Solving Nonlinear Equations: Methods & Root Finding Techniques

1. ChE 250 Numeric Methods Lecture #7, Chapra Chapter 6 (non-linear systems), Chapter 7 20070131

2. Open Methods • Non-Linear Systems of Equations • Fixed Point Method • Talyor Series Linearization Roots of Polynomials • Polynomial tricks and subroutines • Friday • Müller’s Method • Bairstow’s Method • Root location with Excel • Matlab, Scilab

3. Systems of Nonlinear Equations • We will discuss two ways to solve nonlinear equations (p. 153-7) • The first is to use a fixed point method where the equations are manipulated to calculate the next iteration • This method is subject to constraints on the formulation of the equations • The method is sensitive to the initial guesses • The second method is to linearize the system • Linear methods are then used for a solution

4. Systems of Nonlinear Equations • What is a linear equation? • Non-linear equation have transcendental functions like log, exponential, sine, cosine, etc. • Or ‘Mixed’ variables: • xy+y3x-1-zsin(y)=x

5. Systems of Nonlinear Equations • To solve a set of equations, we need a specified set • n equations • n unknowns • We will need initialization values for all variables • If you do not have these…. • Do more work on your model • Make assumptions?

6. Systems of Nonlinear Equations • Fixed Point Iteration • As before, with a single independent variable, we rearrange the functions to isolate one independent variable • Then we can iterate a set of solutions (x,y)

7. Systems of Nonlinear Equations • Fixed Point Method • Advantages • Easy to understand and use • Easy algorithm • Disadvantages • Diverges very easily depending on the formulation • Need a fairly close initialization set of variables e.g. (x,y) • We will use this more for linear systems • Questions??

8. Systems of Nonlinear Equations • Newton-Raphson • ‘Linearize’ the system of equations using the Taylor Expansion • Throw away all higher order terms!

9. Systems of Nonlinear Equations Taylor Expansion for two variables Rearrange and group

10. Systems of Nonlinear Equations • The resultant iteration scheme depends only on • xi, yi • ui, vi evaluated at xi, yi • All the partial differentials evaluated at xi,yi • Easy to iterate! Solve simultaneously for x and y

11. Systems of Nonlinear Equations • Newton method looks complicated and intimidating, but is really quite easy to implement • Strengths • Fast convergence • Weaknesses • Need sufficiently close initialization values • Partials require many calculations contributing to error • As always, curvature…but now in three dimensions • Hard to visualize! No easy graphical method like 2-d • Questions??

12. Roots of Polynomials • First a note on the computation of polynomial equations • Optimum polynomial function evaluation shown on p. 163, • Derivative function on p. 164 • These are best for calculating polynomials IF your software doesn’t have them already built in

13. 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

14. 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?

15. Preparation for Feb 2nd • Reading • Chapra Chapter 8: • Homework due Feb 7th • Chapter 6 • 6.2, 6.7, 6.9, 6.11, 6.12, 6.13 • Chapter 7 • 7.4, 7.5, 7.12, 7.18, 7.19a • Chapter 8 • 8.1, 8.2