Math 467 Numerical Analysis Spring 2009

• (Amazon.com link) Textbook
• Matlab Code for the Text
• Errata for the Text
• Getting started with Matlab

Post Exam 2

1. Week 12
   • Mon, Apr 20 Differentiation (5.1)
     NOTE: We're skipping Chapter 4.
     MATLAB NOTE: Skip Section 5.1.4, we do not have the symbolic toolbox for Matlab
     Homework:
     • p. 251-252: 1, 2, 7, 8*, 9*, 13*
     • p. 253 (Matlab) 1* (Turn in a plot of the result)
       (Update Apr 28: Delete problem 13. Problems 12 and 15 are more like what we need to be able to do.)
   • Wed, Apr 22: Finished Section 5.1.
     • Example: How the does the error behave for the derivative as the stepsize h goes to zero? (Matlab file: diffErrorExamp.m)
     • Example: Richardson Extrapolation on the derivative of f(x)=xexp(x) at x=1. (Matlab file: RichardsonExample.m)
   • Fri, Apr 24: HW 6 DUE
     • Second extrapolation example (RichardsonExample2.m)
     • Example 1, Differentiation formula
     • Example 2, Second derivative
     • Example 3, Numerical Integration
2. Week 13: This week: Finish up integration
   • Mon, Apr 27: For HW 7:
     • Write a Matlab function (diffvec.m) to differentiate a vector of function values, . Assume they are sampled at a fixed rate h.
     • p. 263: 5*, 6*, 7, 8, 10
     • p. 264: 3(c,d)*, 4(c,d)*, and: Approximate the derivative of the integrand using your Matlab code diffvec.m using 16 and 32 points on [0, pi] respectively(*).
     • p. 273: Use the text's code to do exercise 8(*).
   • Wed, Apr 29
     • Summary of formulas for integrals and derivatives
     • How to use Matlab's quad function: Several examples
   • Fri, May 1: HW 7 DUE
     Using Matlab's QUAD function (and perhaps an intro to ODEs)
     • Look at A couple of integrals to do using Matlab's quad function
     • Matlab example function (middle.m)
     • Matlab example function (tripletrialfunc.m)
3. Week 14: Solving Differential Equations
   • Mon, May 4: Intro to IVPs (Chapter 6). HW:
     • Matlab's Quad routine (see previous handout)
     • 1(c), p. 295
     • 1(c), p. 306
     • 1(c), using Runge Kutta order 4 (See below for code)
   • Wed, May 6: Matlab code for IVPs
     • Basic Euler code (eulerBasic.m)
     • Basic RK4 code (solveRK4.m)
     • Driver code for table on p. 322
   • Fri, May 8 (No homework due- Discuss any problems with homework)
     • Summary of using Matlab with ODE solvers
     • Lorenz Equations (lorenzeq.m)
     • Script file to drive the Lorenz solutions using Matlab's built-in solver. (lorenzdriver.m)
     • Script file to drive the Lorenz solutions using our own RK4 code above (lorenzRK4.m)
     • Equations for Predator-Prey (predator1.m)
     • Driver for Predator Prey (driver2.m)
       To use the code, download the files. Simply type the name of the driver file- For example, >> driver2
     • Week 15
       • Mon, May 11: The take home final exam is due by May 19th, 10PM- No exceptions!
         
         • Shooting Method Example (Shooting01.m)
         • File for the BVP equations (BVPeqn.m)
         • Sample error function for the Shooting Method, using Bisection (BVPSample.m)

Post Exam 1

1. Week 7
   • Wed, Mar 4: Vector and matrix norms, introduce Condition Number
     • Homework: Vector handout here and Matrix norm handout here
   • Fri, Mar 6: Finish the norms, LU decomp. and swamping.
     • Homework (Try to finish for Monday's discussion): The Condition Number
     • Use Matlab's lu command (In Matlab, type: help lu) to solve exercise 3(b) and 4(b), p. 109.
     • Matrix B from the condition number homework- As MatrixB.m
     • Script file that has LU decomposition, followed by backsubstitutions (ScriptLU.m)
2. Week 8
   • Mon, Mar 9: Using iterations to solve Ax=b; Jacobi (Section 2.5)
     • Notes from Class- Background theory (New 4/14/09) The Solutions to the background notes exercises
     Homework to turn on Friday: Set 1: 1, 5, 6 (from the vector handout), Set 2: 1, 5, 6 (matrix norm handout), Set 3: 1, 3 (condition number handout)
   • Wed, Mar 11: Jacobi and Gauss-Seidel iterations
     Matlab file to split the matrix A=(D+L+U) for Jacobi iteration (SplitMatrix.m)
     You should also see Programs 2.1 and 2.2 from the text (linked above).
     Example of a driver for the Jacobi iterations (JacobiExample.m)
     
   • Fri, Mar 13: Finish Gauss-Seidel, briefly talk about multidimensional Newton's Method to finish the material we will cover from Chapter 2.
     • Matlab Code and Example (GaussSeidel.pdf)
     • Example: Loops instead of matrix multiplication
     • Multidimensional Newton's Method example in Matlab (MultiNewton.pdf)
3. (Spring Break)
4. Week 9
   • Mon, Mar 30: Finish Newton's Method, start Interpolation. Homework for Friday (only turn in the starred problems):
     • p 99-100: 1(a), 2(a), 5*, 13(a)*, 14(a)
     • p. 121-122: 1(a), 2(a)
     • p. 124: 1*
     • p. 137: 1(a), 2(a), 3(a)*, 4(a)
   • Wed, Apr 1: Newton's Method (finish up), start Interpolation. For homework, use Newton's Method to solve the optimization problem given below. We will set it up in class:
     
     (*) Maximize x+y+z subject to x^2-y^2=z and x^2+z^2=4
     
     Code from class for the homework:
     • Example function, examp.m
     • Newton's Method, multinewton.m
     • The matrix for Exercise 1, p. 124 (Prob1Pg124.m)
     • Program for Gauss-Seidel (GaussSeidel2.m) (This code was in the PDF file in the link above on Mar 13)
     
     
   • Fri, Apr 3: HW 4 DUE. You don't need to turn in the Matlab code, but summarize your answer on your paper with the other solutions.
5. Week 10
   • Mon, Apr 6: Today in-class, we discussed the error in using the Lagrange interpolating polynomial, and began discussing Newton's Divided Difference formulas. We ended the class by looking at the following code:
     Code from class for Section 3.1:
     • newtdd.m
     • nest.m (slightly changed from text)
     • Example of how to use the last two m-files (InterpExamp.m)
     Homework for Section 3.1/3.2: (Turn in starred problems only)
     • page 151-152: 1(a, b), 2(a,b), 3*, 9, 10* (in Matlab- plot the data and the interpolating polynomial).
     • page 153: Computer Problem 1*
     • page 159: 1, 2*, 4
   • Wed, Apr 8: We'll finish Newton's Divided Differences, and begin a little in Chebyshev Polynomials if time.
     Computer Code from today's session:
     • The error function in polynomial interpolation (ClassNewtonError.m)
     • The Runge Phenomenon (RungeExample1.m)
     • Runge, but with Chebyshev points (RungeExample2.m)
   • Fri, Apr 10: HW 5 DUE
     Here are the new Review Questions for Exam 2 and here are Review Set 2 SOLUTIONS
     Homework that was posted:
     • p. 160, Exercises 5, 6
     • p. 169, Exercises 1(d), 2(d), 6
     • p. 170, Exercise 1 (Matlab)
     • Make a piecewise defined polynomial using Matlab: P(x)=3x-5x^2 if 0<=x<=1, 5x-x^3 if 1<=x<=2
6. Week 11
   • Mon, Apr 13: Matlab with Cubic Splines
   • Wed, Apr 15: EXAM 2 (In-class portion, Take Home posted and Due on Monday)
   • Fri, Apr 17: Finished Cubic splines- Applications to parametric curves and antiderivatives.

Handouts from Class

• Class Syllabus
• Class Calendar
• Homework Set 1 (general, not to turn in)

Materials for Exam 1

• Topics for Exam 1
• Review questions.
• Review Solutions (corrections to Questions 11 and 20).
• Take Home Exam is here
• Matlab code for LU decomposition (done in class) is here.

Chapter 1

• Homework Set 1 (DUE FRI, JAN 30)
  Error: On problem 3, the first row of matrix A should be: [1 2 3 x] (as written in the text)
• Matlab Files:
• M-file for Fixed Point Iteration: This version is very basic, but outputs ALL iterates of your starting point. (file: fpi2.m)
• M-file fpiScript.m. The description of the problem is here
• Modified Bisection (for the next example, this is bisection1.m)
• Matlab script for Example 1.7, p 47 (this is M467Temp1_3.m)
• The Wilkinson Polynomial, wilkpoly.m
• In Matlab, we can put a function within a function (see Sample1.m
• Newton's Method with embedded function (newton1.m)
• Newton's Method that examines the error (newtonEx1.m) in a table.
• The secant method as a function (secant1.m)
• Homework Set 2 (DUE FRI, FEB 13)