naginterfaces.library.ode.ivp_​rk_​zero_​simple

naginterfaces.library.ode.ivp_rk_zero_simple(x, xend, y, fcn, tol, relabs, output=None, g=None, data=None)

ivp_rk_zero_simple integrates a system of first-order ordinary differential equations over an interval with suitable initial conditions, using a fixed order Runge–Kutta method, until a user-specified function, if supplied, of the solution is zero, and returns the solution at points specified by you, if desired.

For full information please refer to the NAG Library document for d02bj

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/d02/d02bjf.html

Parameters

xfloat

The initial value of the independent variable $x$.

xendfloat

The final value of the independent variable. If $xend<x$, integration will proceed in the negative direction.

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

The initial values of the solution $y_1,y_2,\dots ,y_{\text{n}}$ at $x=x$.

fcncallable f = fcn(x, y, data=None)

$fcn$ must evaluate the functions $f_i$ (i.e., the derivatives $y_i^{\prime }$) for given values of its arguments $x,y_1,\dots ,y_{\text{n}}$.

Parameters

xfloat

$x$, the value of the independent variable.

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

$y_{\text{i}}$, for $\text{i}=1,2,\dots ,\text{n}$, the value of the variable.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

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

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

tolfloat

A positive tolerance for controlling the error in the integration. Hence $tol$ affects the determination of the position where $g(x,y)=0$, if $g$ is supplied.

ivp_rk_zero_simple 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 ivp_rk_zero_simple 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 ivp_rk_zero_simple with $relabs=\text{'D'}$ and with each of $tol={10.0}^{-p}$ and $tol={10.0}^{-p-1}$ where $p$ correct significant digits are required in the solution, $y$.

The accuracy of the value $x$ such that $g(x,y)=0$ is indirectly controlled by varying $tol$.

You should experiment to determine this accuracy.

relabsstr, length 1

The type of error control. At each step in the numerical solution an estimate of the local error, $\text{est}$, is made. For the current step to be accepted the following condition must be satisfied:

$$\text{est}=max\left(e_i/({\tau }_r\times max(\left|y_i\right|,{\tau }_a))\right)\le 1.0$$

where ${\tau }_r$ and ${\tau }_a$ are defined by

$relabs$	${\tau }_r$	${\tau }_a$
‘M’	$tol$	1.0
‘A’	${ϵ}_r$	$tol/{ϵ}_r$
‘R’	$tol$	${ϵ}_a$
‘D’	$tol$	${ϵ}_a$

where ${ϵ}_r$ and ${ϵ}_a$ are small machine-dependent numbers and $e_i$ is an estimate of the local error at $y_i$, computed internally. If the 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, $relabs$ should be set to ‘A’ on entry, whereas if the error requirement is in terms of the number of correct significant digits, $relabs$ should be set to ‘R’. If you prefer a mixed error test, $relabs$ should be set to ‘M’, otherwise if you have no preference, $relabs$ should be set to the default ‘D’. Note that in this case ‘D’ is taken to be ‘R’.

outputNone or callable xsol = output(xsol, y, data=None), optional

Note: if this argument is None then a NAG-supplied facility will be used.

$output$ 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 ivp_rk_zero_simple 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 $xend$, ivp_rk_zero_simple will integrate to $xend$ with no further calls to $output$; if a call to $output$ is required at the point $xsol=xend$, $xsol$ must be given precisely the value $xend$.

Parameters

xsolfloat

The output value of the independent variable $x$.

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

The computed solution at the point $xsol$.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

xsolfloat

You must set $xsol$ to the next value of $x$ at which $output$ is to be called.

gNone or callable retval = g(x, y, data=None), optional

Note: if this argument is None then a NAG-supplied facility will be used.

$g$ must evaluate the function $g(x,y)$ 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.

Parameters

xfloat

$x$, the value of the independent variable.

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

$y_{\text{i}}$, for $\text{i}=1,2,\dots ,\text{n}$, the value of the variable.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalfloat

The value of $g(x,y)$ at the specified point.

dataarbitrary, optional

User-communication data for callback functions.

Returns

xfloat

If $g$ is supplied by you, it contains the point where $g(x,y)=0$, unless $g(x,y)\ne 0$ anywhere on the range $x$ to $xend$, in which case, $x$ will contain $xend$ (and the error indicator $errno$ = 6 is set); if $g$ is not supplied by you it contains $xend$. However, if an error has occurred, it contains the value of $x$ at which the error occurred.

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

The computed values of the solution at the final point $x=x$.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $1$)

On entry, $x=xend=⟨value⟩$.

Constraint: $x\ne xend$.

(errno $1$)

On entry, $tol=⟨value⟩$.

Constraint: $⟨value⟩\le tol<0.01$.

(errno $1$)

On entry, $relabs=⟨value⟩$.

Constraint: $relabs=\text{'M'}$, $\text{'A'}$, $\text{'R'}$ or $\text{'D'}$.

(errno $2$)

With the given value of $tol$, no further progress can be made across the integration range from the current point $x=x$.

The solution is returned at this point, $x=⟨value⟩$.

(errno $3$)

With the given value of $tol$, no initial progress can be made from the starting point of integration provided.

(errno $4$)

On the first call to $output$, $xsol$ was returned as $⟨value⟩$, which is inconsistent with $(x,xend)$ — $(⟨value⟩,⟨value⟩)$.

(errno $4$)

On the first call to $output$, $xsol$ remained unchanged.

(errno $5$)

On a call to $output$, $xsol$ was returned as $⟨value⟩$, which is inconsistent with previous $xsol$ and $xend$ — $(⟨value⟩,⟨value⟩)$.

(errno $5$)

On a call to $output$, $xsol$ remained unchanged.

(errno $6$)

No change in sign of the function $g(x,y)$ was detected in the integration range.

(errno $7$)

Unexpected internal error in call to zero-finding routine.

The zero-finding routine returned error flag $⟨value⟩$.

(errno $7$)

Unexpected internal error in call to interpolation routine.

The interpolation routine returned error flag $⟨value⟩$.

(errno $8$)

Unexpected internal error in call to step integrator.

The step integrator returned error flag $⟨value⟩$.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

ivp_rk_zero_simple advances the solution of a system of ordinary differential equations

$$y_i^{\prime }=f_i(x,y_1,y_2,\dots ,y_{\text{n}})\text{,}i=1,2,\dots ,\text{n}\text{,}$$

from $x=x$ to $x=xend$ using a fixed order Runge–Kutta method. The system is defined by $fcn$, which evaluates $f_i$ in terms of $x$ and $y=(y_1,y_2,\dots ,y_{\text{n}})$. The initial values of $y=(y_1,y_2,\dots ,y_{\text{n}})$ must be given at $x=x$.

The solution is returned via the $output$ supplied by you and at points specified by you, 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 by $C^1$ interpolation to the solution. It is assumed that $g(x,y)$ is a continuous function of the variables, so that a solution of $g(x,y)=0$ 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 $g(x,y)=0$, is controlled by the arguments $tol$ and $relabs$.

References

Shampine, L F, 1994, Numerical solution of ordinary differential equations, Chapman and Hall