ChE 250 Numeric Methods

1
ChE 250 Numeric Methods

• Lecture 9, Chapra Chapter 7 (Bairstows),
  Chapter 8
• 20070205

2
Roots of Polynomials

• Friday
• Mullers Method
• Today
• Bairstows Method
• Root location with Excel
• Matlab, Scilab

Case Studies

• Chemical Engineering Problems

3
Bairstows Method

• Whereas the Mullers Method is analogous to the
  secant method and is fairly straightforward to
  understand, the Bairstow method is fundamentally
  different
• For Bairstow, we rely on deflation of the
  polynomial
• Guess a quadratic devisor
• Iterate the coefficients of the quadratic
• repeat the division until the quotient is second
  or first order
• The only trick is how we find the devisor

4
Bairstows Method

• Polynomial Deflation
• Removing the found roots from a polynomial so you
  can find the remaining roots
• For this equation, once we find x-1, 4 then we
  want to remove those two and concentrate on the
  others
• The process is then repeated which removes two
  more roots and only one is left

5
Bairstows Method

• Starting with a general polynomial with
  coefficients an
• An initial guess of s, r for the coefficients of
  a quadratic, gives a set of bn that form a new
  deflated polynomial with order n-2

6
Bairstows Method

• Synthetic division is easily implemented in a
  program using this algorithm
• The final b values (0,1) are the remainder term
  which we want to be 0
• Since b0 and b1 are functions of r and s, they
  can be written in a Taylor expansion (first
  order!)
• Set both b0 and b1 equal to zero and solve for
  delta s and r

7
Bairstows Method

• But what about the partials?
• A new coefficient, c, can be used to determine
  the partials
• Use the same sub used for b to determine c
• Only c 1,2,3 are used

Same algorithm
8
Bairstows Method

• Substitute the cs for the partials
• And then solve two equations and two unknowns,
  delta r and delta s
• We then use these deltas to calculate the next
  values, ri1 and si1
• Example 7.3
• Questions?

9
Excel Goal Seek

• Useful for functions of one variable
• Heavily dependant on initialization
• Will abort if it finds a complex situation or
  other errors
• Great for a first try at a solution!

10
Excel Solver

• Powerful tool for finding solutions to
  multivariate problems
• Constraints keep the solution from returning
  nonsensical solutions or find a distinct solution
• First choice for multivariate problems with real
  solutions

11
Matlab

• If you have not used Matlab before or it has been
  a while, please brush up by trying example 7.7
• Understand how to perform calculations and assign
  variable names
• Create a function of several variables
• Lab will be tomorrow night with times per e-mail

12
Preparation for Feb 7th

• Reading
• Chapra Part 3 intro, Chapter 9 Gauss
  Elimination
• Reminder 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