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Merge pull request #108 from SciML/ChrisRackauckas-patch-1
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Improve the motivation for HW1 Problem 1
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ChrisRackauckas authored Jul 15, 2023
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Expand Up @@ -14,16 +14,22 @@ Homework 1 is a chance to get some experience implementing discrete dynamical
systems techniques in a way that is parallelized, and a time to understand the
fundamental behavior of the bottleneck algorithms in scientific computing.

Please email the results to `[email protected]` titled "**Homework 1: (Your Name)**".

## Problem 1: A Ton of New Facts on Newton

In lecture 4 we looked at the properties of discrete dynamical systems to see that running
many systems for infinitely many steps would go to a steady state. This process is used as
a numerical method known as **fixed point iteration** to solve for the steady state of
systems $x_{n+1} = f(x_{n})$. Under a transformation (which we will do in this homework),
it can be used to solve rootfinding problems $f(x) = 0$ to solve for $x$.

In this problem we will look into Newton's method. Newton's method is the
dynamical system defined by the update process:

$$x_{n+1} = x_n - \left(\frac{dg}{dx}(x_n)\right)^{-1} g(x_n)$$

For these problems, assume that $\frac{dg}{dx}$ is non-singular.
For these problems, assume that $\frac{dg}{dx}$ is non-singular. We will prove a few properties
to show why, in practice, Newton methods are preferred for quickly calculating the steady
state.

### Part 1

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