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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

ChE 250 Numeric Methods

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


Open methods
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

Systems of nonlinear equations
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

Systems of nonlinear equations1
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

Systems of nonlinear equations2
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?

Systems of nonlinear equations3
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)

Systems of nonlinear equations4
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??

Systems of nonlinear equations5
Systems of Nonlinear Equations

  • Newton-Raphson

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

  • Throw away all higher order terms!

Systems of nonlinear equations6
Systems of Nonlinear Equations

Taylor Expansion for two variables

Rearrange and group

Systems of nonlinear equations7
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

Systems of nonlinear equations8
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??

Roots of polynomials
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

Roots of polynomials1
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

Roots of polynomials2
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?

Preparation for feb 2nd
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