### About

# Description of the simulation window

The simulation has two windows with an *x, y *coordinate system. As
default the left one shows a power function (parabola) of fourth grade.
Its **roots** are to be calculated (the *x* values for which *y*
is zero). Its formula is visible in a text field:

*y = 0.1(x+4)(x+1)(x-2)(x-3)-0.1 *

*= 0.1(x ^{4} - 3x^{2} + 10x + 23) *

The first formulation indicates that there are 4 roots which are near to
the integers *-4, -1, 2* and *3*. Substituting the constant
element (*-0.1*) by *0,* these integers would be exact
solutions. The roots of the predefined function are irrational numbers.

The range of the *x* coordinate goes from *-x _{max} *to

*+x*. The default value of

_{max}*x*is

_{max}*5*; it can be changed in the number field. The range of the

*y*coordinate goes from -12 to +12. Changing to another

*y*scaling is achieved by using appropriate factors (default

*0,1*) in the editable formula.

The magenta colored point is the starting point of the calculation
(default *x _{0}= - 4.5)*. It can be drawn with the mouse
along the function curve to another abscissa.

*While the blue iteration points wanders along the curve in the positive*

*x*direction, the starting point stays fixed

*till a root has been calculated with the required accuracy*

*delta.*Then it will jump to this root.

The required deviation* *from exactly zero* delta = y -0 *can
be defined in the white number field (default* 0.00001*). The
current deviation during the calculation process is shown in the gray *y*
field below. Two gray number fields retain the values *xo* and *y0
*of the initial point and hence also of the last root found.

The current iteration point is shown in blue. Once a root has been calculated with the required precision, it jumps to the first point of the next iteration.

At high accuracy, the movement of the iteration point will no longer be
recognizable after a few iteration steps. For this reason the right
window shows a zoomed section of the curve with self adjusting scales.
Up o 4 consecutive points of the iteration are retained, the current one
blue, three trailing ones in red (after a *flyback* blue and red
coincide for one point).

In the zoom window one can follow the flybacks of the iteration process
and the reduction of the *x *step by a factor of 10 at such an
event up to the highest resolution. During this process the curve will
appear more and more like a straight line.

The **Start/ Stop **button controls the iteration process. **Restart**
leads back to the default situation. The number of iteration steps per
second is defined by the slider **speed **between** **1 and 25
(default 2).** Default** stops the iteration when a root has been
found; to get the next one shift the initial point with the mouse and **Start
**again. Refusing the Option **One root only **calculates all roots
in consecutve order.

The speed of the Simulation is determined by the stepwise time control of visualization. It has no relation to the very short time interval which the computer would need to calculate a new root without these interruptions

The text field of the **formula is editable**. You can input any
formula to determine its roots. Take care to adjust *x _{max}*
and the ordinate scale to the specifics of the function.

# Iteration algorithm

The equation *f(x) = g(x) *

can be formulated as *f(x) - g(x) = 0 *

Hence the general task to determine conditions of equality of 2
functions for a value *x* is reduced to finding the *roots *of
one equation, where its value is zero. If *g(x)* is a number *q*,
the task is to find a value *x* for which *f(x)* has the value *q*.

Logically the task presents the inversion of the simple task to
calculate the value of *f(x)* for a given *x*. Yet an analytic
solution is possible only for very simple functions, as a linear or a
second order parabola. With trigonometric functions or logarithms, in
classical technique, we use tables of corresponding numbers, without
normally asking how they originated.

Numerically the roots of any function can be calculated by iteration up
to every wanted, finite degree of accuracy. Instead of looking for an *inverting*
calculus, we simply calculate "forward". We take an initial arbitrary
value of *x* and calculate *y = f(x)*. In general *y*
will not be zero, and hence *x* will not be a root. Now we vary *x
*__systematically__ (iterate *x*) until *y* becomes
smaller than a desired accuracy *delta*.

For easy understanding and demonstration this simulation uses a very
simple iteration algorithm in form of a calculating loop. In each step
of the loop the following commands are performed (you will find the code
at the *Evolution page *of the EJS* *model);

*d x = 0.1/n*; defines the step width in dependence on a step
variable *n*

*x = x + dx*; goes one step further

the corresponding *y*- value is calculated with the given formula
for *f(x)* (*Fixed Relations *page)

**if** (*y>0*) ( it is assumed
that initially *y<0*, with changing sign (*y>0*) a root
has been crossed) {

*x = x - 2*dx*; jump back to last point

*n=10*n*; reduce step by factor 10

}

if (*Math.abs(y)<2*delta*) when *y*
is sufficiently small {

_pause(); -stop calculation

}

While this algorithm is very simple and easy to understand, it needs
many steps. One can speed it up considerably using variable step widths,
that adjust themselves to the steepness of the curve or even to its
curvature. In the zoom window one recognizes that the function can
quickly be approximated well by a linear. The root of a linear
determined by 2 consecutive points of the iteration or even of a
parabola determined by 3 of them, will be very close to the root of the
curve itself after just one more step of iteration. The linear
approximation is used in the so called *Newton* algorithm.

With the availability of fast PCs the time needed for iteration
algorithms is so minute that one can use them universally. Standard
programs like *Excel* or *Mathematica* offer ready solutions.

Our simple algorithm is easily extended to calculate all roots in a
given interval in one run. An additional * if-
else-* logic condition is sufficient to adjust the flyback
command to the 2 cases of the initial

*y*value being positive or negative.

**if **(* y0<0*) {

*dx = 0.1/n*;

*x = x + dx*;

if (* y > 0*) {

*x = x - 2*dx*;

*n = 10*n*;

}}

**else** {

*dx = 0.1/n*;

*x = x + dx*;

if (* y < 0*) {

*x = x - 2*dx*;

*n =10*n*;

}}

As the formula field is editable, one can determine roots for any arbitrary function.

**E1: **Start the default situation. Watch the flybacks with
changing step width in the zoom window, and the changes in the number
fields.

**E2:** After **Reset** increase speed to 10. In the zoom window
the consecution of the iteration point, the formation of the trailing
followers and the jump back after crossing a root will be better
recognizable.

**E4: **Change speed to 1. Stop iteration after a flyback. Using
the **Start/Stop** button you can now follow single steps. In the
zoom window you can follow the jumps in scaling of both abcissa and
ordinate.

**E5:** Change speed while the iteration runs.

**E6:** Change the residual value *delta* over a wide range and
control if it is really achieved.

**E7:** Draw the initial point and calculate the 4 roots separately
this way.

**E8: **Delete the constant element in the formula window. Now
the roots should be integers. Control the achieved accuracy!

**E9:** Change the formula in the formula field to *10*sin(x)*
(factor 10 for adjustment to the *y* scale). Calculate the first
root. Repeat the calculation for different values of *delta. *Compare
the results to π = 3.14159 26535 89793...

**E10:** Calculate roots for functions of your own.

### Translations

Code | Language | Translator | Run | |
---|---|---|---|---|

### Software Requirements

Android | iOS | Windows | MacOS | |

with best with | Chrome | Chrome | Chrome | Chrome |

support full-screen? | Yes. Chrome/Opera No. Firefox/ Samsung Internet | Not yet | Yes | Yes |

cannot work on | some mobile browser that don't understand JavaScript such as..... | cannot work on Internet Explorer 9 and below |

### Credits

Dieter Roess - WEH-Foundation; Fremont Teng; Loo Kang Wee

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### Sample Learning Goals

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### For Teachers

## Calculus of Roots by Iteration JavaScript Simulation Applet HTML5

### Instructions

#### Speed Slider and Delta Field Box

#### Function Box

#### Pause at Root Check Box

#### Toggling Full Screen

#### Play/Pause and Reset Buttons

Research

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### Video

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### Version:

### Other Resources

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### end faq

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