Math 128A, Numerical Analysis, Spring 2008.

• Class Information
  • DSP students: The exam will be between 11:00AM and 5:00PM on May 19 in Room 939 Evans.
  • Sample Final ; the actual final exam might have fewer questions, though.
  • Jeff will have reviews on Wed. 5/14 2:30-5:00PM in 3106 Etcheverry, and Sat. 5/17 3:00-5:30PM in 102 Wurster. He has also prepared a list of practice problems at his website /~jdonatel/128A.html.
  • Final will be cumulative, covering all matrials covered in class, through Section 7.3. Monday May 12 will be the last lecture on Chapter 7.
  • Last homework set is for Chapter 7. It will not be collected, though.
  • I will have a review on Wed. 5/14. Time is Noon-2:PM. Room will be 247 Cory Hall.
  • Midterm II will cover all the materials up to and include Friday's lecture (March 14). But the emphasis will be on material since Midterm I.
  • Special announcement for students in the Wed. 11:00AM-12:00Noon section: Please bring your midterms in to section next week.
  • General information about the course can befound here .
  • A syllabus can be found here . This document is where weekly homework and reading assignments can be found. Homeworks are collected Wednesdays during discussion. For those who can not make it, homework can be handed in on Friday before class.
  • For problem 4 of section 4.8 in the homework set due 4/7, only do 3a as the integral, and only use an upper bound on the quadrature error to find values of m and n.
• Exams and Solutions
  • DSP students: the upcoming midterm II will take place at 12:30PM in 961 Evans and finish at 2:30PM, March 17, 2008.
  • DSP students: the upcoming midterm will take place at 11:30AM in 961 Evans and finish at 1:30PM, Feb. 20, 2008.
  • For the upcoming midterm, there will not be any cheat sheet allowed, nor will be any calculators. There will be a brief review on Friday, Feb. 15. The exam will cover all material through 3.2.
  • Sample Midterm II and Solutions .
  • Sample Midterm I and Solutions .
  • Here is a Grade Curve for Midterm I. The average is 73.4 and median 73. The top 1/3 scored 84 and above, and the middle 1/3 scored between 66-84.
  • Here are the Midterm Solutions.
• Programming Homework
  • Programming Assignment #1 , Due Mar. 5, 2008. Note the change in this assignment (Feb. 25, 2008).
  • Programming Assignment #2 , Due April 9, 2008.
  • Programming Assignment #3 . Note that the initial setup in this assignment has been changed. The due date has been moved to May 9, 2008.
• Pictures
  • Here are temperatures downloaded from www.weather.com , for the City of Berkeley, on Monday, Feb. 29. The daily temperatures are from 00:00AM to 23:00PM. We use numerical differentiation schemes in the book to compute temperature changes on the hour temperature changes on the hour. Note the higher order scheme works better, but covers fewer hours for lack of data. You can his is a download matlab program to see how it works.
  • The well-known Gibbs Phenomenon can be illustrated with interpolating the function f(x) = 1/(1+x^2) with a high degree polynomial. In the following experiment, we interpolate f(x) over 81 points on interval [-2,2]. this picture illustrates the accuracy on the interval [-1.5, 1.5] . It is clear that interpolation accuracy is quite satisfactory here. However, the interpolate actually swings wildly near end points (-2 and 2), as can be seen in this picture . Of course, this problem completely goes away with splines
  • Here are pictures obtained from interpolating daily temperatures downloaded from www.weather.com , for the City of Berkeley, on Monday, Feb. 11. The daily temperatures are from 6AM (6:00) to 9PM (21:00) on 3-hour intervals, a grand total of 6 data points. We use these data to compute an interpolating polynomial of degree $5$ that approximates the temperatures throughout the day from 6AM (6:00) to 9PM. We verify the accuracy of this interpolation by comparing the approximate temperatures with the hourly temperatures downloaded from www.weather.com . The interpolation works well during the day. But the error is quite big around 7AM (when the Sun arises) and 9PM, indicating that interpolation sometimes does not do as well as we would hope.
  • Here is an example of Newton's method dancing in cycles without convergence. However, the same method will converge to the root for a slightly different initial guess p0=1.
  • Newton's Method slows down and becomes less accurate in the case of multiple roots. Here is an example of Newton's method accurate only to 8 digits on the double root x = 0.
• Matlab Codes
  • This is a matlab program that plots the forecast temperatures for Berkeley on Feb. 29/08 and estimated temperature changes using numerical differentiation methods in Section 4.1.
  • This is a sample program to do interpolation with Lagrangian polynomials; and this is a sample program that computes the derivative with Lagrangian polynomials.
  • Most of the methods in the book can be downloaded from the Matlab Program site set up by the authors of the text. The URL is http://www.as.ysu.edu/~faires/Numerical-Analysis/DiskMaterial/programs/matlab .
    • Bisection Method .
    • Fixed Point Iteration.
    • Newton Method .
    • Secant Method .
    • Steffensen's Method .
    • Horner's Method .
    • Muller's Method .
    • Neville's Interpolation Method . Note that we have fixed a bug in the original Faires code.
    • Newton's Divided Differences Method .
    • Hermite Interpolation Method .
    • Natural Cubic Spline Method .
    • Clamped Cubic Spline Method .
    • Bezier Curve Method .
  • Two different ways to evaluate polynomials in single precision: Direct evaluation and nested evaluation. Play with these codes and compare these results with the matlab function polyval.
• Web Links
  • A brief history of numerical analysis.
  • A list of famous disasters caused by numerical mistakes.
  • Renaissance Technologies.
  • SPICE, a general-purpose circuit simulation program.