1. Roots of Equations
Open Methods
(Part 2)

2. The following root finding methods will be introduced:
A. Bracketing Methods
A.1. Bisection Method
A.2. Regula Falsi
B. Open Methods
B.1. Fixed Point Iteration
B.2. Newton Raphson's Method
B.3. Secant Method

3. B.2. Secant Method
Newton-Raphson method needs to compute the derivatives.
The secant method approximate the derivatives by finite divided difference.
From Newton-Raphson
method

4. Secant Method

5. Secant Method – Example
Find root of f(x) = e-x - x = 0 with initial estimate of
x-1 = 0 and x0 = 1.0. (Answer: α= ) i
xi-1 xi
f(xi-1)
f(xi)
xi+1 εt 1
8.0 %
0.58 % 2 %
Again, compare this results obtained by the Newton-Raphson method and simple fixed point iteration method.

6. Comparison of the Secant and False-position method
Both methods use the same expression to compute xr.
They have different methods for the replacement of the initial values by the new estimate. (see next page)

7. Comparison of the Secant and False-position method

8. Comparison of the Secant and False-position method

9. Modified Secant Method
Needs only one instead of two initial guess points
Replace xi-1 - xi by δxi and approximate f'(x) as
From Newton-Raphson method,

10. Modified Secant Method
Find root of f(x) = e-x - x = 0 with initial estimate of
x0 = 1.0 and δ=0.01. (Answer: α= ) i xi
xi+δxi
f(xi)
f(xi+δxi)
xi+1 1
1.01 2
Compared with the Secant method i
xi-1 xi
f(xi-1)
f(xi)
xi+1 εt 1
8.0 %
0.58 % 2 %

11. Modified Secant Method – About δ
If δ is too small, the method can be swamped by round-off error caused by subtractive cancellation in the denominator of
If δ is too big, this technique can become inefficient and even divergent.
If δ is selected properly, this method provides a good alternative for cases when developing two initial guess is inconvenient.

12. The following root finding methods will be introduced:
A. Bracketing Methods
A.1. Bisection Method
A.2. Regula Falsi
B. Open Methods
B.1. Fixed Point Iteration
B.2. Newton Raphson's Method
B.3. Secant Method
Can they handle multiple roots?

13. Multiple Roots
A multiple root corresponds to a point where a function is tangent to the x axis.
For example, this function has a double root.
f(x) = (x – 3)(x – 1)(x – 1)
= x3 – 5x2 + 7x - 3
For example, this function has a triple root.
f(x) = (x – 3)(x – 1)(x – 1) (x – 1)
= x4 – 6x3 +12x2 - 10x + 3

14. Odd multiple roots cross the axis. (Figure (b))
Even multiple roots do not cross the axis. (Figure (a) and (c))

15. Difficulties when we have multiple roots
Bracketing methods do not work for even multiple roots.
f(α) = f'(α) = 0, so both f(xi) and f'(xi) approach zero near the root. This could result in division by zero. A zero check for f(x) should be incorporated so that the computation stops before f'(x) reaches zero.
For multiple roots, Newton-Raphson and Secant methods converge linearly, rather than quadratic convergence.

16. Modified Newton-Raphson Methods for Multiple Roots
Suggested Solution 1:
Disadvantage:
work only when m is known.

17. Modified Newton-Raphson Methods for Multiple Roots
Suggested Solution 2:

18. Example of the Modified Newton-Raphson Method for Multiple Roots
Original Newton Raphson method i xi
εt (%)
100 1 57 2 31 3 17 4
8.7 5
4.4 6
2.2
The method is linearly convergent toward the true value of 1.0.

19. Example of the Modified Newton-Raphson Method for Multiple Roots
For the modified algorithm i xi
εt (%)
100 1 11 2
0.31 3

20. Example of the Modified Newton-Raphson Method for Multiple Roots
How about their performance on finding the single root? i
Standard
εt (%)
Modified 4 33 1
3.4 13 12 2
3.1
3.3
6.0 3
0.29
1.3
0.0025
0.05 5
2x10-7
7.7x10-5

21. Modified Newton-Raphson Methods for Multiple Roots
What's the disadvantage of the modified Newton-Raphson Methods for multiple roots over the original Newton-Raphson method?
Note that the Secant method can also be modified in a similar fashion for multiple roots.

22. Summary of Open Methods
Unlike bracketing methods, open methods do not always converge.
Open methods, if converge, usually converge more quickly than bracketing methods.
Open methods can locate even multiple roots whereas bracketing methods cannot. (why?)

23. Study Objectives Understand the graphical interpretation of a root
Understand the differences between bracketing methods and open methods for root location
Understand the concept of convergence and divergence
Know why bracketing methods always converge, whereas open methods may sometimes diverge
Realize that convergence of open methods is more likely if the initial guess is close to the true root.

24. Study Objectives
Understand what conditions make a method converges quickly or diverges
Understand the concepts of linear and quadratic convergence and their implications for the efficiencies of the fixed-point-iteration and Newton-Raphson methods
Know the fundamental difference between the false-position and secant methods and how it relates to convergence
Understand the problems posed by multiple roots and the modifications available to mitigate them