ChE 250 Numeric Methods

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ChE 250 Numeric Methods

• Lecture 6, Chapra Chapter 6
• 20070129

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Open Methods

• Open methods do not require previous knowledge of
  the range where the root is located
• The algorithm must have a starting point provided
  up front, and often success depends on how good
  the starting point is
• Open methods converge (or diverge) more quickly
  than bracket methods

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Open Methods

• Fixed Point Iteration
• Convergence
• Newton-Raphson
• Pitfalls
• Secant Method
• Multiple Roots
• System of Non-Linear Equations

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Fixed Point Iteration

• The simplest approach is to reformulate the
  equation in terms of xi and xi1
• Find the root of f(x)e-x-x
• f(x)0
• xe-x
• xi1e-xi

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Fixed Point Iteration

• Find the root of f(x)ex-x-2
• f(x)0
• xex-2
• xi1exi-2

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Fixed Point Iteration

• Find the root of f(x)ex-x-2
• f(x)0
• xln(x2)
• xi1ln(xi2)

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Fixed Point Iteration

• If the right side of the equation is graphed
  against f(x)x, the intersection would be the
  root
• For convergence, the absolute value of the
  derivative of the right side (evaluated at xi
  must be less than 1

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Fixed Point Iteration

• Advantages
• Only one starting guess required
• Easy to visualize and understand
• Disadvantages
• Sensitive the formulation of the problem as well
  as initial guess
• Not particularly fast
• Questions?

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Newton-Raphson Method

• Most widely used root location method
• Only need xi, f(xi), and f(xi)
• Use the slope of the function to find the next
  approximation

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Newton-Raphson Method

• The Newton-Raphson formula is easily used to
  iterate a root solution

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Newton-Raphson Method

• Example 6.3
• Find the root of f(x)e-x-x
• Start at x00 same as previous example
• f(x0)1, f(x0)-2
• x1.5
• Converges fast

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Newton-Raphson Method

• Pitfalls!
• Because this method uses the derivative, it is
  likely to fail or stall out with functions that
  have
• High curvature
• Asymptotes
• Local maxima, minima
• Any program should check for
• f(x) close to zero
• Divergence
• oscillation
• jumps

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Newton-Raphson Method

• Example for multiple roots
• f(x)x4-1
• Four possible roots
• Extra credit
• Determine roots
• Use Newtons method to solve for the roots, and
  give at least one starting value that finds each
  root
• Submit algorithm and a table showing starting
  value, and root value

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Secant Method

• What if f(x) is not known or very hard to
  compute?
• The Newton-Raphson Method can be modified to use
  a finite difference instead of a derivative
• This requires two initial guesses, x-1, x0 to
  begin the iteration, however unlike a bracket
  method, we do not require that f(x1) and f(x0) be
  opposite signs

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Secant Method

• The Secant Method formula used to iterate a root
  solution based on two initial guesses

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Secant Method

• Example 6.6
• Find the root of f(x)e-x-x
• Start at x01, but now we also need x-10
• f(x-1)1, f(x0)-.632
• x1.6127
• The first iteration
• Closer than Newton
• However, in the second
• Newton was faster

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Method Comparison

• Fixed Point is simple and easy to implement
• Newtons method is very fast and works well for
  systems where the pitfall areas are know and can
  be avoided
• Secant method is useful if the derivative is not
  easily calculated
• Questions?

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Multiple Roots

• Functions having multiple roots present a problem
  because they will have a a derivative that
  approaches zero as the function value approaches
  zero
• This sometimes causes the Newton-Raphson and
  Secant methods to blow up, so in writing an
  algorithm, the value of the derivative must be
  monitored and a modified subroutine called to
  solve in this situation

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Multiple Roots

• Triple and other odd multiple roots can be
  found using bracket methods
• But even multiple roots must be found with open
  methods

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Multiple Roots

• When Newton is applied, the result depends on the
  starting guess, x0.
• Some starting values find the root 3, some find
  1, but if f(x)0 an error will be generated even
  thought the solution has been found!

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Multiple Roots

• The starting guess determines the final result,
  and the number iterations required

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Multiple Roots

• Recognizing and avoiding the pitfall is most
  important to prevent oscillation and blowups
• Questions

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Preparation for Jan 31st

• Reading
• Chapra Chapter 7 Roots of Polynomials
• Homework set 2 is due at the beginning of class
  on Wednesday

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Systems of Nonlinear Equations

• There are basically two ways to solve nonlinear
  equations
• 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

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Systems of Nonlinear Equations

• Linearize a system of equations