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Merge pull request #108 from SciML/ChrisRackauckas-patch-1
Improve the motivation for HW1 Problem 1
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@@ -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. | ||
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Please email the results to `[email protected]` titled "**Homework 1: (Your Name)**". | ||
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## Problem 1: A Ton of New Facts on Newton | ||
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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 $x$. | ||
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In this problem we will look into Newton's method. Newton's method is the | ||
dynamical system defined by the update process: | ||
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$$x_{n+1} = x_n - \left(\frac{dg}{dx}(x_n)\right)^{-1} g(x_n)$$ | ||
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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. | ||
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### Part 1 | ||
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