NAG Toolbox: nag_ode_ivp_stiff_c1_interp (d02xk)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{xsol}$ – double scalar

The point at which the first $m$ components of the solution are to be evaluated. xsol should not be an extrapolation point, that is xsol should satisfy $\left(\mathbf{xsol}-\mathbf{x}\right)\times \mathbf{hu}\le 0.0$. Extrapolation is permitted but not recommended.

2:     $\mathrm{m}$ – int64int32nag_int scalar

The number of components of the solution whose values at xsol are required. The first $m$ components are evaluated.

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

3:     $\mathrm{ysav}\left(\mathit{ldysav},\mathbf{sdysav}\right)$ – double array

ldysav, the first dimension of the array, must satisfy the constraint $\mathit{ldysav}\ge 1$.

The values provided in the argument ysav on return from the integrator.

4:     $\mathrm{acor}\left(\mathbf{neq}\right)$ – double array

The value returned in position $\left(\mathit{ldysav}+50+\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{neq}$, of the argument rwork returned by the integrator. If one of the forward communication D02N functions is being employed and nag_ode_ivp_stiff_c1_interp (d02xk) is to be used in monitr, then $\mathbf{acor}\left(\mathit{i}\right)$ must contain the value given in position $\left(\mathit{i},2\right)$ of the monitr argument acor, for $\mathit{i}=1,2,\dots ,\mathbf{neq}$ (e.g., see nag_ode_ivp_stiff_exp_fulljac (d02nb)).

5:     $\mathrm{x}$ – double scalar

The latest value at which the solution has been computed, as provided in the argument tcur on return from the optional output nag_ode_ivp_stiff_integ_diag (d02ny).

6:     $\mathrm{nqu}$ – int64int32nag_int scalar

The order of the method used up to the latest value at which the solution has been computed, as provided in the argument nqu on return from the optional output nag_ode_ivp_stiff_integ_diag (d02ny).

Constraint: $\mathbf{nqu}\ge 1$.

7:     $\mathrm{hu}$ – double scalar

The last successful step used, that is the step used in the integration to get to x, as provided in the argument hu on return from the optional output nag_ode_ivp_stiff_integ_diag (d02ny).

8:     $\mathrm{h}$ – double scalar

The next step size to be attempted in the integration, as provided in the argument h on return from the optional output nag_ode_ivp_stiff_integ_diag (d02ny).

Optional Input Parameters

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

Default: the second dimension of the array ysav.

The value used for the argument sdysav when calling the integrator.

Constraint: $\mathbf{sdysav}\ge \mathbf{nqu}+1$.

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

Default: the dimension of the array acor.

The value used for the argument neq when calling the integrator.

Constraint: $1\le \mathbf{neq}\le \mathit{ldysav}$.

Output Parameters

1:     $\mathrm{sol}\left(\mathbf{m}\right)$ – double array

The calculated value of the $\mathit{i}$th component of the solution at xsol, for $\mathit{i}=1,2,\dots ,m$.

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

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

If nag_ode_ivp_stiff_c1_interp (d02xk) is to be used for extrapolation, ifail must be set to $1$ before entry. It is then essential to test the value of ifail on exit for $\mathbf{ifail}=\mathbf{1}$ or $\mathbf{2}$.

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.

   $\mathbf{ifail}=1$

On entry,	$\mathbf{m}<1$,
or	$\mathbf{neq}<1$,
or	$\mathit{ldysav}<1$,
or	$\mathbf{neq}>\mathit{ldysav}$,
or	$\mathbf{m}>\mathbf{neq}$,
or	$\mathbf{nqu}<1$,
or	$\mathbf{sdysav}<\mathbf{nqu}+1$,
or	the BDF integrator was not previously used,
or	the Petzold error test, if applicable, was used.

   $\mathbf{ifail}=2$

On entry, $\mathbf{hu}=0.0$ or $\mathbf{h}=0.0$. This error can only occur if h and hu have been changed by you or possibly if the integrator has failed before calling nag_ode_ivp_stiff_c1_interp (d02xk).

W  $\mathbf{ifail}=3$

nag_ode_ivp_stiff_c1_interp (d02xk) has been called for extrapolation. Before returning with this error exit, the value of the solution at xsol is calculated and placed in sol.

   $\mathbf{ifail}=-99$

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: d02xk_example

function d02xk_example


fprintf('d02xk example results\n\n');

% Initialize setup setup variables and arrays.
neq    = int64(3);
neqmax = neq;
maxord = int64(5);
sdysav = maxord+1;
petzld = false;
tcrit  = 0;
hmin   = 1.0e-10;
hmax   = 10;
h0     = 0;
maxstp = int64(200);
mxhnil = int64(5);
const  = zeros(6, 1);
rwork  = zeros(50+4*neq, 1);

% BDF with Newton iterations.
[const, rwork, ifail] = d02nv( ...
                               neqmax, sdysav, maxord, 'Newton', petzld, ...
                               const, tcrit, hmin, hmax, h0, maxstp, ...
                               mxhnil, 'Average-L2', rwork);

% Numerical Jacobian
nwkjac = neqmax*(neqmax + 1);
[rwork, ifail] = d02ns( ...
                        neq, neqmax, 'Numerical', nwkjac, rwork);

% Initialize variables and arrays.
njcpvt = int64(1);
wkjac  = zeros(nwkjac, 1);
ydot   = zeros(neq, 1);
ysave  = zeros(neq, sdysav);
inform(1:23) = int64(0);
jacpvt(1) = int64(0);
algequ = zeros(neq, 1);

% Integrate to tout by overshooting (itask = 1).
% At monitoring stages output intermediate solutions after interpolating.
t     = 0;
tout  = 10.0;
itask = int64(1);
iout  = 1;
xout  = 2;
y     = [1; 0; 0];
itol  = int64(1);
rtol  = [1e-04];
atol  = [1e-07];

% Output header and initial results.
fprintf('\n      x           y(1)       y(2)          y(3)\n');
fprintf(' %8.3f %12.4f %12.2e %11.4f\n', t, y);

% bogus initial entry values for unitialized input parameters.
irevcm = int64(-999); imon = irevcm; inln = irevcm; ires = irevcm;

% Ask for warning messages only.
itrace = int64(0);

% The reverse communication process is controlled by the value of irevcm
% (which is 0 for the final exit).
while (irevcm ~= 0)
  [t, y, ydot, rwork, inform, ysave, wkjac, jacpvt, imon, inln, ires, ...
   irevcm, ifail] = d02nm( ...
                           t, tout, y, ydot, rwork, rtol, atol, itol, ...
                           inform, ysave, wkjac, jacpvt, imon, inln, ...
                           ires, irevcm, itask, itrace, 'neq', neq, ...
                           'sdysav', sdysav, 'nwkjac', nwkjac);

  % Evaluate the derivative, then use irevcm to place it.
  f(1) = -0.04*y(1) + 1.0e4*y(2)*y(3);
  f(2) =  0.04*y(1) - 1.0e4*y(2)*y(3) - 3.0e7*y(2)*y(2);
  f(3) =  3.0e7*y(2)*y(2);
  if (irevcm == 1 || irevcm == 3 )
    lrw = 50+2*neq;
    rwork(lrw+1:lrw+neq) = f;
  elseif (irevcm == 4)
    lrw = 50+neq;
    rwork(lrw+1:lrw+neq) = f;
  elseif (irevcm == 5)
    ydot = f;
  elseif (irevcm == 9 && imon == 1)
    % Extract useful information about the current step.
    tc    = rwork(19);
    hlast = rwork(15);
    hnext = rwork(16);
    nqu   = int64(rwork(10));

    % xout is intermediate output point.
    while (xout <= tc)
      lrw = 50+neq;
      [sol, ifail] = d02xk( ...
                            xout, neq, ysave, rwork(lrw+1:lrw+neq), ...
                            tc, nqu, hlast, hnext);
      % Output interpolated result, and save it for plotting.
      fprintf(' %8.3f %12.4f %12.2e %11.4f\n', xout, sol);
      iout = iout + 1;
      xout = iout*2;
    end
  end
end

d02xk example results


      x           y(1)       y(2)          y(3)
    0.000       1.0000     0.00e+00      0.0000
    2.000       0.9416     2.70e-05      0.0584
    4.000       0.9055     2.24e-05      0.0945
    6.000       0.8793     1.96e-05      0.1207
    8.000       0.8585     1.77e-05      0.1414
   10.000       0.8414     1.62e-05      0.1586