The Numerics Window

Clicking on "Numerical" in the main window, opens up the Numerics Window.

The system of ODEs from the "Main Window" is copied and used here. After opening the Numerics Window, any changes made on the Main Window are updated on the Numerics Window. (This is different from what happens in the Plot 1 Window, Plot 2 Windows, and Plot 4 Windows.)

The Numerics Window

Under "Methods" there are four different numerical routines to choose from: Euler's Method, Heun's Method (sometimes called the Modified Euler's Method), Runge Kutta 4, or Adaptive RK4 (adaptive stepsize Runge Kutta 4). (Adaptive RK4 is under construction.)

Under "View" there are three different options to go with the numerical method, once the "Apply" button is clicked.

"Table of values".
"Result of calculation".
"Comparison by step size".

The option, "Frequency", alongside "Table of values", allows you to select the number of calculations shown in the table. The options are 1, 2, 5, 10, and 100. Selecting "1" shows every line in the table, "2" every second line, "5" every fifth line, and so on.

The option "Number of decimals" allows you to change the number of decimal places displayed from the default of 6 to any number from 1 to 10. However, this is purely for cosmetic purposes. All calculations are carried out to 15 decimal places.

The option "Font size" allows you to change the font size used in "Table of values", "Result of calculation", and "Comparison by step size" from the default of 9 points, to 10, 12, 12,114, 16, 18, or 20. The effect is immediate. Checking "Bold" makes the font bold, after the font size is changed. This option is useful when projecting the screen in a classroom.

The following buttons are active.

"Apply", which solves the system of differential equations numerically based on the selections made in "Methods" and "View", using the current step size.
"Cancel", which closes the Numerics Window, and any of the windows generated from it.
"Help", which shows this help file.
"Main", brings the Main Window into focus.
"Show ODEs", which shows the current ODEs, initial values, parameters in use, etc.
"Close Tables", which closes any open tables.

Table of values

This uses the selected numerical method and step size to create and display a table of values such as

(This is for the ODE

du/dt = u, u(t₀) = 1, t₀ = 0, t₁ = 1,

with s = 0.1 and f(t) = e^t using Runge Kutta 4).

The maximum number of points calculated is 5000.

If a function f(t) is defined, as it is here with f(t) = e^t, then its values are also displayed. If the system consists of one ODE, as it is here, then the "Error" (the difference between the value of f(t) and the numerical approximation to the solution), and "% Error" are also shown.

The following buttons are active.

"Save", which saves the system of ODEs and current table to a text file, with extension TXT.
"Export", which saves the table as a column delimited text file with extension DTA. This is the same as "Save" except no details of the ODEs are included. This is useful for importing into other software.
"Print" which prints the current table and the system of ODEs to the local printer.
"t1 to t0", which continues the table from where it left off.
"Cancel", which closes the table.

Result of calculation

This generates a one line output similar to "Table of values", but shows only the final result. Unlike the "Table of values", it is not limited to 5000 calculations. Depending on the "t1" and the step size, the time to finish can be long, so a progress bar and an "Abort" button are exposed, allowing you to abandon the calculations.

Comparison by step size

This generates an output similar to "Result of calculation" but it reduces the step size by half until the "Number of step size halvings" has been reached. You can choose the number of halvings to be 4, 8, 12, and 16. Because the time to finish grows exponentially as the number of halvings increases, a progress bar and an "Abort" button are exposed, allowing you to abandon the calculations.

Also displayed is the final time at which the calculation occurred, "t1". If (t1 - t0)/s is not an integer then the calculation will stop earlier than the original t1. That did not happen in this case.

If f(t) is defined then "Ratio" represents the ratio of the previous error to the current error. In this example, these ratios are about 16, which suggests that, with Runge Kutta 4, halving the step size increases the accuracy 16 times.

"Comparison by step size" is not available using Adaptive RK4.