EE:211 Computational Techniques in Electrical Engineering Lecture#2 Dr. Mubashir Alam King Saud University

Outline Chapter#3 Rootfinding Bisection Method Newton's Algorithm Secant Method

Introduction

The Bisection Method

Example: 3.1.1

Error Bounds α is the true root and cn is its estimate Error: |α-cn|

Newton’s Algorithm

If x1 is the root of p(x) then p(x1)=0 Repeat the next with x1 as the initial estimate and determine x2 And so on

Newton’s Method

Example: 3.2.1

Stopping criteria

Example n xn f(xn) xn-xn-1 1.5 8.890625 ------------------ 1 1.5 8.890625 ------------------ 1 1.30049088 2.537264 -0.199950912 2 1.18148042 0.53845863 -0.11901046 3 1.13945559 0.04923525 -0.042025 4 1.13477763 5.50373 x 10-3 -4.678 x 10-3 Stopping criteria, ε = 1 x 10-2

Example: 3.2.1

Example: 3.2.2

Secant Method From this perspective, other straight-line approximation to y=f(x) would also lead to methods for approximating a root of f(x). One such straight-line approximation leads to SECANT METHOD

The two pints (x0,f(x0)) and (x1,f(x1)) , on the graph of y=f(x), determine a straight line , called a secant line. This line is an approximation to the graph of y=f(x), and its root x2 is an approximation to true root α.

Equation of lines Equation of line with slope = m and passing through a point (x1,y1): y-y1 = m(x-x1) y = y1+ m(x-x1) Slope of the line between points (x1,y1) and (x2,y2) Slope = m = (y2- y1) / (x2- x1)

Secant Method Find the equation of the line and then its root x2 Having found x2, we can drop x0, and use x1,x2 as a new set of approximate values for α. This will lead to an improved value x3. Continue this process …..

Secant Method General Formula: Two point method, since two approximate values are needed to obtain the next improved value.

Example: 3.3.1

Matlab Function: fzero This function uses ideas involved in the bisection and the secant method. Use: root=fzero(f_name, [a,b]) Produces a root within [a,b], assume f(a)f(b) <= 0 Use: root=fzero(f_name, xo) Find a root near x0.

Matlab Function: fzero Define function for: f(x)=x6-x-1 function f = myfun(x) f = x.^6-x-1; x=fzero(@myfun,[1,2]) x=1.134724138401519 x=fzero(@myfun,1.9)