gibson:teaching:spring-2016:math445:finaltopics
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gibson:teaching:spring-2016:math445:finaltopics [2016/05/13 10:09] gibson [Evaluating expressions] |
gibson:teaching:spring-2016:math445:finaltopics [2016/05/16 06:45] (current) gibson [Plotting] |
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| ====== Math 445 final exam topics ====== | ====== Math 445 final exam topics ====== | ||
| + | The Math 445 final exam will be comprehensive, covering all material presented in lecture and lab (except for the derivation of differential equations from physics presented in lecture). Below is a broad overview but not exhaustive of topics that might be covered on the exam. | ||
| + | ---- | ||
| + | ===== Matlab syntax ===== | ||
| ====Vectors and matrices==== | ====Vectors and matrices==== | ||
| Line 19: | Line 22: | ||
| * ''* \ + - ='' | * ''* \ + - ='' | ||
| + | ---- | ||
| ====Linear algebra versus elementwise operations=== | ====Linear algebra versus elementwise operations=== | ||
| You should know the difference between linear algebra and elementwise operations on matrices and vectors. For example, given two 2 x 2 matrices ''A'' and ''B'', you should be able to compute ''A * B'' and ''A .* B'' by hand. You should also know in what circumstances it's appropriate to use linear algebra operations or elementwise operations. | You should know the difference between linear algebra and elementwise operations on matrices and vectors. For example, given two 2 x 2 matrices ''A'' and ''B'', you should be able to compute ''A * B'' and ''A .* B'' by hand. You should also know in what circumstances it's appropriate to use linear algebra operations or elementwise operations. | ||
| + | ---- | ||
| ==== Solving systems of equations ==== | ==== Solving systems of equations ==== | ||
| Line 47: | Line 51: | ||
| </code> | </code> | ||
| + | ---- | ||
| ==== Plotting ==== | ==== Plotting ==== | ||
| You should know | You should know | ||
| * how to make $xy$-plots of given functions $y=f(x)$ using **linspace**, elementwise operations on vectors, **plot**, **semilogx**, **semilogy**, and **loglog**. | * how to make $xy$-plots of given functions $y=f(x)$ using **linspace**, elementwise operations on vectors, **plot**, **semilogx**, **semilogy**, and **loglog**. | ||
| - | * how to make 2D contour plots with **linspace**, **meshgrid**, elementwise matrix operations, and **contour** or **contourf** | + | * how to make 2D color plots with **pcolor**, using **linspace**, **meshgrid**, elementwise matrix operations as well |
| - | * how to make 3D surface plots with **linspace**, **meshgrid**, elementwise matrix operations, and **surf** | + | * how to make 2D contour plots with **contourf** and **linspace**, **meshgrid** etc. |
| + | * how to make 2D quiver plots with **quiver** and **linspace**, **meshgrid** etc. | ||
| + | * how to make 3D surface plots with **surf** and **linspace**, **meshgrid**, etc. | ||
| * how to load data from a file and plot it | * how to load data from a file and plot it | ||
| * how to label axes, title a plot, color the lines, show markers on data points, display a coordinate grid, show a colorbar, etc. | * how to label axes, title a plot, color the lines, show markers on data points, display a coordinate grid, show a colorbar, etc. | ||
| - | ==== Log-linear relations ==== | + | ---- |
| - | + | ||
| - | You should know how to infer a functional relation $y=f(x)$ given a logarithmic or linear plot | + | |
| - | , and which of **plot**, **semilogx**, **semilogy**, and **loglog** is best for a given relation $y=f(x)$. | + | |
| ==== Evaluating expressions ==== | ==== Evaluating expressions ==== | ||
| Line 78: | Line 82: | ||
| </code> | </code> | ||
| + | ---- | ||
| ==== Functions ==== | ==== Functions ==== | ||
| - | You should be able to write simple Matlab functions to perform specified computations. For example,if asked to write a Matlab function that 5 | + | You should be able to write simple Matlab functions to perform specified computations. For example, if asked to write a Matlab function that, given a value of $N$, evaluates |
| + | |||
| + | \begin{eqnarray*} | ||
| + | \sum_{n=1}^{N} \frac{n}{(n+1)^2} | ||
| + | \end{eqnarray*} | ||
| + | |||
| + | you should respond with | ||
| + | |||
| + | <code matlab> | ||
| + | function s = f(N) | ||
| + | n=1:N; | ||
| + | s = sum(n./(n+2).^2) | ||
| + | end | ||
| + | </code> | ||
| + | |||
| + | ---- | ||
| + | ====for loops==== | ||
| + | |||
| + | You should know how to use ''for'' loops to perform repeated computations. For example, if asked to write a function that computes the above sum using a ''for'' loop (instead of using the ''sum'' function), you would write | ||
| + | |||
| + | <code matlab> | ||
| + | function s = f(N) | ||
| + | s = 0; | ||
| + | for n = 1:N | ||
| + | s = s + n/(n+2)^2 | ||
| + | end | ||
| + | end | ||
| + | </code> | ||
| + | |||
| + | ---- | ||
| + | ==== random numbers ==== | ||
| + | |||
| + | You should know how to get random numbers of various kinds | ||
| + | |||
| + | * **rand**: a random floating-point number between 0 and 1. | ||
| + | * **randi(max)**: a random integer between 1 and max. | ||
| + | * **randn**: a random number in a Gaussian distribution with mean 0 and standard deviation 1. | ||
| + | |||
| + | Each of these random number generators has a matrix version, as well. E.g. | ||
| + | * **rand(m,n)**: an m x n matrix of random floating-point numbers between 0 and 1. | ||
| + | * **randi(m,n,max)**: an m x n matrix of random integers between 1 and max. | ||
| + | * **randn(m,n)**: an m x n matrix of a random numbers in a Gaussian distribution with mean 0 and standard deviation 1. | ||
| + | |||
| + | ==== if-else statements ==== | ||
| + | |||
| + | You should know how to write simple ''if-else'' statements such as | ||
| + | |||
| + | <code matlab> | ||
| + | x = randn(); | ||
| + | if x < 0 | ||
| + | fprintf('%d is negative\n', x) | ||
| + | elseif x == 0 | ||
| + | fprintf('%d is zero\n', x) | ||
| + | else | ||
| + | fprintf('%d is positive\n', x) | ||
| + | end | ||
| + | </code> | ||
| + | |||
| + | ---- | ||
| + | ===== Lab material ==== | ||
| + | |||
| + | You should have a good grasp on the mathematics and Matlab programming of the lab material. For example, | ||
| + | |||
| + | ---- | ||
| + | ==== log-linear relations ==== | ||
| + | |||
| + | You should know how to infer a functional relation $y=f(x)$ given a logarithmic or linear plot | ||
| + | , and which of **plot**, **semilogx**, **semilogy**, and **loglog** is best for a given relation $y=f(x)$. | ||
| + | |||
| + | ---- | ||
| + | ==== writing functions ===== | ||
| + | |||
| + | You should know how to write Matlab functions that do basic computations, like matrix-vector multiplication. | ||
| + | |||
| + | ---- | ||
| + | ==== hamster dynamics / Google Page Rank ==== | ||
| + | |||
| + | Given a graph of nodes and one-way links between them, you should be able to write a system of equations that governs random walks through the network of links, and then write Matlab code that would calculate the steady-state distribution. | ||
| + | |||
| + | ---- | ||
| + | ==== nonlinear equations and Newton's method ==== | ||
| + | |||
| + | You should know the mathematics behind Newton's method for solving nonlinear equations, how to code it in Matlab, and how to use Matlab's built-in solver **fsolve** to solve nonlinear equations. | ||
| + | |||
| + | ---- | ||
| + | ==== differential equations ==== | ||
| + | |||
| + | You should know now how to write an anonymous function for a system of first-order differential equations $d\vec{x}/dt = \vec{f}(t, \vec{x})$, and how to solve that system of equations numerically using Matlab's **ode45**. And given a quiver plot of a 2-d differential equation, you should be able to draw an approximate solution of the equation starting from a given initial condition, by tracing out a curve that is everywhere tangent to the arrows. | ||
| - | * **fprintf**: how to do basic formatted output with **fprintf**. | ||
| - | * **function**: how to write your own functions. | ||
| - | * scripts: how to write scripts to perform a given sequence of commands. | ||
| - | * hamster dynamics / Google Page Rank: how to translate a graph of nodes and links to a transition matrix and then calculate the steady-state distribution. | ||
gibson/teaching/spring-2016/math445/finaltopics.1463159373.txt.gz · Last modified: 2016/05/13 10:09 by gibson
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