Part 1 Lectures In basic computational numerical analysis has contents: Numerical linear algebra, solution of nonlinear equations, approximation theory. | (m) −1 )] ƒ(x ) [J(ƒ (m+1) x =x − (m) (m) D f f 0 i = +1 − f i −1 i 2h LECTURES IN BASIC COMPUTATIONAL NUMERICAL ANALYSIS J. M. McDonough University of Kentucky Lexington, KY 40506 E-mail: jmmcd@ ƒ( y′ = y,t) LECTURES IN BASIC COMPUTATIONAL NUMERICAL ANALYSIS J. M. McDonough Departments of Mechanical Engineering and Mathematics University of Kentucky c 1984, 1990, 1995, 2001, 2004, 2007 Contents 1 Numerical Linear Algebra Some Basic Facts from Linear Algebra . . . . . . . . . . . . . . . Solution of Linear Systems . . . . . . . . . . . . . . . . . . . . . . Numerical solution of linear systems: direct elimination . Numerical solution of linear systems: iterative methods . Summary of methods for solving linear systems . . . . . . The Algebraic Eigenvalue Problem . . . . . . . . . . . . . . . . . The power method . . . . . . . . . . . . . . . . . . . . . . Inverse iteration with Rayleigh quotient shifts . . . . . . . The QR algorithm . . . . . . . . . . . . . . . . . . . . . . Summary of methods for the algebraic eigenvalue problem Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 5 5 16 24 25 26 29 30 31 32 2 Solution of Nonlinear Equations Fixed-Point Methods for Single Nonlinear Equations . . . . . . . . . Basic fixed-point iteration . . . . . . . . . . . . . . . . . . . . Newton iteration . . . . . . . . . . . . . . . . . . . . . . . . . Modifications to Newton’s Method . . . . . . . . . . . . . . . . . . . The secant method . . . . . . . . . . . . . . . . . . . . . . . . The method of false position . . . . . . . . . . . . . . . . . . Newton’s Method for Systems of Equations . . . . . . . . . . . .
