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