MTH 471 Numerical Analysis

Instructor:     Prof. Orlando Merino     merino@math.uri.edu

Semester:     Fall 99

Text:            Numerical Analysis, by D. Kincaid and W. Cheney -Second Edition

Evaluation:    Homework and other assignments (60%) and 2 Exams (20% each)

About the course

This is an introductory course to the algorithms and methods used in scientific computing, for students of Mathematics, Sciences,  Engineering, Computer Science.  You should be familiar with calculus, linear algebra, and one programming language (e.g. matlab, maple, mathematica, fortran, etc.)   Algorithms will be discussed in pseudocode, so they are easy to write in any computer language. The exposition is going to be mathematical, and it will include the statement and proofs of theorems. 

Topics

Mathematical Preliminaries
Basic Concepts and Taylor's Theorem. Order of Convergence.
Computer Arithmetic.
Floating Point Numbers and Roundoff Errors. Absolute and Relative Errors. Stable and Unstable Computations. Conditioning.
Solution of Nonlinear Equations.
Bisection Method. Newton's Method.  Secant Method.
Solving Systems of Linear Equations.
Matrix Algebra.   The LU and Cholesky Factorizations. Pivoting and Constructing an Algorithm. Norms and the Analysis of Errors. Neumann Series and Iterative Refinement.
Selected Topics.
Eigenvalues and the Power Method. Schur's and Gershgorin's Theorems.
Approximating Functions
Polynomial Interpolation.   Divided Differences. Hermite Interpolation. Spline Interpolation.
Numerical Differentiation and Integration.
Numerical Differentiation and Richardson Extrapolation. Numerical Integration Based on Interpolation. Gaussian Quadrature. Romberg Integration. Adaptive Quadrature.