Transcript Chapter 1 Computing Tools

Newton’s Method,
Root Finding with MATLAB and Excel
Chapter 6
Finding the Roots of
Equations
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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
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
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
Newton’s Method
• Newton’s Method (also know as the NewtonRapshon 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
Newton’s Method
y(xi)
Tangent
Line: Slope =
y’(xi)
Initial guess xi
Engineering Computation: An Introduction Using MATLAB and Excel
Newton’s Method
New guess for x:
xn = xi - y(xi)/y’(xi)
y(xi)
xi
y(xi)/y’(xi)
Engineering Computation: An Introduction Using MATLAB and Excel
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
Newton’s Method Example
• For x = 10:
• This is the new value of x
Engineering Computation: An Introduction Using MATLAB and Excel
Newton’s Method Example
y = 1350
y’ = 580
New x-value =
7.6724
Initial
guess =
10
Engineering Computation: An Introduction Using MATLAB and Excel
Newton’s Method Example
• For x = 7.6724:
Engineering Computation: An Introduction Using MATLAB and Excel
Newton’s Method Example
y = 368.494
y’ = 279.621
New x-value =
6.3546
Initial
value =
7.6724
Engineering Computation: An Introduction Using MATLAB and Excel
Newton’s Method Example
• Continue iterations:
Method quickly converges to this root
Engineering Computation: An Introduction Using MATLAB and Excel
Newton’s Method Example
• To find the
other roots,
use different
initial guess
values
Engineering Computation: An Introduction Using MATLAB and Excel
Newton’s Method Example
• Three roots found:
x = -2.0764
x = 1.4187
x = 5.6577
Engineering Computation: An Introduction Using MATLAB and Excel
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
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
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
MATLAB Code
• MATLAB functions defining the function and its
derivative:
Engineering Computation: An Introduction Using MATLAB and Excel
MATLAB Code
Engineering Computation: An Introduction Using MATLAB and Excel
MATLAB Results
>> Newton
Enter initial guess
10
Root found: 5.6577
>> Newton
Enter initial guess
0
Root found: 1.4187
>> Newton
Enter initial guess
-10
Root found: -2.0764
Engineering Computation: An Introduction Using MATLAB and Excel
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
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
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
Goal Seek Example
Engineering Computation: An Introduction Using MATLAB and Excel
Goal Seek Example
Engineering Computation: An Introduction Using MATLAB and Excel
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
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])
ans =
5.6577
Engineering Computation: An Introduction Using MATLAB and Excel
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
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
Graphing to Help Find Roots
• Equation 1:
3 real roots
Equation 2:
1 real root
Engineering Computation: An Introduction Using MATLAB and Excel
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
roots Example
>> A = [3, -15, -20, 50];
>> roots(A)
ans =
5.6577
-2.0764
1.4187
Engineering Computation: An Introduction Using MATLAB and Excel
roots Example
• Now find roots of
>> B = [3, -5, -20, 50];
>> roots(B)
ans =
-2.8120
2.2393 + 0.9553i
Two complex roots
2.2393 - 0.9553i
Engineering Computation: An Introduction Using MATLAB and Excel
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