naginterfaces.library.ode.ivp_​rkts_​interp

naginterfaces.library.ode.ivp_rkts_interp(n, twant, ideriv, nwant, f, comm, data=None)

ivp_rkts_interp computes the solution of a system of ordinary differential equations using interpolation anywhere on an integration step taken by ivp_rkts_onestep().

For full information please refer to the NAG Library document for d02ps

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/d02/d02psf.html

Parameters

nint

$n$, the number of ordinary differential equations in the system to be solved by the integration function.

twantfloat

$t$, the value of the independent variable where a solution is desired.

iderivint

Determines whether the solution and/or its first derivative are to be computed

$ideriv=0$

compute approximate solution.

$ideriv=1$

compute approximate first derivative.

$ideriv=2$

compute approximate solution and first derivative.

nwantint

The number of components of the solution to be computed. The first $nwant$ components are evaluated.

fcallable yp = f(t, y, data=None)

$f$ must evaluate the functions $f_i$ (that is the first derivatives $y_i^{\prime }$) for given values of the arguments $t,y_i$.

It must be the same procedure as supplied to ivp_rkts_onestep().

Parameters

tfloat

$t$, the current value of the independent variable.

yfloat, ndarray, shape $\left(n\right)$

The current values of the dependent variables, $y_{\text{i}}$, for $\text{i}=1,2,\dots ,\text{n}$.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

ypfloat, array-like, shape $\left(n\right)$

The values of $f_{\text{i}}$, for $\text{i}=1,2,\dots ,\text{n}$.

commdict, communication object, modified in place

Communication structure.

This argument must have been initialized by a prior call to ivp_rkts_setup().

dataarbitrary, optional

User-communication data for callback functions.

Returns

ywantfloat, ndarray, shape $\left(nwant\right)$

An approximation to the first $nwant$ components of the solution at $twant$ if $ideriv=0$ or $2$. Otherwise $ywant$ is not defined.

ypwantfloat, ndarray, shape $\left(nwant\right)$

An approximation to the first $nwant$ components of the first derivative at $twant$ if $ideriv=1$ or $2$. Otherwise $ypwant$ is not defined.

Raises

NagValueError

(errno $1$)

You cannot call this function after the integrator has returned an error.

(errno $1$)

You cannot call this function before you have called the step integrator.

(errno $1$)

You cannot call this function when you have specified, in the setup function, that the range integrator will be used.

(errno $1$)

$\text{method}=-3$ or $3$ in setup, but interpolation is not available for this method. Either use $\text{method}=-2$ or $2$ in setup or use reset function to force the integrator to step to particular points.

(errno $1$)

On entry, $n=⟨value⟩$, but the value passed to the setup function was $\text{n}=⟨value⟩$.

(errno $1$)

On entry, $ideriv=⟨value⟩$.

Constraint: $ideriv=0$, $1$ or $2$.

(errno $1$)

On entry, $nwant=⟨value⟩$ and $n=⟨value⟩$.

Constraint: $1\le nwant\le n$.

(errno $1$)

On entry, a previous call to the setup function has not been made or the communication arrays have become corrupted, or a catastrophic error has already been detected elsewhere.

You cannot continue integrating the problem.

Notes

ivp_rkts_interp and its associated functions (ivp_rkts_setup(), ivp_rkts_onestep(), ivp_rkts_reset_tend(), ivp_rkts_diag() and ivp_rkts_errass()) 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 (see Brankin et al. (1991)), integrate

$$y^{\prime }=f(t,y)\text{given}y\left(t_0\right)=y_0$$

where $y$ is the vector of $\text{n}$ solution components and $t$ is the independent variable.

ivp_rkts_onestep() computes the solution at the end of an integration step. Using the information computed on that step ivp_rkts_interp computes the solution by interpolation at any point on that step. It cannot be used if $\text{method}=3$ or $-3$ was specified in the call to setup function ivp_rkts_setup().

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