1. Newton’s Method, Root Finding with MATLAB and Excel
Chapter 6 Finding the Roots of Equations
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2. Review: 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

3. Review: 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

4. Review: 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
Engineering Computation: An Introduction Using MATLAB and Excel

5. Newton’s Method
Newton’s Method (also know as the Newton-Rapshon Method) is another widely-used algorithm for finding roots of equations
In this method, the slope (derivative) of the function is calculated at the initial guess value and projected to the x-axis
The corresponding x-value becomes the new guess value
The steps are repeated until the answer is obtained to a specified tolerance
Engineering Computation: An Introduction Using MATLAB and Excel

6. Newton’s Method y(xi) Tangent Line: Slope = y’(xi) Initial guess xi
Engineering Computation: An Introduction Using MATLAB and Excel

7. Newton’s Method New guess for x: xn = xi - y(xi)/y’(xi) y(xi) xi
Engineering Computation: An Introduction Using MATLAB and Excel

8. Newton’s Method Example
Find a root of this equation:
The first derivative is:
Initial guess value: x = 10
Engineering Computation: An Introduction Using MATLAB and Excel

9. Newton’s Method Example
For x = 10:
This is the new value of x
Engineering Computation: An Introduction Using MATLAB and Excel

10. Newton’s Method Example
y = 1350
y’ = 580
New x-value =
Initial guess = 10
Engineering Computation: An Introduction Using MATLAB and Excel

11. Newton’s Method Example
For x = :
Engineering Computation: An Introduction Using MATLAB and Excel

12. Newton’s Method Example
y =
y’ =
Initial value =
New x-value =
Engineering Computation: An Introduction Using MATLAB and Excel

13. Newton’s Method Example
Continue iterations:
Method quickly converges to this root
Engineering Computation: An Introduction Using MATLAB and Excel

14. Newton’s Method Example
To find the other roots, use different initial guess values
Engineering Computation: An Introduction Using MATLAB and Excel

15. Newton’s Method Example
Three roots found:
x =
x =
x =
Engineering Computation: An Introduction Using MATLAB and Excel

16. Newton’s Method - Comments
Usually converges to a root much faster than the bisection method
In some cases, the method will not converge (discontinuous derivative, initial slope = 0, etc.)
In most cases, however, if the initial guess is relatively close to the actual root, the method will converge
Don’t necessarily need to calculate the derivative: can approximate the slope from two points close to the x-value. This is called the Secant Method
Engineering Computation: An Introduction Using MATLAB and Excel

17. In-Class Exercise Draw a flow chart of Newton’s Method
Write the MATLAB code to apply Newton’s Method to the previous example:
Engineering Computation: An Introduction Using MATLAB and Excel

18. Define converge tolerance tol
Input initial guess x
Calculate f(x)
while abs(f(x)) > tol
Calculate slope fpr(x)
x = x – f(x)/fpr(x)
Calculate f(x)
Output root x
Engineering Computation: An Introduction Using MATLAB and Excel

19. MATLAB Code MATLAB functions defining the function and its derivative:
Engineering Computation: An Introduction Using MATLAB and Excel

20. MATLAB Code
Engineering Computation: An Introduction Using MATLAB and Excel

21. MATLAB Results
>> Newton Enter initial guess 10 Root found: Root found: Root found:
Engineering Computation: An Introduction Using MATLAB and Excel

22. Excel and MATLAB Tools
Bisection method or Newton’s method can be easily programmed
However, Excel and MATLAB have built-in tools for finding roots of equations
Engineering Computation: An Introduction Using MATLAB and Excel

23. Excel and MATLAB Tools
General non-linear equations:
Excel: Goal Seek, Solver
MATLAB: fzero
Polynomials:
MATLAB: roots
Graphing tools are also important to locate roots approximately
Engineering Computation: An Introduction Using MATLAB and Excel

24. Excel Tools
We have seen how to use Solver for a system of equations; we can easily apply it to find roots of a single equation
There is a tool in Excel even easier to use for a single equation: Goal Seek
With Goal Seek, we instruct Excel to change the value of a cell containing the independent variable to cause the value of the cell containing the function to equal a given value
Engineering Computation: An Introduction Using MATLAB and Excel

25. Goal Seek Example
Engineering Computation: An Introduction Using MATLAB and Excel

26. Goal Seek Example
Engineering Computation: An Introduction Using MATLAB and Excel

27. fzero Example
The MATLAB function fzero finds the root of a function, starting with a guess value
Example: function fun1 defines this equation:
Engineering Computation: An Introduction Using MATLAB and Excel

28. fzero Example
The function fzero has arguments of the name of the function to be evaluated and a guess value:
>> fzero('fun1',10)
ans =
5.6577
Or the name of function to be evaluated and a range of values to be considered:
>> fzero('fun1',[4 10])
Engineering Computation: An Introduction Using MATLAB and Excel

29. fzero Example
If a range is specified, then the function must have opposite signs at the endpoints
> fzero('fun1',[0 10])
??? Error using ==> fzero at 292
The function values at the interval endpoints must differ in sign.
Engineering Computation: An Introduction Using MATLAB and Excel

30. Graphing to Help Find Roots
Making a graph of the function is helpful to determine how many real roots exist, and the approximate locations of the roots
Example: Consider these two equations:
How many roots does each have?
Engineering Computation: An Introduction Using MATLAB and Excel

31. Graphing to Help Find Roots
Equation 1: Equation 2:
3 real roots
1 real root
Engineering Computation: An Introduction Using MATLAB and Excel

32. roots Example
For polynomials, the MATLAB function roots finds all of the roots, including complex roots
The argument of roots is an array containing the coefficients of the equation
For example, for the equation
the coefficient array is [3, -15, -20, 50]
Engineering Computation: An Introduction Using MATLAB and Excel

33. roots Example
>> A = [3, -15, -20, 50]; >> roots(A) ans =
Engineering Computation: An Introduction Using MATLAB and Excel

34. roots Example Now find roots of Two complex roots
>> B = [3, -5, -20, 50];
>> roots(B)
ans = i i
Two complex roots
Engineering Computation: An Introduction Using MATLAB and Excel

35. Summary
The bisection method and Newton’s method (or secant method) are widely-used algorithms for finding the roots of equations
When using any tool to find the roots of non-linear equations, remember that multiple roots may exist
The initial guess value will affect which root is found
Engineering Computation: An Introduction Using MATLAB and Excel