = 2;
nwant  = int64(2);

fprintf('\n  T       Y(1)       Y(2)\n');
fprintf('%4.1f %10.5f %10.5f\n', t, y(1), y(2));

% Integrate by steps and interpolate onto selected values
while (t < tend && ifail == 0)
  [t, y, yp, ydp, rwork, ifail] = d02la(@fcn, t, tend, y, yp, ydp, rwork);
  while (tnext <= t && ifail == 0)
    [ywant, ypwant, ifail] = d02lz(t, y, yp, nwant, tnext, rwork);
    fprintf('%4.1f %10.5f %10.5f\n', tnext, ywant(1:2));
    tnext = tnext + 2;
  end
end

if (ifail == 0)
  [hnext, hused, hstart, nsucc, nfail, natt, thres, thresp, ifail] = ...
    d02ly(nwant, rwork);

  fprintf('\n Number of successful steps = %d\n', nsucc);
  fprintf(' Number of failed steps     = %d\n', nfail);
end



function ydp = fcn(neq, t, y)
% Evaluate second derivatives.
r      = sqrt(y(1)^2+y(2)^2)^3;
ydp(1) = -y(1)/r;
ydp(2) = -y(2)/r;

d02lz example results


  T       Y(1)       Y(2)
 0.0    0.50000    0.00000
 2.0   -1.20573    0.61357
 4.0   -1.33476   -0.47685
 6.0    0.35748   -0.44558
 8.0   -1.03762    0.73022
10.0   -1.42617   -0.32658
12.0    0.05515   -0.72032
14.0   -0.82880    0.81788
16.0   -1.48103   -0.16788
18.0   -0.26719   -0.84223
20.0   -0.57803    0.86339

 Number of successful steps = 108
 Number of failed steps     = 16