ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 8 Roots of Equations Open Methods

Last Time The Problem Define Function c must satisfy c is the ROOT of the equation

Last Time Classification Methods BracketingOpen Graphical Bisection Method False Position Fixed Point Iteration Newton-Raphson Secand

Last Time Bisection Method Repeat until convergence xlxl xuxu x r =0.5(x l +x u )

Last Time False Position Method f(x l ) f(x u ) xlxl xuxu xrxr

Last Time Bisection Method Check Convergence Root = If Error

Last Time Convergence

Objectives OPEN Methods –Fixed Point Iteration –Newton Raphson –Secant

Open Methods Bracketing Methods Two Initial Estimates Needed that bracket the root Always Converge Open Methods ONE Initial Estimate Needed Sometimes Diverge

Fixed Point Iteration X root x is a root if f(x) = 0

Fixed Point Iteration X X f 1 (X) f 2 (X) +x+x +x+x f 1 (X)f 2 (X) root

Fixed Point Iteration X f 1 (X) f 2 (X) f 1 (X)f 2 (X) root x is a root if f 1 (x) = f 2 (x)

Fixed Point Iteration X f 1 (X) f 2 (X) Initial Guess New Guess root New Guess

Fixed Point Iteration X f 1 (X) f 2 (X) Initial Guess New Guess root New Guess Method Diverges

Condition for Convergence X f 1 (X) f 2 (X) New Guess

Newton Raphson X g(x) Initial Guess New Guess New Guess g’(x i )

Newton Raphson

Inflection Point in Vicinity of Root

Newton Raphson Persistent Oscillations near local max or min

Newton Raphson Initial guess close to root jumps several roots away

Newton Raphson Zero Derivative

Newton Raphson No Convergence Criteria Depends on Nature of Function Depends on Initial Guess Use Initial Guess Sufficiently Close to Root It converges very fast!! (when it does)

Homework (a),(b),(c) Due Date: September 22