1. ChE 250 Numeric Methods Lecture #7, Chapra Chapter 7 20070202

2. Roots of Polynomials • We discussed on Wednesday Polynomial tricks and why we need the roots • Iterative Methods • Müller’s Method • Bairstow’s Method • Root location with Excel • Matlab, Scilab

3. Roots of Polynomials • Why polynomials • There are many engineering models that use linear differential equations • The ODE then must be solved and polynomials come into play • Finding the roots of the characteristic equation is the first step in understanding the behavior of the system Review

4. Roots of Polynomials • Second order ODE example • We know from DiffEq that y=ert is the form of the solution • We solve for r, the value of the roots • The roots tells us the nature of the solution • Real or complex? • Positive or negative? Review

5. Roots of Polynomials • For a linear equation y=ax+b we have an analytical solution that is easy to calculate • For a quadratic equation, y=ax2+bx+c the solution is slightly more complex, but still easily calculated • But for higher order equations, the analytical solutions (if you can figure them out) are harder to calculate • But using numeric methods, the same algorithm can be used on any polynomial and the roots found with only a few iterations

6. Müller’s Method • This method is analogous to the secant method, but now use three points and a parabolic curve fit • Three initialization values are required • The quadratic equation is used to calculate x3 based on x0, x1, and x2

7. Müller’s Method • To solve for xi+1 first use three equations based on the known points • So we have constructed a parabola relative to position x2 • Which leaves two equations and two unknowns that are solved algebraically

8. Müller’s Method • After solving for a and b • Which can be shown using some difference equations

9. Müller’s Method • We now have the formula for the parabola, so we solve for the f(xi+1-xi)=0 • Use the quadratic equation (slightly different formulation to avoid round-off error) • Now we have our iteration method in terms of xi+1

10. Müller’s Method • So now we have two possible calculations, which do we use? • Choose the sign to be the same sign as b • This will make the denominator the largest absolute value and xi+1 closer to xi • What happens if the approximation is complex?

11. Müller’s Method • Now we have a new approximation, which of the previous three points do we toss out? • Two approaches for two objectives • To find only a real root • Throw out the furthest point • For any root • Just throw out the lowest index • Example 7.2 Excel • Questions??

12. Preparation for Feb 2nd • Reading • Chapra Chapter 7: Bairstow’s Method • Chapra Chapter 8: Case Studies: Roots of Equations • On Monday we will briefly go over Bairstow’s Method and do some examples from Chapter 8 to solidify your understanding of these methods