Math 128B, Numerical Analysis, Spring 2010.
 Class Information

Unconventional ODE schemes by W. Kahan and
R.C. Li. Here is an
email from R.C. on the same subject.

Brent's method for root finding. We will discuss Brent's
method later in the week of March 15.
 Sample
Midterm Exam.
 Makeup class on Friday, Feb. 12 will be from
9:0010:00AM in Room 6 Evans.
 This semester, Ryan Hynd is teaching a oneunit credit
course in matlab, Math
98 during the first 5 weeks of the semester. You are
encouraged to take this oneunit course if you are not quite
familiar with matlab programming. Note that this course is
offered on a firstcome firstserved basis. So sign up as
quickly as you can.
 General information about the course can be found at
http://www.math.berkeley.edu/~mgu/MA128BSpring2010/general.html.
 The class syllabus will be discussed in class. My
current plan is to cover most of the material from Chapter 7
to Chapter 12 in Burden and Faires, plus two topics
of interest to the class. We will probably discuss
 Possible Term Final Projects
 Nonsymmetric QR Algorithm : In this project,
we extend the symmetric QR algorithm to solve the
nonsymmetric eigenvalue problem. This will involve
Hessenberg reduction, and single and double shift QR
algorithm.
 Preconditioned Conjugate Gradient Method : In
this project, we will develop the preconditioned version
of the CG method, and test the effectiveness of various
preconditioners. See Cojugate
gradient method on Wikipedia.
 Variations of the Fast Foruier Transform : The
FFT comes in many different forms and variations. In this
project, we study these variations and develop
corresponding matlab code for doing these transforms. See
Discrete cosine transform and
Discrete sine transform on Wikipedia.
 Brent's Method: We study convergence behavior
of
Brent's method for solving nonlinear equations, and
compare it with the matlab function fzero.
 Quiz and Exam Schedule:
 Feb. 1 (quiz)
 Feb. 22 (quiz)
 March 8 (Midterm)
 Mar. 29 (quiz)
 Apr. 12 (quiz)
 Apr. 26 (quiz)
 Homework and Programming Assignment

Homework #1, due Feb. 1 in discussion.

Homework #2, due Feb. 8 in discussion.

Homework #3, due Feb. 17 in class.

Homework #4, due Feb. 22 in discussion.

Homework #5, due March 1 in discussion.

Homework #6, due March 8 in discussion.

Homework #7, due March 15 in discussion.

Homework #8, due March 29 in discussion.

Homework #9, due April 5 in discussion.

Homework #10, due April 12 in discussion.

Homework #11, due April 19 in discussion.

Homework #12, due April 26 in discussion.

Homework #13, due May 3 in review.

Programming Assignment #1, due Feb. 17 by Midnight.

Programming Assignment #2, due March 15 by Midnight.

Programming Assignment #3, due April 12 by Midnight.

Programming Assignment #4, due May 3 by Midnight. You
will need the
van der Pol solver,
van der Pol function and
boundary condition function for this Assignment.

Take Home Final Exam, due May 12 by Midnight.
 Matlab Codes and Plots

CrankNicolson method for the heat equation. Here is a
Demo program for CrankNicolson method. It requires a
test function.
 Shooting methods can be useful, but are sometimes
unstable. Here is a
Linear Shooting Demo program and a
Nonlinear Shooting Demo program. They call the
Linear Shooting Method and the
Nonlinear Shooting Method . You will need the following
programs to run the demos:
 Steepest Descent Method can have a hard time deciding
whether to converge to a point that is nearly, but not,
zero.
Here is such a function. Note that there is a local
minimum on
the squared function that is not a zero of the original function.

Homotopy Method demo program that demonstrates convergence
behavior of different initial guesses. Here is a
demo function and its
Jacobian matrix. This is the same function used in Section
10.2 in the book. Here is
Homotopy method that solves the nonlinear equations by
using the matlab ODE solver ode45, and here is the
wrapper function called by ode45.

Newton Method for several variables demo program that demonstrates convergence
behavior of different initial guesses. Here is a
demo function and its
Jacobian matrix. This is the function used in Section
10.2 in the book, which turns out to have two solutions, not
one. Here is
Newton's method for systems of nonlinear equations.

rootfinder demo program that demonstrates convergence
behavior of different rootfinders, including the matlab builtin
function fzero,
Newton's method,
Secant method, and
Bisection method. You should run the demo program with
different values of rho. Here is a
demo function and its
derivative.

Fourier Approximations to sin x and Errors. This is a
nonsmooth function, and the approximation is poor near
points x =pi, 0, pi.

This is a matlab program to test discrete and continuous
Fourier Approximations. You will need to download
DLS.m and
EvalFourier.m to run the test program.

Fourier Approximations to exp(cos(x)) and Errors. This
is a smooth and periodic function, and the Fourier
approximations work very well.

Power2Chebyshev.m . This code returns a matrix $C$ such
that a polynomial P(x) = a_0 + a_1 x + ... a_n x^n is
converted into P(x) = b_0 + b_1 T_1(x) + ... b_n T_n(x)
where (b_1, b_1, ..., b_n) = (a_0, a_1, ..., a_n)*C.
 Here is an example with
Newton's Method
to find roots of nonlinear equations.
 Here is the
QR algorithm
for finding eigenvalues of a symmetric tridiagonal
matrix. Last modified April 9, 2009 to fix a bug in deflation.
 Here is an unstable version of the
Householder Transformation.
 The
Power Method
and the
Inverse Power Method codes.

Redix 2 and 3 FFT .
 The FFT
and the
Inverse FFT codes.

This is a demo code for performing Chebyshev
rational approximation on an
example function. Here is the
Chebyshev polynomial evaluation function.

This code converts a power series into a Chebyshev
series. For the function f(x) = 1/(1x/2)
on [1, 1], the Chebyshev series converges much faster than
the power series, requiring almost 40% fewer terms in the
expansion for double precision accuracy.
This matlab plot illustrates this point.

Conjugate Gradient with long recursions. This matlab
code performs an iteration that is mathematically equivalent
to Conjugate Gradient, but does so by explicitly performing
Aconjugation against all previous Aconjudate vectors at
every step. It also plots the nonzero patterns of the
conjudate coefficients.

Preconditioned Conjugate Gradient.

Conjugate Gradient without Preconditioning .
 Student Projects

SunEarthSat.m . This code was written by Eric Keldrauk
in the Fall of 2008 as his final project for planetary
simulations.
 This is a
planetary simulation package written by Daniel Andrews
and Marco Marquez in the Spring of 2009 for their final
project.
 Web Links