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ChE 250 Numeric Methods - PowerPoint PPT Presentation

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

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ChE 250 Numeric Methods

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

20070131

• 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

• Easy to understand and use

• Easy algorithm

• 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?

• 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