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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
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
• 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

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

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