Mathematica’s Manipulate function, introduced in version 6 is a powerful and easy to use function making it easy to develop a user interface (UI) application.
Manipulate basic logic ﬂow is simple: It evaluates its expression and displays the result of this evaluation. It does this each and every time any one of the control variables changes value.
The Manipulate expression can be as complicated as we make it. It is in general a function of the control variables. Therefore each time any control variable changes value by the use of a slider or other UI components, the eﬀect of this change is immediately reﬂected in the display of Manipulate. Manipulate evaluates the expression using the new value of the variable and updates the display automatically.
The following diagram helps illustrate the above logic ﬂow
Because the evaluation of the expression occurs when any one of the control variable changes, this might not be a very eﬃcient use of resources.
Instead, we would like to know which speciﬁc variable changed in order to customise which part of the expression to evaluate. In the following section, we show a model to use to detect which control variable changed and to only perform an action based on that speciﬁc change.
By running the Manipulate expression logic as a state machine which detects the speciﬁc variable which changed and updates an expression based on that event only we can make better use of resources.
The event will be the change of a speciﬁc control variable. The state of the machine is stored in a Manipulate control of type None
The state of the machine consists of the event name and any other state information needed.
This setup is similar to how UI programming is done in traditional setting where a speciﬁc event triggers a callback associated with the event.
To simulate event driven UI, an inner dynamic is added, with a Refresh option, inside the Manipulate expression with its own TrackedSymbols option.
Each control variable will have its own Dynamic with a TrackedSymbols for its own variable. Inside this Dynamic, the event control variable is set to indicate which control variable has just triggered.
Since an inner Dynamic can have its own TrackedSymbols, we have eﬀectively moved the task of detecting the change of the control variables from Manipulate down to the inner Dynamics.
The event control variable of type None will be used by the state machine to check which event has just occurred and then perform an action based on the event.
After the action is completed, the event is reset to special reset value.
The above logic repeats again each time a control variable changes value. Refresh is needed to be used inside the inner Dynamic and within the Row construct as will be shown below.
The above method is illustrated with examples, starting with a very simple annotated example, showing how to program a Manipulate which displays the name of the variable which was last changed using a slider.
In these examples, a string is used to indicate the event that occurred, but this can be changed, and other type of values can be used.
When moving the 'a' slider in the above example, the following message will be displayed on the Manipulate window
A similar display results for 'b' when its value changes.
To illustrate the beneﬁt of this approach, we will now use it to show how to numerically solve an ODE. We will have one control variable for the initial condition, where each time the initial condition variable is changed, NDSolve have to be called to obtain a new numerical solution. In addition, we will have a second variable to indicate which color to use to plot the solution with.
Clearly, there is no need to call NDSolve when the color variable is changed as the initial condition did not change.
Using event driven UI programming we are now able to do this. The following digram shows the solution to the above. Notice that the solution is saved in a Manipulate control variable as well as the event name.
Now the above solution is compared to the solution without using event driven model as is normally done.
We notice that the solution above is much simpler, however it is not eﬃcient on resources. The whole expression is evaluated whenever any one of the variables changes value. In other words, NDSolve is called each time the color variable has changed, which is not required.
Mathematica Manipulate can be used to build UI very quickly. Using event driven model can improve the eﬃciency of using Manipulate by giving the user more control of what action to do based on which variable changes.
Using event driven UI is much more eﬃcient on resources, but it requires more logic to be added to the Manipulate expression.
Using event driven UI programming makes using Manipulate more similar to the event/callback model used by other UI systems where the event is viewed as the change of a speciﬁc control variable, and the callback function is the inner Dynamic which detects the change in that speciﬁc variable.