ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 7 Roots of Equations Bracketing 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 Graphical Methods c f(c) v=10 m/s t=3 sec m=65 kg g=9.81

Last Time Graphical Methods No Roots Even Number of Roots Lower and Upper Bounds of interval yield values of same sign

Last Time Graphical Methods Lower and Upper Bounds of interval yield values of opposite sign Odd number of Roots

Last Time Bisection Method Choose Lower, x l and Upper x u guesses that bracket the root xlxl xuxu

Last Time Bisection Method Calculate New Estimate x r and f(x r ) xlxl xuxu x r =0.5(x l +x u )

Last Time Bisection Method Define New Interval that Brackets the Root Check sign of f(x l )*f(x r ) and f(x u )*f(x r ) xlxl xuxu Previous Guess xuxu

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

Last Time Bisection Method Check Convergence Root = If Error

Objectives Master methods to compute roots of equations Assess reliability of each method Choose best method for a specific problem REGULA FALSI Method (False Position)

False Position Method xlxl xuxu x r =0.5(x l +x u ) Recall Bisection Method No consideration on function values

False Position Method f(x l ) f(x u ) xlxl xuxu xrxr NEW ESTIMATE

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

False Position Method f(x l ) f(x u ) xlxl xuxu xrxr From Similar Triangles

False Position Method

Add and subtract New Estimate

Loop x old =x r Error=100*abs(x-x old )/x r Sign=f(x l )*f(x r ) Sign x u =x r f u =f(x u ) x l =x r f l =f(x l ) Error=0 Error<E all ROOT=x r FALSE <0>0 f u =f(x u ), f l =f(x l )

False Position Typically Faster Convergence than Bisection

False Position Not Efficient in this Case