Math 151B, Spring 2008

Applied Numerical Methods

  • Final exam: Monday, June 9, 2008, 11:30am- 2:30pm, in MS 6229.
  • You are responsible for all sections for the final.
    Lectures: MWF 11:00am - 11:50am, MS 6229

    Instructor: Virginia B. Pasour
    Office: MS 6617B
    Office hours: Wed 9-10am, Thurs 4-5pm, or by appointment.
    E-mail: pasour@math.ucla.edu

    Discussion Section: Thursday, 11:00am - 11:50am, MS 6229

    Teaching Assistant: Frances Hammock
    Office: MS 6139
    Office hours: Thursdays (12pm-2pm)
    E-mail: @math.ucla.edu

    Textbook: R.L. Burden and J.D. Faires, Numerical Analysis, 8th ed. , Brooks/ Cole.
    Introduction to numerical methods with emphasis on algorithms, analysis of algorithms, and computer implementation issues. Solution of nonlinear equations. Numerical differentiation, integration, and interpolation. Direct methods for solving linear systems.

    Errata for the textbook

    A copy of the textbook will be placed on reserve at the SEL Library.

    Requisites: courses 151A.

    Useful Links:
  • PIC Lab: Boelter Hall 2817
    http://www.pic.ucla.edu/piclab/
  • MATLAB documentation
  • More about matlab
  • Another MATLAB Documentation, thanks to Prof. C. Anderson, UCLA
  • Numerical Recipes in C

    Class Web Page: http://www.math.ucla.edu/~pasour/151b.1.08s/index.html

    Homework Assignments:
    Homework assignments consist of both theoretical ("pencil-and-paper" type) and computational work.
    For computational work, please provide your program as well as the program output.
    The homework assignments will be assigned and collected at the beginning of Friday lecture.
    No late homework will be accepted.
    Please check the Class Web Page for the current schedule and homework.
    It is part of your duty to work additional problems from the textbook, and not only those assigned in the homeworks.

    Computing:
    Enrolled students will have accounts in the computer labs in Boelter Hall 2817.
    Students are encouraged to use Matlab for the programming assignments. If you would like to use another programming language, please see me.

    Examinations: One midterm exam and one final exam.
    Midterm: Monday, May 5th, 11am-11:50am (lecture).
    Final: Tuesday, June 9th, 11:30am- 2:30am, in MS 6229.
    The midterm examination is closed-book and closed-note.
    No exams at a time other than the designated ones will be allowed (exceptions for illness with document proof, or emergency).

    Grading Policy: Homework assignments: 30%. Midterm: 30%. Final: 40%.
    Your lowest homework score will not be counted towards your final homework grade.


    LECTURES/HOMEWORK ASSIGNMENTS
    March 31: Overview

    April 2: Section 5.1: Introduction to the solution of initial value problems.

    April 4: Section 5.2: Derivations of Euler's method. Definition of convergence.
    April 7: Sections 5.2, 5.3: Error bounds and asymptotic error estimate for Euler's method. Local truncation error, global error.

    April 9: Sections 5.2, 5.3, 5.4: Convergence proof for Euler's method. Derivation of Runge-Kutta methods.

    April 11: Section 5.4: Runge-Kutta methods (continued). Derivation of the general second order Runge-Kutta methods.
    HW 1 (complete): 5.1: 1(abd), 2(d), 3, 6, 7, 8(a,b)
    5.2: 1, 5, 7, 9, 15 (look at 17 - not turn in)

    HW 1 Even Answers
    April 14: Another source: Timestep estimation. Model problem analysis; intervals of absolute stability.

    April 16: Another source: Timestep estimation for general equations.

    April 18: Section 5.11: Implicit methods (Trapezoidal rule, Backward Euler). Comparison of ODE methods.
    HW 2 (complete): 5.4: 15, 23.
    Additional RK4 Problems
    Implement the adaptive time step version of RK4 in Matlab and use it to solve (again) Section 5.4, problem 15.

    HW 2 Selected Answers
    Adaptive Runge-Kutta 4 with no do over
    Adaptive Runge-Kutta 4 with do over
    April 21: Section 5.11: Implicit methods. Solving the implicit equations. Stiff differential equations.

    April 23: Section 5.6: Overview of multistep methods.

    April 25: Section 5.9: Numerical methods for systems of ODE's.
    HW 3 (complete): 5.11: 1(a,b), 3(a,b), 7(a,b), 13(a,b).
    Additional HW 3 Problems

    HW 3 Selected Answers
    Backward (Implicit) Euler method
    April 28: Section 5.9: Results on numerical methods for systems; convergence results, error bounds, asymptotic error estimates. Regions of absolute stability.

    April 30: Section 11.3: Two point boundary value problems. Finite difference approximation.

    May 2: Sections 7.1, 11.3: Review of vector and matrix norms. Error estimates for linear two-point boundary value problem.
    HW 4 (complete): 5.10: 1, 3, 5, 7.
    Additional HW 4 Problem
    Additional HW4 Problem Answers
    May 5: Midterm.

    May 7: Section 11.4: Midterm discussion. Programming considerations for two point boundary value problems.

    May 9: Section 7.3: Iterative methods for the solution of linear systems of equations. Gauss-Jacobi.
    HW 5 (complete): 5.9: 1, 9.
    7.1: 1, 2, 3(c,d), 4(c,d), 5(a,b), 7, 9, 13.
    HW 5 Selected Answers
    Midterm Answers
    May 12: Section 7.3: Iterative methods (continued). Gauss-Seidel. Error analysis for iterative methods.

    May 14: Sections 7.1, 7.3, 7.4: Error analysis for iterative methods (continued). Relationship between error and the residual. Stopping criterion.

    May 16: Sections 7.3, 7.4: Convergence results for iterative methods. Condition number.
    HW 6 (complete): 11.3: 1, 5, 7, 9.
    Answer to 11.3, #9
    May 19: Section 8.1: Discrete least squares approximation. Construction of the normal equations.

    May 21: Sections 8.1, another source: Derivation of the normal equations. Matrix/vector formulation of the discrete least squares problem.

    May 23: Another source: Using the QR decomposition to solve normal equations. Relation of QR to Gram-Schmidt.
    HW 7 (complete): 7.3: 5(a,d), 7(a,d), 17, 18, 21 (for part b, use 1(a)), 22, 24, 26.
    7.4: 1(a,b), 3(a,b), 7, 9.
    Selected Answers for HW7
    May 26: Memorial Day

    May 28: Section 8.5: Introduction to discrete Fourier approximation.

    May 30: Section 8.5: Fourier approximation (continued). Complex form of the discrete Fourier approximation.
    HW 8 (complete): 8.1: 3, 5, 9, 11, 13. + verify results for QR factorization problem from Friday's class
    June 2: Section 8.6: The fast Fourier transform.

    June 4: Review.

    June 6: More review.