Math 270C: Numerical Analysis (Part C) - Bo Li

Spring quarter, 2009

Math 270C: Numerical Analysis (Part C)

General Course Information

Prerequisites Calculus and linear algebra.
Textbook There will be no textbooks. Lecture notes will be distributed.
Lectures 12:00 - 12:50, Mondays, Wednesdays, and Fridays. AP&M 5829.
Instructor Bo Li
Office: AP&M 5723. Office phone: (858) 534-6932. E-mail:
Office hours: 1:00 - 2:00 Wednesdays, or by appointment.
Teaching assistant Chris Deotte
Office: AP&M 5760. E-mail:
Office hours: 3:00 pm - 5:00 pm, Wednesdays.
Class web page
Homework Assigned, collected, and graded regularly. Weekly homework paper should be placed into the homework dropbox located on the 6th floor of AP&M. Each homework is due by 6:00 pm, Friday.
Exams There will be one midterm exam and one final exam. The final exam will be cumulative. The time and place of the final exam will be the same as the qualifying exam. More information later.
Grading The final course grade will be determined based on the homework and exams with the weight: homework - 30%, midterm exam 20%, and final exam - 50%.

Lecture Notes: PDF


Assignment 1, due 6:00 pm, Friday, 4/3/09:
Chapter 1: 1, 2 (1), 4, 9, 12, 14.
Assignment 2, due 6:00 pm, Friday, 4/10/09:
Chapter 1: 16, 31, 33.
Assignment 3, due 6:00 pm, Friday, 4/17/09:
Chapter 1: 21, 22, 23, 24, 25.
Assignment 4, due 6:00 pm, Friday, 4/24/09:
Chapter 1: 41, 42, 43, 44.
Assignment 5, due 6:00 pm, Friday, 5/1/09:
Chapter 2: 1, 2, 3, 5, 7, 9.
Assignment 6, due 6:00 pm, Friday, 5/8/09:
Chapter 2: 11, 12, 13.
Assignment 7, due 6:00 pm, Friday, 5/15/09:
Chapter 3: 4, 9, 16.

Homework Solution

Course Outline

Polynomial approximation
  • The Weierstrass Theorem and Bernstein polynomials
  • best uniform approximations: existence, uniqueness, characterization
  • Chebyshev polynomials of first kind
  • trigonometric polynomials, second Weierstrass Theorem
  • best uniform approximations by trigonometric polynomials
  • modulus of continuity, Lipschitz functions, Jackson Theorems and their consequences
  • least-squares approximations
  • orthogonal polynomials: definition, basic properties, the Gram-Schmidt orthogonalization
  • more properties of orthogonal polynomials
  • Legendre polynomials

Polynomial interpolation
  • Lagrange interpolation: definition and Lagrange formula
  • Lagrange interpolation: Newton formula and divided differences, iterated linear interpolation
  • Lagrange interpolation: remainder, optimal interpolation points, Peano kernals
  • Hermite interpolation and divided differences with repeated points. Hermite-Gennochi formula
  • convergence of interpolation polynomials
  • piecewise polynomial interpolation. splines
  • trigonometric polynomial interpolation
  • fast Fourier transforms

Numerical quadrature
  • The basics: degree of precision, method of undetermined coefficients, basic quadrature and their composite rules
  • interpolatory quadrature, Peano Kernel Theorem, Newton-Cotes formulas
  • Euler-Maclaurin formula, Richardson extrapolation, Romberg integration
  • weighted Gaussian quadrature, Gauss-Legendre quadrature
  • convergence of sequences of integral approximations
  • singular integrals, adaptive numerical integration

Numerical solution of ordinary differential equations: initial-value problems
  • review of ODE theory: existence, uniqueness, and stability, finite-time blow-up for nonlinear equations
  • review of ODE theory: high-order equations, solutions to linear equations, Gronwall inequality, a lemma
  • Euler's method: derivation, truncation error and consistency, convergence and error estimates, numerical stability, and asymptotic expansion
  • linear multistep methods: examples, local discretization error, consistency, convergence, and stability
  • linear multistep methods: necessary and sufficient for consistency and error control, examples, convergence
  • One step methods
  • Runge-Kutta methods
  • Stiffness

Numerical solution of ordinary differential equations: two-point boundary-value problems
  • shooting methods
  • finite difference methods
  • weak formulation and the simplest finite element method

A List of References

  1. A. K. Atkinson, An Introduction to Numerical Analysis, Wiley, 1978.
  2. E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Dover, 1994.
  3. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 3rd ed., Springer, 2004.
  4. E. Suli and D. Mayers, An Introduction to Numerical Analysis, Cambridge Univ. Press, 2003.

Approximation theory and methods
  1. E. W. Cheney, Introduction to Approximation Theory, McGraw-Hill, New York, 1966.
  2. P. J. Davis, Interpolation and Approximation, Dover, 1975.
  3. P. J. Davis and P. H. Rabinowitz, Methods of Numerical Integration, Academic Press, 1975.
  4. P. M. J. D. Powell, Approximation Theory and Methods, Cambridge University Press, 1981.
  5. T. J. Rivlin, Introduction to the Approximation of Functions, Dover, 1987.
  6. G. Szgo, Orthogonal Polynomials, 3rd ed., Amer. Math. Soc., 1967.

Numerical solution of ordinary differential equations
  1. J. Butcher, The Numerical Analysis of Ordinary Differential Equations, Wiley, 1987.
  2. J. D. Lambert, Numerical Methods for Ordinary Differential Systems The Initial Value Problem, John Wiley & Son, 1991.
  3. L. F. Shampine, Numerical Solution of Ordinary Differential Equations, CRC Press, 1994.
  4. H. Stetter, Analysis of Discretization Methods for Ordinary Differential Equations, Springer-Verlag, 1973.

Last updated by Bo Li on April 6, 2009.