1. Today’s class Roots of equation Finish up incremental search
Open methods
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

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
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

3. False Position Method Numerical Methods, Prof. Jinbo Bi CSE, UConn
Lecture 5
Prof. Jinbo Bi CSE, UConn

4. False Position Method Modified False Position Method
Detect when you get stuck and use a bisection method
Can get you to convergence faster
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

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
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

6. Incremental Searches Numerical Methods, Prof. Jinbo Bi CSE, UConn
Lecture 5
Prof. Jinbo Bi CSE, UConn

7. Open Methods
Bracket methods depend on knowing the interval in which the root resides
What if you don’t 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
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

8. Open Methods Numerical Methods, Prof. Jinbo Bi CSE, UConn Lecture 5
As shown in this Figure, Bracket methods shrinks the search interval, by repeating this process, it always results in better estimate of true root. This means convergence.
Now for open method, we start from one single estimate, or perhaps two estimates that do not necessarily bracket the root. Then we apply a formula to get next estimate. This does not guarantee convergence. Look at this figure.
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

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 doesn’t work, just add x to both sides
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

10. Fixed-point iteration
The function transformation allows us to use g(x) to calculate a new guess of x
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

11. Example Find root of f(x)=e-x-x Transform f(x)=0 to x=g(x)=e-x
Start with an estimate of x0=0
x1=g(x0)=e-0=1
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

12. Example true value of the root: 0.56714329 Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

13. Example Numerical Methods, Prof. Jinbo Bi CSE, UConn Lecture 5
Here are the two graphical methods to determine the root of the function f(x). One method is the standard way that we draw the curve that corresponds to the function itself, then we look at where this curve intersects with the x-axis to find a root.
The other method is to examine the function so we find finding the root of f(x) is the same as finding the value of x so that this e(-x) = x, which means we look at the intersection of the two functions, one is e(-x), the other is f(x) = x.
So the fixed point method follows the idea of the second graphical method.
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

14. Fixed-point iteration
Convergence properties
If converge, much faster than bracketing methods
May not converge
Depends on the curve characteristics
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

15. Fixed-point iteration
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

16. Fixed-point iteration
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

17. Fixed-point iteration
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

18. Fixed-point iteration
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

19. Convergence Analysis Assume xr is the true root
Combine with the iterative relationship
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

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.
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

21. Newton-Raphson Method
Similar idea to False Position Method
Use tangent to guide you to the root
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

22. Example Find root of f(x)=e-x-x Start with an estimate of x0=0
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

23. Example true value of the root: 0.56714329 Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

24. Newton-Raphson Method
Convergence analysis
First-order Taylor series expansion
At root
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

25. Newton-Raphson Method
Newton-Raphson method is quadratically convergent
If Newton-Raphson method does converge, The error at each iteration is roughly proportional to the square of the previous error. This means that the number of correct decimal places approximately doubles with each iteration.
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

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
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

27. Newton-Raphson Method
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

28. Newton-Raphson Method
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

29. Newton-Raphson Method
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

30. Newton-Raphson Method
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

31. Newton-Raphson Method
Algorithm should guard against slow convergence or divergence
If slow convergence or divergence detected, use another method
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

32. Secant method
Newton-Raphson method requires calculation of the derivative
Instead, approximate the derivative using backward finite divided difference
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

33. Secant method From Newton-Raphson method
Replace with backward finite difference approximation
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

34. Example Find root of f(x)=e-x-x
Start with an estimate of x-1=0 and x0=1
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

35. Example true value of the root: 0.56714329 Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

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
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

37. Secant Method vs. False-Position Method
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

38. Modified Secant Method
Instead of using backward finite difference to estimate the derivative, use a small delta
Substitute back into Newton-Raphson formula
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

39. Example Find root of f(x)=e-x-x
Start with an estimate of x0=1 and δ=0.01
Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

40. Example true value of the root: 0.56714329 Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn

41. Next class Polynomial roots Read Chapter 7 Numerical Methods,
Lecture 5
Prof. Jinbo Bi CSE, UConn