Convergences: Function convergence
In CGRAPH, complex functions can be examined for convergence or divergence through iteration. A graph shows the course of the iteration.
After each iteration, the programme checks whether the absolute value of the function is moving further away from or closer to the starting point. Each function value is plotted in the graph.
Unlike many other graphics, the settings and graphics are located in the same window.
Only functions containing the variables z and param are permitted, whereby the parameter param is mandatory. Enter the desired function in the input field for the function, or select one of the existing functions by clicking on the arrow to the right of the input field.
Complex numbers are permitted for the parameter. The decision limit, on the other hand, is the distance from the respective starting point, i.e. an amount. Therefore, only real numbers greater than zero are permitted here.
The value Convergence at describes the difference between the input and output values of the function that must be exceeded in order for the calculation to be terminated. When this difference is reached, you will receive a message that allows you to perform further calculations.
The value Divergence at describes the difference between the input and output values of the function that must be exceeded in order for the calculation to be terminated and divergence to be determined. When this difference is reached, you will receive a message that allows you to perform further calculations.
After clicking the Create graph button, CGRAPH displays the course of the function and indicates above the graph whether the function converges or diverges.
The image shows a converging spiral for the parameter 0.4+0.14i. of the function zn+1= param * (4-zn2). The default values can be viewed in the tree on the left-hand side. An explanation of how to change the parameter graphically can be found in the notes on changing the parameter in the graphic display.
The values can be changed retrospectively. The graphic is adjusted automatically, at the latest when the Create graphic button is clicked.
The display of the axes is also part of the tree, as the graphics for this type of graphic are independent of each other and can be set separately.