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

## Math 270C: Numerical Mathematics (Part C)

### General Course Information

 Prerequisites Calculus and linear algebra. Textbook There will be no textbooks. Lecture notes will be distributed. Lectures Mondays: 2:00 pm - 2:50 pm, AP&M B412. Instructor Bo Li Office hours: 4:30 pm - 5:00 pm Mondays and Wedensdays, or by appointment. Teaching assistant Hieu Nguyen Class web page http://www.math.ucsd.edu/~bli/teaching/math270Cs08/ Homework Assigned and graded regularly. 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 - 40%, midterm exam 20%, and final exam - 40%.

### Homework

Assignment 1, due Monday, 4/7/08: Chapter 1: 1(a, n = 2), 2, 4, 8, 14.
Assignment 2, due Monday, 4/14/08: Chapter 1: 15, 30, 32, 36.
Assignment 3, due Wednesday, 4/23/08: Chapter 1: 34, 38, 39, 40.
Assignment 4, due Wednesday, 4/30/08: Chapter 2: 2, 3, 5, 7, 8.
Assignment 5, due Friday, 5/9/08: Chapter 2: 11, 12, 13, 17.
Assignment 6, due Friday, 5/16/08: Chapter 2: 3, 4, 8, 9, 13.

### A Tentative Course Outline

Polynomial approximation
• Weierstrass Theorem and Bernstein's polynomials
• best uniform approximation: existence, uniqueness, and characterization
• Chebyshev polynomials of first kind
• least-squares approximations: existence, uniqueness, and characterization
• Gram-Schmidt orthogonalization, orthogonal polynomials: properties and examples

Polynomial interpolation
• Lagrange interpolation: Lagrange formula, Newton formula and divided differences, Iterated linear interpolation
• Lagrange interpolation: remainder, optimal interpolation points, Peano kernals
• convergence of Lagrange and piecewise Lagrange interpolation polynomials
• Hermite interpolation

• degree of precision, method of undetermined coefficients, basic quadrature and their composite rules
• Peano Kernel Theorem, interpolatory quadrature, Newton-Cotes formulas
• weighted Gaussian quadrature: formula, error, and convergence

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

### Syllabus for Qualifying Exam

Polynomial approximation: Weierstrass Theorem and Bernstein's polynomials, best uniform approximations, least-squares approximations, orthogonal polynomials.

Polynomial interpolation: Lagrange interpolation, remainder, Peano kernals, iterated linear interpolation, convergence of Lagrange interpolation polynomials.

Numerical quadrature: degree of precision, method of undetermined coefficients, interpolatory quadrature, Newton-Cotes formulas, Peano kernals, Gaussian quadrature.

Numerical solution of ordinary differential equations: Euler's method, linear multistep methods, one step methods, Runge-Kutta methods.

### A List of References

General
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 March 25, 2008.