This Sage quickstart tutorial was developed for the MAA PREP Workshop “Sage: Using Open-Source Mathematics Software with Undergraduates” (funding provided by NSF DUE 0817071).

Invaluable resources are the Sage wiki http://wiki.sagemath.org/interact (type “sage interact” into Google), http://interact.sagemath.org (a collection of contributed interacts), and the interact documentation.

How would one create an interactive cell? First, let’s focus on a new thing to do! Perhaps we just want a graph plotter that has some options.

So let’s start by getting the commands for what you want the output to look like. Here we just want a simple plot.

```
sage: plot(x^2,(x,-3,3))
Graphics object consisting of 1 graphics primitive
```

Then abstract out the parts you want to change. We’ll be letting the
user change the function, so let’s make that a variable `f`.

```
sage: f=x^3
sage: plot(f,(x,-3,3))
Graphics object consisting of 1 graphics primitive
```

This was important because it allowed you to step back and think about what you would really be doing.

Now for the technical part. We make this a `def` function - see the
*programming tutorial*.

```
sage: def myplot(f=x^2):
....: show(plot(f,(x,-3,3)))
```

Let’s test the `def` function `myplot` by just calling it.

```
sage: myplot()
```

If we call it with a different value for `f`, we should get a
different plot.

```
sage: myplot(x^3)
```

So far, we’ve only defined a new function, so this was review. To make
a “control” to allow the user to interactively enter the function, we just preface the function with
`@interact`.

```
sage: @interact
sage: def myplot(f=x^2):
....: show(plot(f,(x,-3,3)))
```

Note

Technically what `@interact` does is wrap the function, so the
above is equivalent to:

```
def myplot(..): ...
myplot=interact(myplot)
```

Note that we can still call our function, even when we’ve used
`@interact`. This is often useful in debugging it.

```
sage: myplot(x^4)
```

We can go ahead and replace other parts of the expression with
variables. Note that `_` is the function name now. That is a just
convention for throw-away names that we don’t care about.

```
sage: @interact
sage: def _(f=x^2, a=-3, b=3):
....: show(plot(f,(x,a,b)))
```

If we pass `('label', default_value)` in for a control, then the
control gets the label when printed. Here, we’ve put in some text for
all three of them. Remember that the text must be in quotes! Otherwise
Sage will think that you are referring (for example) to some variable
called “lower”, which it will think you forgot to define.

```
sage: @interact
sage: def _(f=('$f$', x^2), a=('lower', -3), b=('upper', 3)):
....: show(plot(f,(x,a,b)))
```

We can specify the type of control explicitly, along with options.
See *below* for more detail on the possibilities.

```
sage: @interact
sage: def _(f=input_box(x^2, width=20, label="$f$")):
....: show(plot(f,(x,-3,3)))
```

Here we demonstrate a bunch of options. Notice the new controls:

`range_slider`, which passes in*two*values,`zoom[0]`and`zoom[1]``True`/`False`gets converted to checkboxes for the end user

```
sage: @interact
sage: def _(f=input_box(x^2,width=20),
....: color=color_selector(widget='colorpicker', label=""),
....: axes=True,
....: fill=True,
....: zoom=range_slider(-3,3,default=(-3,3))):
....: show(plot(f,(x,zoom[0], zoom[1]), color=color, axes=axes,fill=fill))
```

There is also one button type to *disable automatic updates*.

The previous interact was a bit ugly, because all of the controls were
stacked on top of each other. We can control the layout of the widget
controls in a grid (at the top, bottom, left, or right) using the
`layout` parameter.

```
sage: @interact(layout=dict(top=[['f', 'color']],
....: left=[['axes'],['fill']],
....: bottom=[['zoom']]))
sage: def _(f=input_box(x^2,width=20),
....: color=color_selector(widget='colorpicker', label=""),
....: axes=True,
....: fill=True,
....: zoom=range_slider(-3,3, default=(-3,3))):
....: show(plot(f,(x,zoom[0], zoom[1]), color=color, axes=axes,fill=fill))
```

There are many potential types of widgets one might want to use for interactive control. Sage has all of the following:

- boxes
- sliders
- range sliders
- checkboxes
- selectors (dropdown lists or buttons)
- grid of boxes
- color selectors
- plain text

We illustrate some more of these below. For complete detail, see the official interact documentation.

```
sage: @interact
sage: def _(frame=checkbox(True, label='Use frame')):
....: show(plot(sin(x), (x,-5,5)), frame=frame)
```

```
sage: var('x,y')
sage: colormaps=sage.plot.colors.colormaps.keys()
sage: @interact
sage: def _(cmap=selector(colormaps)):
....: contour_plot(x^2-y^2,(x,-2,2),(y,-2,2),cmap=cmap).show()
```

```
sage: var('x,y')
sage: colormaps=sage.plot.colors.colormaps.keys()
sage: @interact
sage: def _(cmap=selector(['RdBu', 'jet', 'gray','gray_r'],buttons=True),
sage: type=['density','contour']):
....: if type=='contour':
....: contour_plot(x^2-y^2,(x,-2,2),(y,-2,2),cmap=cmap, aspect_ratio=1).show()
....: else:
....: density_plot(x^2-y^2,(x,-2,2),(y,-2,2),cmap=cmap, frame=True,axes=False,aspect_ratio=1).show()
```

By default, ranges are sliders that divide the range into 50 steps.

```
sage: @interact
sage: def _(n=(1,20)):
....: print factorial(n)
```

You can set the step size to get, for example, just integer values.

```
sage: @interact
sage: def _(n=slider(1,20, step_size=1)):
....: print factorial(n)
```

Or you can explicitly specify the slider values.

```
sage: @interact
sage: def _(n=slider([1..20])):
....: print factorial(n)
```

And the slider values don’t even have to be numbers!

```
sage: @interact
sage: def _(fun=('function', slider([sin,cos,tan,sec,csc,cot]))):
....: print fun(4.39293)
```

Matrices are automatically converted to a grid of input boxes.

```
sage: @interact
sage: def _(m=('matrix', identity_matrix(2))):
....: print m.eigenvalues()
```

Here’s how to get vectors from a grid of boxes.

```
sage: @interact
sage: def _(v=('vector', input_grid(1, 3, default=[[1,2,3]], to_value=lambda x: vector(flatten(x))))):
....: print v.norm()
```

As a final problem, what happens when the controls get so complicated that it would counterproductive to see the interact update for each of the changes one wants to make? Think changing the endpoints and order of integration for a triple integral, for instance, or the example below where a whole matrix might be changed.

In this situation, where we don’t want any updates until we specifically
say so, we can use the `auto_update=False` option. This will create a
button to enable the user to update as soon as he or she is ready.

```
sage: @interact
sage: def _(m=('matrix', identity_matrix(2)), auto_update=False):
....: print m.eigenvalues()
```