MATH 175: Numerical Analysis II Lecturer: Jomar Fajardo Rabajante IMSP, UPLB 2nd Sem AY 2012-2013

3rd Method: Regula Falsi/False Position/Inverse Linear Interpolation (also a bracketing method) We will still use the Intermediate Zero Theorem The function should be continuous The function values of the two endpoints of the line should be of opposite signs

3rd Method: Regula Falsi y-axis x-axis Approximate root

3rd Method: Regula Falsi We will use inverse linear interpolation (in slope-intercept form): To get x3, set f(x3)=0 (from the zero of the line):

3rd Method: Regula Falsi y-axis Set new x1 = old x3 New Approximate root x-axis Old Approximate root

3rd Method: Regula Falsi y-axis Old Approximate root x-axis Set new x2 = old x3 New Approximate root

3rd Method: Regula Falsi y-axis 3rd 2nd x-axis 1st

3rd Method: Regula Falsi Try this at home: create a flowchart of the Regula Falsi method Hint: use bisection algorithm, however change the formula for x3 to

3rd Method: Regula Falsi Like bisection method, Regula Falsi is guaranteed to converge to the root (assuming IZT is met). The rate at which this method converges will depend on how nearly linear f(x) is near its zero. If f(x) is sufficiently differentiable then it is well approximated by a straight line over small intervals. Regula Falsi is often (not always) faster than bisection method, but still the order of convergence is linear. Bullet 1: global convergence

3rd Method: Regula Falsi We can use abs(x3,k-x3,k-1)<tol as our stopping criterion but not (b-a)/2k<tol (where [a,b] is the initial bracket) since the width of the bracket may not converge to zero like in bisection. But a better stopping criterion is (Let x3,k be the approximate root at iteration k, and λ be the asymptotic error constant. Invoking the error evolution equation: ) Tol=10^(-m): accurate at least up to m decimal places where

There’s an improved Regula Falsi method called the Modified Regula Falsi. y-axis 1/2 3rd 2nd x-axis 1st

Bisection, Regula Falsi and Modified Regula Falsi methods are called Interval/Bracketing Methods. Next topics: Iterative methods, such as secant method, Newton’s method, fixed point iteration, Mϋller’s method, Bairstow’s method Usually, faster methods require more assumptions and offer fewer guarantees.