Lecture 11 Rootfinding – Newton’s and secant methods 1 Lecture 11  More root finding methods  Newton’s method  Very fast way to find roots  Requires taking the derivative of f(x)  Can be unstable if ‘unattended’  Secant method  Similar to Newton’s method, but derivative is numerical not analytical

Lecture 11 Rootfinding – Newton’s and secant methods 2 Newton’s method

Lecture 11 Rootfinding – Newton’s and secant methods 3 Define slope: X old = 4 f(x old ) = x old = 2.64 X old = f(x old ) = x old =7.955 X new = X new = 4.839

Lecture 11 Rootfinding – Newton’s and secant methods 4 Newton Method Calculations

Lecture 11 Rootfinding – Newton’s and secant methods 5 Newton’s method Answer depends on where you start. x init = 2.00 x root = steps = 3 x init = 4.00 x root = steps = 4 x init = x root = steps = 4

Lecture 11 Rootfinding – Newton’s and secant methods 6 Function for Newton’s Method newtonexample.cpp code can be found in the Examples page.

Lecture 11 Rootfinding – Newton’s and secant methods 7 X 0 = 4 f(x 0 ) = Secant method X 2 = X 1 = 6 f(x 1 ) = 9.88 X 3 = x 1 and x 0 don’t have to bound solution f(x 2 ) = f(x 3 ) = X 4 = X2X2

Lecture 11 Rootfinding – Newton’s and secant methods 8 n xn-1xn f(xn-1) f(xn)xn Secant method table

Lecture 11 Rootfinding – Newton’s and secant methods 9 Secant method function secantexample.cpp code can be found in the Examples page.

Lecture 11 Rootfinding – Newton’s and secant methods 10 Comparing methods: Iterations required to reach a tolerance of MethodInitial xx Root#Iterations substitution bisection4.0, Newton’s Secant4.0,