ChE 250 Numeric Methods - PowerPoint PPT Presentation

ChE 250 Numeric Methods

Lecture #7, Chapra Chapter 6 (non-linear systems), Chapter 7

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• 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

• 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

• 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

• 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?

• 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)

• 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??

• Newton-Raphson

• ‘Linearize’ the system of equations using the Taylor Expansion

• Throw away all higher order terms!

Taylor Expansion for two variables

Rearrange and group

• 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

• 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??

• 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

• 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

• 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?

• 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