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ROOTS OF EQUATIONSStudent Notes
ENGR 351 Numerical Methods for EngineersSouthern Illinois University CarbondaleCollege of EngineeringDr. L.R. ChevalierDr. B.A. DeVantier
Applied Problem
The concentration of pollutant bacteria C in a lakedecreases according to:
Determine the time required for the bacteria to be reduced to 10 ppm.
C e et t 80 202 0 1.
You buy a $20 K piece of equipment for nothing downand $5K per year for 5 years. What interest rate are you paying? The formula relating present worth (P), annualpayments (A), number of years (n) and the interest rate(i) is:
A Pi i
i
n
n
1
1 1
Applied Problem
Quadratic Formula
xb b ac
a
f x ax bx c
2
2
4
2
0( )
This equation gives us the roots of the algebraic functionf(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 precisionTrial and error
-2
0
2
4
6
8
10
-2 -1 0 1 2
x
f(x)
Overview of Methods
Bracketing methods Bisection method False position
Open methods Newton-Raphson Secant method
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
Specific Study Objectives
Bracketing Methods
Bisection method False position method (regula falsi
method)
Graphically Speaking
xl xu
1. Graph the function2. Based on the graph, select two
x values that “bracket the root”3. What is the sign of the y value?4. Determine a new x (xr) based
on the method5. What is the sign of the y value
of xr?6. Switch xr with the point that
has a y value with the same sign
7. Continue until f(xr) = 0
xr
x
f(x)
x
f(x)
x
f(x)
x
f(x)
consider lowerand upper boundsame sign,no roots or even # of roots
opposite sign,odd # of roots
Theory Behind Bracketing Methods
Bisection Method
xr = (xl + xu)/2Takes advantage of sign changingThere is at least one real root
x
f(x)
Graphically Speaking
xl xu
1. Graph the function2. Based on the graph, select two
x values that “bracket the root”3. What is the sign of the y value?4. xr = (xl + xu)/25. What is the sign of the y value
of xr?6. Switch xr with the point that
has a y value with the same sign
7. Continue until f(xr) = 0
xr
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
100
approxpresent
approxpreviousapproxpresenta
Let’s consider an example problem:
• f(x) = e-x - x• xl = -1 xu = 1
Use the bisection method to determine the root
Example
STRATEGY
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
-4
-2
0
2
4
6
8
10
12
f(x) = 3.718
f(x) = -0.632
x
f(x)
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 ea 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 estimateReplacing the curve by a straight
line gives the “false position”
xl
xu
f(xl)
f(xu)
next estimate,
xr
f x
x x
f x
x x
x xf x x x
f x f x
l
r l
u
r u
r uu l u
l u
Based on similar triangles
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
f(x) = x3 - 98
-40
-30
-20
-10
0
10
20
30
4 4.5 5
x
f(x)
Example
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 ea 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
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
?
xl xu f(xl) f(xu) xr f(xr) f(xl)f(xr)
0.01 0.10 -549.03 592.15 0.06 3.58 -1964.96
Example Spreadsheet
Pitfalls of False Position Method
f(x)=x10-1
-5
05
10
15
2025
30
0 0.5 1 1.5
x
f(x)
Open Methods
Newton-Raphson methodSecant methodMultiple roots In the previous bracketing
methods, the root is located within an interval prescribed by an upper and lower boundary
Newton Raphsonmost widely used
f(x)
x
Newton Raphson
tangent
dy
dxf
f xf x
x x
rearrange
x xf x
f x
ii
i i
i ii
i
'
'
'
0
1
1
f(xi)
xi
tangent
xi+1
Newton Raphson
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
ii
ii xf
xfxx
'1
i x f(x) f’(x)
0 A B C
1 D
2
Solution can “overshoot”the root and potentiallydiverge
x0
f(x)
x
x1x2
Newton RaphsonPitfalls
Use the Newton Raphson method to determine the root off(x) = x2 - 11 using an initial guess of xi = 3
Example
STRATEGY
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
-12
-10
-8
-6
-4
-2
0
2
4
6
x
f(x)
StrategyStart a table to track your solution
i xi f(xi) f’(xi)
0 x0
Calculate f(x) and f’(x)Estimate the next xi based on the
governing equationUse es to determine when to stopNote: use of subscript “0”
Secant method
f x
f x f x
x xi i
i i
'
1
1
Approximate derivative using a finite divided difference
What is this? HINT: dy / dx = Dy / Dx
Substitute this into the formula for Newton Raphson
Secant method
ii
iiiii
i
iii
xfxfxxxf
xx
xfxf
xx
1
11
1 '
Substitute finite difference approximation for thefirst derivative into this equation for Newton Raphson
Secant method
Requires two initial estimates f(x) is not required to change signs,
therefore this is not a bracketing method
ii
iiiii xfxf
xxxfxx
1
11
Secant method
new estimate initial estimates
slopebetweentwoestimates
f(x)
x{
Determine the root of f(x) = e-x - x using the secant method. Use the starting points x0 = 0 and x1 = 1.0.
Example
STRATEGY
0 0.5 1 1.5 2 2.5
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Series1; 1.000
-0.632
xf(x)
StrategyStart a table to track your results
i xi f(xi) ea
0 0 Calculate
1 1 Calculate
2 Calculate
Note: here you need two starting points!
Use these to calculate x2
Use x3 and x2 to calculate ea at i=3
Use es
Comparison of False Position and Secant Method
x
f(x)
x
f(x)
1
1
2
new est.new est.
2
Multiple Roots
Corresponds to a point where a function is tangential to the x-axis
i.e. double root f(x) = x3 - 5x2 + 7x -3 f(x) = (x-3)(x-1)(x-1) i.e. triple root f(x) = (x-3)(x-1)3 -4
-2
0
2
4
6
8
10
0 1 2 3 4
x
f(x) multiple root
Difficulties
Bracketing methods won’t work Limited to methods that may
diverge
-4
-2
0
2
4
6
8
10
0 1 2 3 4
x
f(x) multiple root
f(x) = 0 at root f '(x) = 0 at root Hence, zero in the
denominator for Newton-Raphson and Secant Methods
Write a “DO LOOP” to check is f(x) = 0 before continuing
-4
-2
0
2
4
6
8
10
0 1 2 3 4
x
f(x) multiple root
Multiple Roots
x xf x f x
f x f x f xi i
i i
i i i
1 2
'
' ' '
-4
-2
0
2
4
6
8
10
0 1 2 3 4
x
f(x) multiple root
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
constantConsider 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 = 0v(x,y) = y + 5xy - x3 = 0The solution would be the values of
x and y that would make the functions u and v equal to zero
Recall the Taylor Series
ii
nni
n
iiiii
xxsizestephwhere
Rhn
xf
hxf
hxf
hxfxfxf
1
......
321
!
!3
'''
!2
'''
Write a first order Taylor series with respect to u and v
iii
iii
ii
iii
iii
ii
yyy
vxx
x
vvv
yyy
uxx
x
uuu
111
111
The root estimate corresponds to the point whereui+1 = vi+1 = 0
Therefore
THE DENOMINATOROF EACH OF THESEEQUATIONS ISFORMALLYREFERRED TOAS THE DETERMINANTOF THEJACOBIAN
This is a 2 equation version of Newton-Raphson
xv
yu
yv
xu
xv
uxu
vyy
xv
yu
yv
xu
yu
vyv
uxx
iiii
ii
ii
ii
iiii
ii
ii
ii
1
1
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
Example
STRATEGY0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
5
10
15
20
25
x
f(x)
StrategyRewrite 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
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
Specific Study Objectives
The concentration of pollutant bacteria C in a lakedecreases according to:
Determine the time required for the bacteria to be reduced to 10 using Newton-Raphson method.
C e et t 80 202 0 1.
Applied Problem
You buy a $20 K piece of equipment for nothing downand $5K per year for 5 years. What interest rate are you paying? The formula relating present worth (P), annualpayments (A), number of years (n) and the interest rate(i) is:
A Pi i
i
n
n
1
1 1
Use the bisection method
Applied Problem