ROOTS OF EQUATIONS
Student Notes
ENGR 351
Numerical Methods for Engineers
Southern Illinois University Carbondale
College of Engineering
Dr. L.R. Chevalier
Dr. B.A. DeVantier
Applied Problem
The concentration of pollutant bacteria C in a lake
decreases according to:
C  80e2 t  20e0.1t
Determine the time required for the bacteria to be
reduced to 10 ppm.
Applied Problem
You buy a $20 K piece of equipment for nothing down
and $5K per year for 5 years. What interest rate are you
paying? The formula relating present worth (P), annual
payments (A), number of years (n) and the interest rate
(i) is:
i1  i
A P
n
1  i  1
n
Quadratic Formula
 b  b 2  4 ac
x
2a
f ( x)  ax 2  bx  c  0
This equation gives us the roots of the algebraic function
f(x)
i.e. the value of x that makes f(x) = 0
How can we solve for f(x) = e-x - x?
Roots of Equations
 Plot the function and determine where
it crosses the x-axis
 Lacks precision
 Trial and error
f(x)
10
8
6
4
2
0
-2
-2
-1
0
x
1
2
Overview of Methods
 Bracketing methods


Bisection method
False position
 Open methods


Newton-Raphson
Secant method
Specific Study Objectives
 Understand the graphical interpretation of
a root
 Know the graphical interpretation of the
false-position method (regula falsi
method) and why it is usually superior to
the bisection method
 Understand the difference between
bracketing and open methods for root
location
Specific Study Objectives
 Understand the concepts of convergence
and divergence.
 Know why bracketing methods always
converge, whereas open methods may
sometimes diverge
 Know the fundamental difference between
the false position and secant methods and
how it relates to convergence
Specific Study Objectives
 Understand the problems posed by multiple
roots and the modification available to
mitigate them
 Use the techniques presented to find the root
of an equation
 Solve two nonlinear simultaneous equations
using techniques similar to root finding
methods
Bracketing Methods
 Bisection method
 False position method (regula falsi
method)
Graphically Speaking
1.
2.
3.
4.
5.
6.
7.
Graph the function
Based on the graph, select two x
values that “bracket the root”
What is the sign of the y value?
Determine a new x (xr) based on
the method
What is the sign of the y value of
xr?
Switch xr with the point that has a
y value with the same sign
Continue until f(xr) = 0
xl
xr
xu
Theory Behind Bracketing Methods
f(x)
f(x)
x
consider lower
and upper bound
same sign,
no roots or
even # of roots
x
f(x)
f(x)
opposite sign,
odd # of roots
x
x
Bisection Method
 xr = (xl + xu)/2
 Takes advantage of sign changing
 There is at least one real root
f(x)
x
Graphically Speaking
1.
2.
3.
4.
5.
6.
7.
Graph the function
Based on the graph, select two x
values that “bracket the root”
What is the sign of the y value?
xr = (xl + xu)/2
What is the sign of the y value of
xr?
Switch xr with the point that has a
y value with the same sign
Continue until f(xr) = 0
xl
xr
xu
Algorithm
 Choose xu and xl. Verify sign change
f(xl)f(xu) < 0
 Estimate root
 xr = (xl + xu) / 2
 Determine if the estimate is in the lower or upper
subinterval
 f(xl)f(xr) < 0
then xu = xr RETURN
 f(xl)f(xr) >0
then xl = xr RETURN
 f(xl)f(xr) =0
then root equals xr - COMPLETE

Error
present approx previous approx
a 
100
present approx
Let’s consider an example problem:
Example
Use the bisection method to determine the root
-x
xu = 1
10
8
6
f(x) = 3.718
f(x)
•f(x) =
•xl = -1
e-x
4
2
0
-2
-1.5
-1
-0.5
0
-2
-4
0.5
1
1.5
f(x) = -0.632
x
STRATEGY
2
Strategy
 Calculate f(xl) and f(xu)
 Calculate xr
 Calculate f(xr)
 Replace xl or xu with xr based on the sign
of f(xr)
 Calculate a based on xr for all iterations
after the first iteration
 REPEAT
False Position Method
 “Brute Force” of bisection method is
inefficient
 Join points by a straight line
 Improves the estimate
 Replacing the curve by a straight line
gives the “false position”
next
estimate,
xr
f(xu)
xl
f(xl)
xu
Based on
similar
triangles
f  xl 
f  xu 

xr  xl xr  xu
f  xu  xl  xu 
xr  xu 
f  xl   f  xu 
Example
Determine the root of the following equation
using the false position method starting with
an initial estimate of xl=4.55 and xu=4.65
30
f(x) = x3 - 98
20
f(x)
10
0
-1 0
-20
-30
-40
4
4.5
5
x
STRATEGY
Strategy
 Calculate f(xl) and f(xu)
 Calculate xr
 Calculate f(xr)
 Replace xl or xu with xr based on the sign
of f(xr)
 Calculate a based on xr for all iterations
after the first iteration
 REPEAT
Example Spreadsheet
 Use of IF-THEN statements
 Recall in the bi-section or false position




methods.
If f(xl)f(xr)>0 then they are the same sign
Need to replace xu with xr
If f(xl)f(xr)< 0 then they are opposite signs
Need to replace xl with xr
Example Spreadsheet
xl
xu
f(xl)
f(xu) xr
f(xr)
0.01
0.10
-549.03 592.15 0.06 3.58
f(xl)f(xr)
-1964.96
?
If f(xl)f(xr) is negative, we want to leave xu as xu
If f(xl)f(xr) is positive, we want to replace xu with xr
The EXCEL command for the next xu entry follows the logic
If f(xl)f(xr) < 0, xu,xr
Example Spreadsheet
Pitfalls of False Position
Method
f(x)
f(x)=x1 0-1
30
25
20
15
10
5
0
-5
0
0.5
1
x
1 .5
Open Methods
 Newton-Raphson method
 Secant method
 Multiple roots
 In the previous bracketing methods, the
root is located within an interval
prescribed by an upper and lower
boundary
Newton Raphson
most widely used
f(x)
x
Newton Raphson
tangent
f(xi)
dy
tangent 
 f'
dx
f  xi   0
f '  xi  
xi  xi1
xi+1
xi
rearrange
f  xi 
xi1  xi 
f '  xi 
Newton Raphson
xi 1  xi 
i
0
1
f xi 
f ' xi 
x
A
D
f(x)
B
f’(x)
C
2





A is the initial estimate
B is the function evaluated at A
C is the first derivative evaluated at A
D= A-B/C
Repeat
Newton Raphson
Pitfalls
Solution can “overshoot”
the root and potentially
diverge
f(x)
x2
x1
x0
x
Use the Newton
Raphson method
to determine the
root of
f(x) = x2 - 11
using an initial
guess of xi = 3
6
f(x)
Example
4
2
0
-2
x
0
1
2
3
4
-4
-6
-8
-10
-12
STRATEGY
5
Strategy
 Start a table to track your solution
i
xi
0
x0
f(xi)
f’(xi)
 Calculate f(x) and f’(x)
 Estimate the next xi based on the
governing equation
 Use s to determine when to stop
 Note: use of subscript “0”
Secant method
Approximate derivative using a finite divided difference
f  xi 1   f  xi 
f '  x 
xi 1  xi
What is this? HINT: dy / dx = Dy / Dx
Substitute this into the formula for Newton Raphson
f  xi 
xi1  xi 
f '  xi 
f  xi  xi 1  xi 
xi1  xi 
f  xi 1   f  xi 
Substitute finite
difference approximation
for the
first derivative into this
equation for Newton
Raphson
Secant
method
Secant method
f  xi  xi1  xi 
xi1  xi 
f  xi1   f  xi 
 Requires two initial estimates
 f(x) is not required to change signs, therefore
this is not a bracketing method
Secant method
f(x)
{
slope
between
two
estimates
new estimate
x
initial estimates
Example
Determine the root of f(x) = e-x - x using the
secant method. Use the starting points x0 = 0
and x1 = 1.0.
1.5
0, 1.000
1.0
f(x)
0.5
0.0
-0.5
-1.0
0
0.5
1
1.5
2
2.5
x
1, -0.632
-1.5
-2.0
-2.5
STRATEGY
Strategy
 Start a table to track your results
i
xi
f(xi)
0
0
Calculate
1
1
Calculate
2
Calculate
a
 Note: here you need two starting
points!
 Use these to calculate x2
 Use x3 and x2 to calculate a at i=3
 Use s
Comparison of False Position and
Secant Method
2
f(x)
f(x)
2
1
x
1
new est.
x
new est.
Multiple Roots
 Corresponds to a point
where a function is
tangential to the x-axis
 i.e. double root



5x2
f(x) = + 7x -3
f(x) = (x-3)(x-1)(x-1)
i.e. triple root
f(x) = (x-3)(x-1)3
8
6
f(x)

x3
10
4
m ultiple root
2
0
-2
-4
0
1
2
x
3
4
Difficulties
 Bracketing methods won’t work
 Limited to methods that may diverge
10
8
f(x)
6
4
m ultiple root
2
0
-2
-4
0
1
2
x
3
4
denominator for
Newton-Raphson
and Secant Methods
 Write a “DO LOOP”
to check is f(x) = 0
before continuing
10
8
6
f(x)
 f(x) = 0 at root
 f '(x) = 0 at root
 Hence, zero in the
4
m ultiple root
2
0
-2
-4
0
1
2
x
3
4
Multiple Roots
 f '  xi   f  xi  f ' '  xi 
2
10
8
6
f(x)
xi  1  xi 
f  xi  f '  xi 
4
m ultiple root
2
0
-2
-4
0
1
2
x
3
4
Systems of Non-Linear Equations
 We will later consider systems of linear
equations


f(x) = a1x1 + a2x2+...... anxn - C = 0
where a1 , a2 .... an and C are constant
 Consider the following equations


y = -x2 + x + 0.5
y + 5xy = x3
 Solve for x and y
Systems of Non-Linear Equations
cont.
 Set the equations equal to zero


y = -x2 + x + 0.5
y + 5xy = x3
 u(x,y) = -x2 + x + 0.5 - y = 0
 v(x,y) = y + 5xy - x3 = 0
 The solution would be the values of x
and y that would make the functions u
and v equal to zero
Recall the Taylor Series
f ' '  xi  2 f ' ' '  xi  3
f  xi 1   f  xi   f '  xi h 
h 
h 
2!
3!
n
f  xi  n
......
h  Rn
n!
where h  step size  xi 1  xi
Write a first order Taylor series with
respect to u and v
 ui
 ui
xi 1  xi  
 yi 1  yi 
ui 1  ui 
x
y
 vi
 vi
xi 1  xi    yi 1  yi 
vi 1  vi 
x
y
The root estimate corresponds to the point where
ui+1 = vi+1 = 0
Therefore
vi
ui
ui
 vi
y
y
xi 1  xi 
ui vi ui vi

x y y x
ui
vi
vi
 ui
x
x
yi 1  yi 
ui vi ui vi

x y y x
THE DENOMINATOR
OF EACH OF THESE
EQUATIONS IS
FORMALLY
REFERRED TO
AS THE DETERMINANT
OF THE
JACOBIAN
This is a 2 equation version of Newton-Raphson
Example
 Determine the roots of the following
nonlinear simultaneous equations


x2+xy=10
y + 3xy2 = 57
 Use and initial estimate of x=1.5, y=3.5
25
20
f(x)
15
10
5
STRATEGY
0
0
1
2
3
x
4
5
Strategy
 Rewrite equations to get


u(x,y) = 0 from equation 1
v(x,y) = 0 from equation 2
 Determine the equations for the partial
of u and v with respect to x and y
 Start a table!
i
xi
yi
u (x,y)
v(x,y)
du/dx
du/dy
dv/dx
dv/dy
J
Specific Study Objectives
 Understand the graphical interpretation of
a root
 Know the graphical interpretation of the
false-position method (regula falsi
method) and why it is usually superior to
the bisection method
 Understand the difference between
bracketing and open methods for root
location
Specific Study Objectives
 Understand the concepts of convergence
and divergence.
 Know why bracketing methods always
converge, whereas open methods may
sometimes diverge
 Know the fundamental difference between
the false position and secant methods and
how it relates to convergence
Specific Study Objectives
 Understand the problems posed by multiple
roots and the modification available to
mitigate them
 Use the techniques presented to find the root
of an equation
 Solve two nonlinear simultaneous equations
Applied Problem
The concentration of pollutant bacteria C in a lake
decreases according to:
C  80e2 t  20e0.1t
Determine the time required for the bacteria to be
reduced to 10 using Newton-Raphson method.
Applied Problem
You buy a $20 K piece of equipment for nothing down
and $5K per year for 5 years. What interest rate are you
paying? The formula relating present worth (P), annual
payments (A), number of years (n) and the interest rate
(i) is:
i1  i
A P
1  i n  1
n
Use the bisection method