1. Chapter 6 Finding the Roots of Equations
The Bisection Method
Chapter 6 Finding the Roots of Equations
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

2. Finding Roots of Equations
In this chapter, we are examining equations with one independent variable.
These equations may be linear or non-linear
Non-linear equations may be polynomials or generally non-linear equations
A root of the equation is simply a value of the independent variable that satisfies the equation
Engineering Computation: An Introduction Using MATLAB and Excel

3. Classification of Equations
Linear: independent variable appears to the first power only, either alone or multiplied by a constant
Nonlinear:
Polynomial: independent variable appears raised to powers of positive integers only
General non-linear: all other equations
Engineering Computation: An Introduction Using MATLAB and Excel

4. Finding Roots of Equations
As with our method for solving simultaneous non-linear equations, we often set the equation to be equal to zero when the equation is satisfied
Example:
If we say that
then when f(y) =0, the equation is satisfied
Engineering Computation: An Introduction Using MATLAB and Excel

5. Solution Methods Linear: Easily solved analytically
Polynomials: Some can be solved analytically (such as by quadratic formula), but most will require numerical solution
General non-linear: unless very simple, will require numerical solution
Engineering Computation: An Introduction Using MATLAB and Excel

6. The Bisection Method
In the bisection method, we start with an interval (initial low and high guesses) and halve its width until the interval is sufficiently small
As long as the initial guesses are such that the function has opposite signs at the two ends of the interval, this method will converge to a solution
Example: Consider the function
Engineering Computation: An Introduction Using MATLAB and Excel

7. Bisection Method Example
Consider an initial interval of ylower = -10 to yupper = 10
Since the signs are opposite, we know that the method will converge to a root of the equation
The value of the function at the midpoint of the interval is:
Engineering Computation: An Introduction Using MATLAB and Excel

8. Bisection Method Example
The method can be better understood by looking at a graph of the function:
Interval
Engineering Computation: An Introduction Using MATLAB and Excel

9. Bisection Method Example
Now we eliminate half of the interval, keeping the half where the sign of f(midpoint) is opposite the sign of f(endpoint)
In this case, since f(ymid) = -6 and f(yupper) = 64, we keep the upper half of the interval, since the function crosses zero in this interval
Engineering Computation: An Introduction Using MATLAB and Excel

10. Bisection Method Example
Now we eliminate half of the interval, keeping the half where the sign of f(midpoint) is opposite the sign of f(endpoint)
In this case, since f(ymid) = -6 and f(yupper) = 64, we keep the upper half of the interval, since the function crosses zero in this interval
Engineering Computation: An Introduction Using MATLAB and Excel

11. Bisection Method Example
The interval has now been bisected, or halved:
New Interval
Engineering Computation: An Introduction Using MATLAB and Excel

12. Bisection Method Example
New interval: ylower = 0, yupper = 10, ymid = 5
Function values:
Since f(ylower) and f(ymid) have opposite signs, the lower half of the interval is kept
Engineering Computation: An Introduction Using MATLAB and Excel

13. Bisection Method Example
At each step, the difference between the high and low values of y is compared to 2*(allowable error)
If the difference is greater, than the procedure continues
Suppose we set the allowable error at As long as the width of the interval is greater than 0.001, we will continue to halve the interval
When the width is less than 0.001, then the midpoint of the range becomes our answer
Engineering Computation: An Introduction Using MATLAB and Excel

14. Bisection Method Example
Excel solution:
Initial Guesses
Evaluate function at lower and mid values.
Is interval width narrow enough to stop?
If signs are same (+ product), eliminate lower half of interval.
Engineering Computation: An Introduction Using MATLAB and Excel

15. Bisection Method Example
Next iteration:
Evaluate function at lower and mid values.
New Interval (if statements based on product at the end of previous row)
If signs are different (- product), eliminate upper half of interval.
Is interval width narrow enough to stop?
Engineering Computation: An Introduction Using MATLAB and Excel

16. Bisection Method Example
Continue until interval width < 2*error (16 iterations)
Answer:
y = 0.857
Engineering Computation: An Introduction Using MATLAB and Excel

17. Bisection Method Example
Or course, we know that the exact answer is 6/7 ( )
If we wanted our answer accurate to 5 decimal places, we could set the allowable error to
This increases the number of iterations only from 16 to 22 – the halving process quickly reduces the interval to very small values
Even if the initial guesses are set to -10,000 and 10000, only 32 iterations are required to get a solution accurate to 5 decimal places
Engineering Computation: An Introduction Using MATLAB and Excel

18. Bisection Method Example - Polynomial
Now consider this example:
Use the bisection method, with allowed error of
Engineering Computation: An Introduction Using MATLAB and Excel

19. Bisection Method Example - Polynomial
If limits of -10 to 0 are selected, the solution converges to x = -2
Engineering Computation: An Introduction Using MATLAB and Excel

20. Bisection Method Example - Polynomial
If limits of 0 to 10 are selected, the solution converges to x = 4
Engineering Computation: An Introduction Using MATLAB and Excel

21. Bisection Method Example - Polynomial
If limits of -10 to 10 are selected, which root is found?
In this case f(-10) and f(10) are both positive, and f(0) is negative
Engineering Computation: An Introduction Using MATLAB and Excel

22. Bisection Method Example - Polynomial
Which half of the interval is kept?
Depends on the algorithm used – in our example, if the function values for the lower limit and midpoint are of opposite signs, we keep the lower half of the interval
Engineering Computation: An Introduction Using MATLAB and Excel

23. Bisection Method Example - Polynomial
Therefore, we converge to the negative root
Engineering Computation: An Introduction Using MATLAB and Excel

24. In-Class Exercise
Draw a flow chart of the algorithm used to find a root of an equation using the bisection method
Write the MATLAB code to determine a root of
within the interval x = 0 to 10
Engineering Computation: An Introduction Using MATLAB and Excel

25. Input lower and upper limits low and high
Define tolerance tol
Input lower and upper limits low and high
while high-low > 2*tol
mid = (high+low)/2
Evaluate function at lower limit and midpoint:
fl = f(low), fm = f(mid)
Keep lower half of range:
high = mid
fl*fm > 0?
Keep upper half of range:
low = mid NO
YES
Display root (mid)

26. MATLAB Solution Consider defining the function
as a MATLAB function “fun1”
This will allow our bisection program to be used on other functions without editing the program – only the MATLAB function needs to be modified
Engineering Computation: An Introduction Using MATLAB and Excel

27. MATLAB Function Check values at x = 0 and x = 10:
function y = fun1(x)
y = exp(x) - 15*x -10;
Check values at x = 0 and x = 10:
>> fun1(0)
ans = -9
>> fun1(10)
2.1866e+004
Different signs, so a root exists within this range
Engineering Computation: An Introduction Using MATLAB and Excel

28. Set tolerance to 0.00001; answer will be accurate to 5 decimal places
Engineering Computation: An Introduction Using MATLAB and Excel

29. Engineering Computation: An Introduction Using MATLAB and Excel

30. Engineering Computation: An Introduction Using MATLAB and Excel

31. Engineering Computation: An Introduction Using MATLAB and Excel

32. Engineering Computation: An Introduction Using MATLAB and Excel

33. Engineering Computation: An Introduction Using MATLAB and Excel

34. Engineering Computation: An Introduction Using MATLAB and Excel

35. Find Root
>> bisect Enter the lower limit 0 Enter the upper limit 10 Root found:
Engineering Computation: An Introduction Using MATLAB and Excel

36. What if No Root Exists? Try interval of 0 to 3: >> bisect
Enter the lower limit
Enter the upper limit
Root found:
This value is not a root – we might want to add a check to see if the converged value is a root
Engineering Computation: An Introduction Using MATLAB and Excel

37. Modified Code
Add “solution tolerance” (usually looser than convergence tolerance):
Add check at end of program:
Engineering Computation: An Introduction Using MATLAB and Excel

38. Check Revised Code
>> bisect Enter the lower limit 0 Enter the upper limit 3 No root found
Engineering Computation: An Introduction Using MATLAB and Excel

39. Numerical Tools
Of course, Excel and MATLAB have built-in tools for finding roots of equations
However, the examples we have considered illustrate an important concept about non-linear solutions:
Remember that there may be many roots to a non-linear equation. Even when specifying an interval to be searched, keep in mind that there may be multiple solutions (or no solution) within the interval.
Engineering Computation: An Introduction Using MATLAB and Excel