Math 270C: Numerical Mathematics (Part C) - Bo Li
Spring quarter, 2007
Math 270C: Numerical Mathematics (Part C)
Instructor: Bo Li
Office: AP&M 5723. Office phone: (858) 534-6932. Email: bli@math.ucsd.edu
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
- Numerical quadrature
- 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
Last updated by Bo Li on May 4, 2007.