1. Lecture Notes Dr. Rakhmad Arief Siregar Universiti Malaysia Perlis
Chapter 6
Open Methods
Lecture Notes
Dr. Rakhmad Arief Siregar
Universiti Malaysia Perlis
Applied Numerical Method
for Engineers

3. Review of previous method
In the previous chapter, the root is located within an interval
Lower bound
Upper bound
Repeated application if these methods always result in closer estimates of true value of the root
These methods are said to be CONVERGENT because they move closer to the truth as computation progresses

4. Open Methods
In contrast, the open methods are based on formulas that require only a single or two of x.
Do not bracket the root.
It is called as diverge or move away from the true root as the computation progresses. Fig (b)
However, when it is converge (Fig. c), they do not so much more quickly than the bracketing method

5. Simple Fixed-Point Iteration
This method also called as, simple fixed-point iteration, or one point iteration or successive substitution method
Open methods employ formula to predict the root.
The formula can be developed by rearranging the function f(x) = 0 so that x is on the left-hand side of the equation:
Example: can be rearranged to:

6. Simple Fixed-Point Iteration
The utility of is to provides a formula to predict a new value of x as function of an old value of x
Thus given an initial guess at the root xi, it can be used to compute a new estimate xi+1 as expressed by the iterative formula:
Thus the approximation error (as in the previous chapters) can be calculated as

7. Ex. 6.1 Simple Fixed-Point Iteration
Use simple fixed-point iteration to locate the root of
Solution
The function can be separated directly and expressed in the form of
With an initial guess of x0 =0, then the iterative equation can be calculated as

8. Ex. 6.1 Simple Fixed-Point Iteration
an initial guess of x0 =0, true root:

9. Convergence Why this result can be called as convergence?
Reading assignment:
Read section pp. 135

10. Ex. 6.2 Two-curve graphical method
Separate the equation into two parts and determine its root graphically
Solution
Reformulate the equation as: and

11. In graphical methods
The intersection of the two curves indicates a root estimate of approximately x =0.57

12. Exercise Use simple fixed-point iteration to locate the root of
Use an initial guess of x0 = 0.5

13. Solution
The first iteration is

14. Solution
The remaining iterations are

15. The Newton-Raphson Methods
The most widely used of all root-locating formula is the Newton-Raphson equation.
If the initial guess at the root is xi, a tangent can be extended from the point [xi, f(xi)]
The point where this tangent crosses the x axis usually represents an improved estimate of the root.

16. The Newton-Raphson Methods
The Newton-Raphson method can be derived on the basis of this geometrical interpretation
The first derivative at x is equivalent to the slope:
which can be rearranged to yield

17. Ex. 6.3 Newton-Raphson Method
Use the Newton-Raphson method to estimate the root of
Employing an initial guess of x0 = 0
Solution
The first derivative of the function can be evaluated as
Which can be substituted along with the original function

18. Ex. 6.3 Newton-Raphson Method
Starting with an initial guess of x0 = 0, this iterative equation can be applied to compute:
The approach rapidly converges on the true root.
Faster that by using simple fixed-point iteration

19. Termination Criteria and Error Estimates
The approximate percent relative error can also be used as a termination criterion.
Other criteria can also be obtained such as discussed in Example 6.4

20. Pitfalls of the Newton-Raphson Method
Although the Newton-Raphson method is often very efficient, there are situations where it performs poorly.
A special case – multiple roots – will be addressed later
However, even when dealing with simple roots, difficulties can also arise, as in the following example

21. Ex. 6.5 Slowly Converging Function with Newton-Raphson
Determine the positive root of f(x) = using the Newton-Raphson method and an initial guess of x = 0.5
Solution
The Newton-Raphson formula for this case is
Which can be used to compute:

22. Iteration results

23. Pitfalls of the Newton-Raphson Method

24. Pitfalls of the Newton-Raphson Method

25. The Secant Method
A potential problem in implementing the Newton-Raphson method is the evaluation of the derivative.
For polynomials and many other function this is not convenient
There are certain function the derivative may be extremely difficult or inconvenient to evaluate

26. The Secant Method
For these cases, the derivative can be approximated by a backward finite divided difference
The secant method can be formulated as:

27. Graphical depiction of the secant method

28. Ex. 6.6
Use the secant method to estimate the root of f(x) = e-x – x. Start with the estimates of xi-1 = 0 and x0 = 1.0.
Solution (true root = …)
First iteration
Xi-1 = f(xi-1)=1.0
X0 = f(x0) =
Calculate t = 8.0%

29. Ex. 6.6 Solution (true root = 0.56714329…) Second iteration
X0 = f(x0)=
X1 = f(x1) =

30. Ex. 6.6 Solution (true root = 0.56714329…) Third iteration
X0 = f(x1)=
X1 = f(x1) =

31. Ex. 6.6x
Use the secant method to estimate the root of f(x) = e-x – 2x. Start with the estimates of xi-1 = 0 and x0 = 1.0.
Solution (true root = xxxx)
First iteration
Xi-1 = f(xi-1)=xx
X0 = f(x0) = xxxx
Calculate t = 8.0%

32. Next week
Reading assignment:
Read pp. 146 – 157
Read pp. 160 – 174

33. Quiz (60 Minutes)
What do you know about mathematical model in solving engineering problem? (10 marks)
Use zero through third order Taylor series expansions to predict f(2.5) for f(x) = ln x using a base point at x = 1. Compute the true percent relative error for each approximation. (15 marks)
Determine the real root of f(x)= 5x3-5x2+6x-2 using bisection method. Employ initial guesses of xl = 0 and xu = 1. iterate until the estimated error a falls below a level of s = 15% (20 marks)

34. The difference between the Secant and False-Position Methods
Similarity of Secant and False-Position Methods
The approximate root equation are identical on a term by term basis
Both of them use two initial estimates to compute an approximation of the slope.
Critical difference:
How in both methods the new estimate replaces the initial value
False Position Method
Secant Method 

36. The false-position method
Ex. 6.7
Use the false-position and secant methods to estimate the root of f(x) = ln x.
Start the computation with value of
xl = xi-1 = 0.5
xu = xi = 5.0
Solution
The false-position method
The secant method

38. Modified Secant Method
Original Secant Method
Modified Secant Method

39. Ex. 6.8
Use the secant method to estimate the root of f(x) = e-x – x. Use a value of 0.01 for  and start with x0 = 1.0.
Solution (true root = …)
First iteration
x0 = f(x0) =
x1+ x0 = f(x1+ x0) =
Calculate t = 5.3%

40. Ex. 6.8 Solution (true root = 0.56714329…) Second iteration
x0 = f(x0) =
x1+ x0 = f(x1+ x0) =
Calculate t = %

41. Ex. 6.8 Solution (true root = 0.56714329…) Third iteration
x0 = f(x0) =
x1+ x0 = f(x1+ x0) =
Calculate t = 2.365x10-5 %

42. Multiple Roots

43. Multiple Roots

44. Modified Newton-Raphson
A modifications have been proposed by Ralston and Rabinowitz (1978)
Original Newton-Raphson
Modified Newton-Raphson

45. Ex. 6.9
Original Newton-Raphson
Use both the standard ad modified Newton-Raphson methods to evaluate the multiple root of f(x) = (x-3)(x-1)(x-1) with initial guess of x0 = 0.
Solution (true value: 1.0)
First derivative: f’(x) = 3x2 - 10x + 7
Second derivative f’’(x) = 6x - 10
Modified Newton-Raphson

46. Original Newton-Raphson Method

47. Modified Newton-Raphson Method

48. Ex. 6.9
For both methods to search for single root at x = 3 using initial guess of x0 = 4
Both methods converge quickly, with the standard method being somewhat more efficient

49. Modified secant method
For multiple root secant method can be modified as:

50. System of Nonlinear Equations
We have discussed on the determination of roots of single equation
How about locating roots of a set of simultaneous equations
Nonlinear equations is algebraic and transcendental equations that do not fit following format

51. By Fixed-Point Iteration
Non linear equations:
Two simultaneous nonlinear equation with two unknown, x and y.

52. 6.10
Use fixed-point iteration to determine the root of Eq. (6.16). Note that a correct pair of roots is x = 2 and y = 3. Initiate the computation with guesses of the x = 1.5 and y = 3.5.
Solution

53. 6.10
Solution
On the basis of the initial value x = 1.5 and y = 3.5

54. 6.10
Solution
On the basis of the initial value x = 1.5 and y = 3.5

55. 6.10
Solution
On the basis of the initial value x = 1.5 and y = 3.5

56. 6.10

57. By Newton-Raphson

58. Assignment
Solve problem 6.2