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ChE 250 Numeric Methods

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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
- 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)

- Advantages
- 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

- Chapter 6