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Numerical Methods,

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Today s class Roots of equation Finish up incremental search Open methods Numerical Methods, Lecture 5 Prof. Jinbo Bi CSE, UConn * Find root of f(x)=e-x-x Start ... – PowerPoint PPT presentation

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Title: Numerical Methods,


1
Todays class
  • Roots of equation
  • Finish up incremental search
  • Open methods

2
False Position Method
  • Although the interval a,b where the root
    becomes iteratively closer with the false
    position method, unlike the bisection method, the
    size of the interval does not necessarily
    converge to zero.
  • Sometimes it can cause the false position to
    converge slower than bisection

3
False Position Method
4
False Position Method
  • Modified False Position Method
  • Detect when you get stuck and use a bisection
    method
  • Can get you to convergence faster

5
Incremental Searches
  • Dependent on knowing the bracket in which the
    root falls
  • Can use bracketed incremental search to speed up
    exhaustive search
  • How big a bracket or increment can determine how
    long the search will take
  • Too small increment and it will take too long
  • Too big increment may miss roots, in partular,
    the multiple roots

6
Incremental Searches
7
Open Methods
  • Bracket methods depend on knowing the interval in
    which the root resides
  • What if you dont know the upper and lower bound
    on the root?
  • Open methods
  • Use a single estimate of the root
  • Use two starting points but not bracketing the
    root
  • May not converge on root

8
Open Methods
9
Open Methods
  • Fixed-Point Iteration
  • One-point iteration
  • Successive substitution
  • Start with equation f(x) 0 and rearrange so x
    is on left hand side.
  • If algebraic manipulation doesnt work, just add
    x to both sides

10
Fixed-point iteration
  • The function transformation allows us to use g(x)
    to calculate a new guess of x

11
Example
  • Find root of f(x)e-x-x
  • Transform f(x)0 to xg(x)e-x
  • Start with an estimate of x00
  • x1g(x0)e-01

12
Example
  • true value of the root 0.56714329

13
Example
14
Fixed-point iteration
  • Convergence properties
  • If converge, much faster than bracketing methods
  • May not converge
  • Depends on the curve characteristics

15
Fixed-point iteration
16
Fixed-point iteration
17
Fixed-point iteration
18
Fixed-point iteration
19
Convergence Analysis
  • Assume xr is the true root
  • Combine with the iterative relationship

20
Fixed-point iteration
  • Use derivative mean-value theorem
  • If the derivative is less than 1, the error will
    get smaller with each iteration (monotonic or
    oscillating).
  • If the derivative is greater than 1, the error
    will get larger with each iteration.

21
Newton-Raphson Method
  • Similar idea to False Position Method
  • Use tangent to guide you to the root

22
Example
  • Find root of f(x)e-x-x
  • Start with an estimate of x00

23
Example
  • true value of the root 0.56714329

24
Newton-Raphson Method
  • Convergence analysis
  • First-order Taylor series expansion
  • At root

25
Newton-Raphson Method
  • Newton-Raphson method is quadratically convergent

26
Newton-Raphson Method
  • Problems and Pitfalls
  • Slow convergence when initial guess is not close
    enough
  • May not converge at all
  • Problems with multiple roots

27
Newton-Raphson Method
28
Newton-Raphson Method
29
Newton-Raphson Method
30
Newton-Raphson Method
31
Newton-Raphson Method
  • Algorithm should guard against slow convergence
    or divergence
  • If slow convergence or divergence detected, use
    another method

32
Secant method
  • Newton-Raphson method requires calculation of the
    derivative
  • Instead, approximate the derivative using
    backward finite divided difference

33
Secant method
  • From Newton-Raphson method
  • Replace with backward finite difference
    approximation

34
Example
  • Find root of f(x)e-x-x
  • Start with an estimate of x-10 and x01

35
Example
  • true value of the root 0.56714329

36
Secant Method vs. False-Position Method
  • False-Position method always brackets the root
  • False-Position will always converge
  • Secant method may not converge
  • Secant method usually converges much faster

37
Secant Method vs. False-Position Method
38
Modified Secant Method
  • Instead of using backward finite difference to
    estimate the derivative, use a small delta
  • Substitute back into Newton-Raphson formula

39
Example
  • Find root of f(x)e-x-x
  • Start with an estimate of x01 and d0.01

40
Example
  • true value of the root 0.56714329

41
Next class
  • Polynomial roots
  • Read Chapter 7
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