**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.

Course Notes

Datatypes: double precision floating point numbers, creating arrays, complex arrays, function handles

IEEE Standard 754 floating point numbers (an article on number representation)

How MATLAB evaluates a function

Plotting on rectangular grids, in three dimensions

Simulation - simulating discrete and continuous probability distributions, random walks, assembling data from simulation bar and hist plots.

Notes:

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 ax^{2}+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.

9/14 Images on the web: JPEG image of a bulding lobby. GIF image of a sunflower and a GIF precipitation forecast from the NWS.

Cellular automata: heat1.m This is a simple cellular automaton inspired by the heat equation in a bar or, more generally, “diffusion” processes.

Some “cellular” images: seashell.jpg wolframdress2.jpg

Cellular automata posts:

http://golly.sourceforge.net/ a program to simulate cellular automata

http://pcgamingtips.blogspot.com/2010/05/life-goes-on-further-ruminations-on.html

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

9/28

Some data to download:

http://www1.tiaa-cref.org/public/performance/retirement/data/index.html

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)=e^{3x}(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:

function y=mypolyval(p,x)

n=length(p);

y=p(n)*ones(size(x));

for k=n-1:-1:1

y=p(k)+x.*y;
%basic step: multiply by x and add next coeff

end

10/3 Written notes on iteration, fixed points, stability/instability, the “arrow” graphical approach to following iteration

10/10 Basic implementation of Newton’s method. Graphical interpretation of Newton’s method: following down the tangent. Error analysis: quadratic convergence. Written notes.

10/12 More Newton’s method – following convergence
to multiple roots with one iteration. Written notes:
the formula for the rate of quadratic convergence: e_{n+1}~[f’’(x*)/(2f’(x*))](e_{n})^{2}

10/19 majority rule function majrule.m . Various worked problems: notes10_19.m

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 4^{th} 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