Hi Scholars!

This week we consider Banach’s fixed point theorem, for which we use the simplified variant below.

Theorem: If a functionfmaps a subsetDof real numbers back intoDandfis a contraction, thenfhas a unique fixed point and each sequence, with initial iterate inDand successive iterates generated by repeated applyingf,will converge to the unique fixed point off.

This classic result, although powerful, can be difficult to visualize. The primary matter to address is how to show the repeated application of an operation for a 1D function *on the same 2D plot.* We took inspiration from others’ efforts online and decided to use cosine and plot (x^k, x^(k+1)), and then (x^(k+1), x^(k+1)), and then (x^(k+1), x^(k+2)). The idea here is to help the reader think about switching the horizontal and vertical components since the vertical component for a current iterate will be the horizontal component for the next iterate. This process appears more intuitive (to me) upon including the yellow line y = x. Having said this, the diagram would be *greatly* aided by a couple minutes of discussion with a student describing this process, particularly noting the iterate tick marks that are being added on the axes in the first few frames of the animation.

Another difficulty with creating this visual is the fact the sequence converges at a geometric rate. Thus, only a handful of iterations can be seen at a given scale. A “zoom in” camera is used so that more of the sequence could be visualized. Also, without the zoom-in feature, the tick marks and iterate labels on the horizontal and vertical axes quickly begin to overlap (which is hidden in the zoom-in version). The camera zoom is set to be slower than the convergence rate so that it would still be clear that the sequence is approaching the fixed point (i.e. rectangles appear to get smaller).

Lastly, the fixed point residual at the top numerically shows how quickly the sequence goes to zero. I think this helps give better scale since it still feels rather abstract only seeing the rectangles “get a little smaller” with each step when the convergence is quite fast.

That’s all for now. If you have suggestions for future topics/diagrams, let me know!

Stay Awesome.

Howard