gibson:teaching:fall-2016:math753:exam1topics
====== Differences ====== This shows you the differences between two versions of the page.
| Next revision | Previous revision | ||
|
gibson:teaching:fall-2016:math753:exam1topics [2016/10/10 09:08] gibson created |
gibson:teaching:fall-2016:math753:exam1topics [2016/10/10 09:27] (current) gibson [Solving systems of equations] |
||
|---|---|---|---|
| Line 12: | Line 12: | ||
| ==== Solving nonlinear equations ==== | ==== Solving nonlinear equations ==== | ||
| - | Understand | + | We studied two methods for finding roots of the 1d equation $f(x) = 0$. |
| - | * the bisection algorithm | + | |
| - | * the Newton algorithm | + | Bisection algorithm: understand |
| - | * the convergence rates of bisection and Newton algorithms | + | * how it works |
| + | * what it requires, in terms of the function $f(x)$ and the starting conditions for the iteration | ||
| + | * the linear convergence rate | ||
| + | |||
| + | Newton method: understand | ||
| + | * how it works, mathematically and geometrically | ||
| + | * what it requires, in terms of the function $f(x)$ and the starting conditions for the iteration | ||
| + | * the quadratic convergence rate and the assumptions necessary to get quadratic converegence | ||
| + | |||
| + | You should be able to write pseudo code or Julia code for either method. You're not expected to get the Julia syntax perfectly. | ||
| + | |||
| + | ==== Solving systems of equations ==== | ||
| + | |||
| + | We've studied two methods for solving systems of equations $Ax=b$. | ||
| + | |||
| + | LU decomposition: understand | ||
| + | * how LU decomposition works | ||
| + | * the formulae for the multipliers $\ell_{ij}$, why the formulae are what they are | ||
| + | * the backsubstition and forward substitution algorithms (formulae and where they came from) | ||
| + | * how you piece together LU decomp, forward substitution, and backward substitution to solve $Ax=b$ | ||
| + | |||
| + | QR decomposition | ||
| + | * how the Gram-Schimidt algorithm computes a QR factorization | ||
| + | * the formulae for $q_j$, $r_{ij}$, and $r_{jj}$, and where the formulae came from | ||
| + | * how to use a QR decomp to solve an $m \times m$ (square) $Ax=b$ problem | ||
| + | * how to use a QR decomp to solve an $m \times n$ (oblong) least-squares $Ax=b$ problem | ||
| + | |||
| + | In addition we covered some basic linear algebra theory, including | ||
| + | * transposes | ||
| + | * inner product | ||
| + | * the 2-norm, for both vectors and matrices | ||
| + | * orthogonal matrices | ||
| + | * how the inner product and 2-norm of both vectors and matrices are preserved under orthogonal transformation | ||
| + | |||
| + | About formulae: Rather than memorizing them, focus on understanding their meaning and being able | ||
| + | to derive them or the first few instances of a general formula. You should be able to write | ||
| + | pseudocode or Julia code (again, not necessarily perfect syntax) to implement a given formula. | ||
| - | ==== Solving systems of linear equations ==== | ||
gibson/teaching/fall-2016/math753/exam1topics.1476115712.txt.gz · Last modified: 2016/10/10 09:08 by gibson
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 3.0 Unported
