Math 222 Numerical and Symbolic Methods in MATH/STAT
Exam 1, Friday Sept. 21
Exam 2 Monday October 22
Things you should know about iteration.
Exam 3 Monday Dec. 3
Final Exam : Thursday, 3-5 PM
Review topics (also see more detailed reviews above)
You can find some old final exams here via links near the bottom of the page.
Datatypes: double precision floating point numbers, creating arrays, complex arrays, function handles
IEEE Standard 754 floating point numbers (an article on number representation)
Plotting on rectangular grids, in three dimensions
Simulation - simulating discrete and continuous probability distributions, random walks, assembling data from simulation bar and hist plots.
8/31 Discussion of MATLAB script files and functions/function files. These are both files with the extension ‘.m’. A script file is just a series of commands that are invoked by giving the file name (without the extension) in the command window, and they are executed as if they were entered one-by-one in the command window. Developing MATLAB functions is the central goal of this course. We created the function quadroots.m to calculate the two roots of a quadratic ax2+bx+c given the coefficients a,b,c as arguments (inputs).
9/5 Loops and sums. If and while statements. Finding the sum of a fixed number of terms; finding a sum while the terms are “large enough”. Defining the Hilbert matrix using nested loops: myhilb.m The same idea can be used to create and analyze general matrices.
9/7 ang.m This is a MATLAB program to “solve a triangle” given its sides. The input is an nx3 array of side length specifications for n triangles.
9/10 Two more functions: ulam.m creates the sequence invented by the mathematician Ulam. Starting with a value n, the next term is the sequence is 3n+1 if n is odd, and is n/2 if n is even. The function bbsort.m sorts a one-dimensional array using the so-called bubble sort method and displays the evolution of the sorting process graphically.
Note: When plotting inside a function, the plot will not be executed until the function has finished UNLESS you include the ‘drawnow’ command, or a ‘pause’ command.
Cellular automata posts:
http://golly.sourceforge.net/ a program to simulate cellular automata
Game of Life as a computer
9/26 Two cellular automata programs:
wolfram.m implements Wolfram’s one dimensional cellular automata
gol.m implements the game of life
Here are two interesting initial states for the game of life. Download them into your current folder and then load them into your MATLAB workspace by saying load acorn, and load gun.
acorn.mat A long-lived game of life starting configuration
gun.mat The Gosper gun
Some data to download:
crefstock.xlsx you can download into your current folder and load into MATLAB using xlsread
sierpinskir.m A recursive construction of the Sierpinski triangle.
10/1 Iteration, fixed points, chaos
Looking for fixed points: plot(x,f(x),x,x) the intersections are the fixed points (why?)
Stability of fixed points: |f’(x*)|<1 ensures a stable fixed point; |f’(x*)|>1 means fixed point is unstable.
Where are the fixed points of f(x)=1/(1+20exp(-6x)). How do you know by looking which ones are stable and which ones are unstable?
What about the function f(x)=e3x(2-x)/28. Which fixed points are stable and which are unstable? How do you know? (You may need to do some calculations for this one…)
Written notes The graphical interpretation of iteration
chaos.m Simulates iterations for the family of functions f(x)=a*x*(1-x). Plots “long term” behavior of the iterations, so that we can “see” stable fixed points, period two orbits, period 4 orbits, ..etc, and then chaos.
Horner’s method for evaluating polynomials:
y=p(k)+x.*y; %basic step: multiply by x and add next coeff
10/3 Written notes on iteration, fixed points, stability/instability, the “arrow” graphical approach to following iteration
10/26 Interpolating values from a given function. Here, high degree polynomials are capable of very good approximation under certain conditions, but they are very sensitive to any errors in the data. Here if we add “noise” of size .01 to the data, the interpolant of degree 12 changes by 20 or 30 times the size of the noise; the phenomenon gets worse for higher degrees.
10/29 The Lagrange fundamental polynomials – directly show how the interpolant can be constructed and show how the interpolant depends on each y data value. For high degree these polynomials have wild swings near the boundary. Written notes here.
11/2 Two programs we developed: polyinterp.m “cans” the standard technique we use for polynomial interpolation. cubeinterp.m is a function to do piecewise cubic interpolation using data from two points on either side of a given point in the domain.
11/5 Some notes: deriving the error expression for the nearest-neighbors cubic interpolation procedure. Then, introduction to least squares: derivation of the normal equations from the orthogonal projection principle. notes11_5.m Performance of our cubic interpolation routine. Least-squares approximation from data input by hand.
11/7 Written notes: review of normal equations for least squares, introduction to meshgrid statement for preparing rectangular coordinate arrays. notes11_7.m least squares approximation of data (entered by user input), approximation of functional data, approximation of a parametric curve. The function pcolor for pseudocolor plots of a function defined at gridpoints.
11/9 pcolor, contour, surf, plot3 (for curves), using the 4th argument of surf for coloring a surface
11/12 Spherical coordinates geometric picture. Various plotting applications: a sphere via spherical coordinates, punching holes in the sphere, a cone in cylindrical coordinates, a general parametric surface.
11/14 Applications: putting holes in a sphere; a geometric pattern formed from random circles, each point colored according to how many circles it falls inside of; the moebius strip as a parametric surface; embedding an image onto a surface. Example tiger.jpg is embedded into a sphere.
Pucci scarf A scarf pattern similar to that shown in the notes
moebius.m The script file to animate the formation of a moebius strip
chance.m Dropping a “ball” through a maze to wind up in a random position
11/28 Review of homework – chebyshev polynomial interpolation, interpolation and least squares problems, plotting surfaces on rectangular and circular domains, plotting a curve
11/30 Some review material
Material on simulation will be added later
Homework 1 Due Friday Sept. 7 by 11:59 PM
Homework 2 Due Monday, Sept. 17 by 11:59 PM
Homework 3 Due Monday, Oct. 15 by 11:59 PM
Homework 4 Due Thursday, Nov. 8 by 11:59 PM
Homework 5 Due Tuesday Nov. 27 by 11:59 PM
hourglass.m computed coefficients, the surface plot
myhourglass.fig the computed outline
myhourglass2.fig in three dimensions
ptpicker.m Auxiliary script file
see notes 11/28 above for worked solutions