1. Roots of Equations
Bracketing Methods

2. Root
We are given f(x), a function of x, and we want to find α such that
f(α) = 0
α is called the root of the equation f(x) = 0, or the zero of the function f(x)

3. Example: Interest Rate
Suppose you want to buy an electronic appliance from a shop and you can either pay an amount of 12,000 or have a monthly payment of 1,065 for 12 months. What is the corresponding interest rate?
We know the payment formulae is:
A is the monthly payment
P is the loan amount
x is the interest rate per period of time
n is the loan period
To find the yearly interest rate, x, you have to find the zero of

4. Finding Roots Graphically
Not accurate
However, graphical view can provide useful info about a function.
Multiple roots?
Continuous?
etc.

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

6. Bracketing Methods
Theorem: If a function f(x) is continuous in the interval [a, b] and f(a)f(b) < 0, then the equation f(x) = 0 has at least one real root in the interval (a, b).

7. f(a)f(b) > 0 implies zero or even number of roots
Usually
f(a)f(b) > 0 implies zero or even number of roots
[figure (a) and (c)]
f(a)f(b) < 0 implies odd number of roots
[figure (b) and (d)]

8. Functions that discontinue within the interval
Exceptional Cases
Multiple roots
Roots that overlap at one point.
e.g.: f(x) = (x-1)(x-1)(x-2) has a multiple root at x=1.
Functions that discontinue within the interval

9. Algorithm for bracketing methods
Step 1: Choose two points xl and xu such that f(xl)f(xu) < 0
Step 2: Estimate the root xr (note: xl < xr < xu)
Step 3: Determine which subinterval the root lies:
if f(xl)f(xr) < 0 // the root lies in the lower subinterval
set xu to xr and goto step 2
if f(xl)f(xr) > 0 // the root lies in the upper subinterval
set xl to xr and goto step 2
if f(xl)f(xr) = 0
xr is the root

10. How to select xr in step 2? Bisection Method
Guess without considering the characteristics of f(x) in (xl, xu)
False Position Method (Regula Falsi)
Use "average slope" to predict the location of the root

11. A.1. Bisection Method
Each guess reduce the search interval by half

12. Bisection Method – Example
Find the root of f(x) = 0 with an approximated error below 0.5%. (True root: α= )

13. Example (continue) n xl xr xu f(xl) f(xr) f(xu) f(xl)f(xu) εa 12 1612 16
6.067
-2.269 1 14
1.569
> 0 2 15
-0.425
< 0
6.667% 3
14.5
0.552
3.448% 4
14.75
0.0590
1.695% 5
14.875
-0.184
0.840% 6
0.422%

14. Error Bounds The true root, α, must lie between xl and xu.xr xu
Let xr(n) denotes xr in the nth iteration
After the 1st iteration, the solution, xr(1), should be within an accuracy of
xl(1)
xu(1)
xr(1)

15. Error Bounds Suppose the root lies in the lower subinterval.
xl(2)
xr(2)
xu(2)
xu(1)
After the 2nd iteration, the solution, xr(2), should be within an accuracy of

16. Error Bounds
In general, after the nth iteration, the solution, xr(n), should be within an accuracy of
If we want to achieve an absolute error of no more than Eα

17. Implementation Issues
The condition f(xl)f(xr) = 0 (in step 3) is difficult to achieve due to errors.
We should repeat until xr is close enough to the root, but we don't know what the root is!
One possible solution is to estimate the error as
and repeat until ea < es (acceptable error)

18. Bisection Method (as C function)
// xl, xu: Lower and upper bound of the interval
// es: Acceptable relative percentage error
// iter_max: Maximum # of iterations
double Bisect(double xl, double xu, double es,
int iter_max) {
double xr; // Est. root
double xr_old; // Est. root in the previous step
double ea; // Est. error
int iter = 0; // Keep track of # of iterations
xr = xl; // Initialize xr in order to
// calculating "ea". Can also be "xu".
do {
iter++;
xr_old = xr;

19. xr = (xl + xu) / 2; // Estimate root
if (xr != 0)
ea = fabs((xr – xr_old) / xr) * 100;
test = f(xl) * f(xr);
if (test < 0)
xu = xr;
else
if (test > 0)
xl = xr;
ea = 0;
} while (ea > es && iter < iter_max);
return xr; }

20. Additional Implementation Issues
Function call is a relatively slow operation.
In the previous example, function f() is called twice in each iteration. Is it necessary?
We only need to update one of the bounds (see step 3 in the algorithm for the bracketing method).

21. Revised Bisection Method (as C function)
double Bisect(double xl, double xu, double es,
int iter_max) {
double xr; // Est. Root
double xr_old; // Est. root in the previous step
double ea; // Est. error
int iter = 0; // Keep track of # of iterations
double fl, fr; // Save values of f(xl) and f(xr)
xr = xl; // Initialize xr in order to
// calculating "ea". Can also be "xu".
fl = f(xl);
do {
iter++;
xr_old = xr;
xr = (xl + xu) / 2; // Estimate root
fr = f(xr);

22. if (xr != 0)
ea = fabs((xr – xr_old) / xr) * 100;
test = fl * fr;
if (test < 0)
xu = xr;
else
if (test > 0) {
xl = xr;
fl = fr; }
ea = 0;
} while (ea > es && iter < iter_max);
return xr;

23. Additional Implementation Issues
Testing if f(xl)f(xr) is positive or negative directly could result in underflow or overflow.
How can we address this problem?
We can test if both the values have the same sign

24. Comments on Bisection Method
The method is guaranteed to converge.
However, the convergence is slow as we gain only one binary digit in accuracy in each iteration.

25. A.2. Regula Falsi Method
Also known as the false-position method, or linear interpolation method.
Unlike the bisection method which divides the search interval by half, regula falsi interpolates f(xu) and f(xl) by a straight line and the intersection of this line with the x-axis is used as the new search position.
The slope of the line connecting f(xu) and f(xl) represents the "average slope" (i.e., the value of f'(x)) of the points in [xl, xu ].

27. False-position vs Bisection
False position in general performs better than bisection method.
Exceptional Cases:
(Usually) When the deviation of f'(x) is high and the end points of the interval are selected poorly.
For example,

28. Bisection Method (Converge quicker)
Iteration xl xu xr
εa (%)
εt (%) 1
1.3
0.65 35 2
0.975
33.3 25 3
1.1375
14.3
13.8 4
7.7
5.6 5
4.0
1.6
False-position Method
Iteration xl xu xr
εa (%)
εt (%) 1
1.3
90.6 2
48.1
81.8 3
30.9
73.7 4
22.3
66.2 5
17.1
59.2

30. Summary Bracketing Methods Bisection Method False-position method
f(x) has the be continuous in the interval [xl, xu] and f(xl)f(xu) < 0
Always converge
Usually slower than open methods
Bisection Method
Slow but guarantee the best worst-case convergent rate.
False-position method
In general performs better than bisection method (with some exceptions).