GENG 300 NUMERICAL METHODS

Dr. Mohammad Aman Ullah

2

Roots of Equations

Chapter 5

Lecture 2

TOPICS COVERED FROM CHAPTER 1 & 47/2/2015

3

Num

erical Methods

1. Why numerical methods?2. Mathematical Modelling concept3. Error Analysis:

a. Significant figuresb. Accuracy and precisionc. Error definitions

i. For known true valueii. For approximations

d. Major errorsi. Round offii. Truncation

WHAT WILL BE COVERED FROM CHAPTER 57/2/2015

4

Roots in engineering and science1. Graphical methods2. Bracketing methods

a. Bisectionb. False position

3. Open methodsa. Simple fixed-point iterationb. Newton-Raphsonc. Secant methods

5

Relatively easy: linear, quadratic equations

Difficult: nonlinear

aacbbxcbxax

240

22

5 4 3 2 0 ?sin 0 ?

sin cos 0 ?

ax bx cx dx ex f xx x x

a x b x cx x

ROOTS IN ENGINEERING AND SCIENCE:FINDING ROOTS…

6

Non

line

ar E

quat

ion

Solv

ers

Graphical

BracketingBisection

False Position

Open Methods

Fixed-point

Newton-Raphson

Secant

FINDING ROOTS…

All Iterative

7

Make a plot of the function and observe where it crosses the -axis, i.e. 0

Not very practical but can be used to obtain rough estimates for roots

These estimates can be used as initial guesses for numerical methods that we’ll study here.

1. GRAPHICAL APPROACH

MatlabExcel

8

Use the graphical approach to determine the mass of the bungee jumper with a drag coefficient of 0.25 / to have a velocity of 36 after 4s of free fall.

Note: The acceleration of gravity is 9.81 / .

tanh

tanh 0

369.810.25 tanh

9.81 0.254 0

Assume different values of m and find

1. GRAPHICAL APPROACH…EXAMPLE 5.1

9

The function crosses the axis between 140 and

150 . Visual inspection of the plot provides a rough estimate of the root of 145 . The validity of the graphical estimate can be checked by substituting it into Eq.

tanh

36.0456 /

1. GRAPHICAL APPROACH…EXAMPLE 5.1

Root

10

(aka Two point methods for finding roots) Two initial guesses for the root are required. These guesses must “bracket” or be on either side of the root.If one root of a real and continuous function, 0, is bounded by values

, and then 0.

(The function changes sign on opposite sides of the root)

2. BRACKETING METHODS

11

2. BRACKETING…

12

Odd

and

eve

n nu

mbe

r of

roo

ts

Exce

ptio

ns

2. BRACKETING METHODS…

13

sin10 cos3

2. BRACKETING METHODS…

14

For the arbitrary equation of one variable, 0

1. Pick and such that they bound the root of interest, check if . 0.2. Estimate the root by evaluating /2 .

2.A. THE BISECTION METHOD

15

3. Find the pair a. If . /2 <0, root lies in the lower interval, then

/2 and go to step 2

b. If . /2 0, root lies in the upper interval, then

/2, go to step 2.

c. If . /2 =0, then root is /2 and terminate.

4. Compare with

5. If , stop. Otherwise repeat the process.

%100

2

2

%100

2

2

ul

ulu

ul

ull

xx

xxx

or

xx

xxx

2.A. THE BISECTION METHOD

16

2. A. THE BISECTION METHOD

17

ProsEasyAlways finds rootsNumber of iterations required to attain an absolute error can be computed a priori.

ConsSlowKnowing and that bound rootMultiple rootsNo account is taken of

and , if is closer to zero, it is likely that root is closer to .

2.A. EVALUATION OF BISECTION METHOD

18

Using Bisection method determine the mass of the bungee jumper with a drag coefficient of 0.25 / to have a velocity of 36 after 4s of free fall. Note: The acceleration of gravity is 9.81 / .

tanh

tanh 0

369.810.25 tanh

9.81 0.254 0

? ,Given that true value of 142.7376

2.A. BISECTION METHOD…EXAMPLE 5.3, 5.4

19

2.A. BISECTION METHOD: EG. 5.3, 5.4:

Initial guess: 50, 200

50, 200,50 200

2 125

142.7376 125142.7376 100% 12.43%

50 125 4.578 0.409 1.871

Root must be located above interval between 125 and 200

125 2002 162.5 13.85%

125 162.5 0.409 0.359 0.147

Root must be located in lower interval between 125 and 162.5

125 162.52 143.75 0.709

20

2.A. BISECTION METHOD: EXAMPLE

How many Iterations will it take?

Δ

1 :Δ2

2 :Δ2

: ,Δ2

21

2.A. BISECTION METHOD: TRY YOURSELF

Δ 200 50 150

,Δ2 ⇒

log Δ,

log 2

. =8 iterations

If thedesirederror, , , 0.41in Example 5.4 how many iterations will it take?

Find the root using bisection method2 0 : 3.52050

22

●The bisection method divides the interval to in half not accounting for the magnitudes of and

. For example if is closer to zero than , then it is more likely that the root will be

closer to

●False position method is an alternative approach where and are joined by a straight line; the intersection of which with represent and improved estimate of the root.

2.B. THE FALSE-POSITION METHOD(Regula-Falsi or linear interpolation method)

23

If a real root is bounded by and of 0, then we can approximate the solution by doing a linear interpolation between the points , and ,

to find the value such that 0, is the linear approximation of .

2.B. THE FALSE-POSITION METHOD(Regula-Falsi or linear interpolation method)

24

Step 1: Find a pair of values of such that

Step 2: Estimate the value of the root from the following formula:

and evaluate .

2.B. FALSE-POSITION…PROCEDURE

25

Step 3: Use the new point to replace one of the original points, keeping the two points on opposite sides of the axis.●If 0 then ⇒

●If 0 then ⇒

●If 0 then you have found the root and need go no further!

Step 4: See if the new and are close enough for convergence to be declared. If they are not go back to step 2.

2.B. FALSE-POSITION…PROCEDURE

26

●Faster

●Always converges for a single root

2.B. FALSE-POSITION: WHY IS THIS METHOD?

27

Using False-Position method determine the mass of the bungee jumper with a drag coefficient of 0.25 / to have a velocity of 36 after 4s of free fall.

Note: The acceleration of gravity is 9.81 / .

tanh

tanh 0

369.810.25 tanh

9.81 0.254 0

? ,Given that true value of 142.7376

2.B. FALSE-POSITION METHOD: EX. 5.5

28

2.B. FALSE-POSITION METHOD…EXAMPLE5.5

29

Slow convergence, better use the bisection method

IS FALSE-POSITION ALWAYS BETTER THANBISECTION? EXAMPLE 5.5Use bisection and false position to locate the root of

1 between 0 and 1.3.

30

WHAT IS COVERED FROM CHAPTER 5

Roots in engineering and science1. Graphical methods

2. Bracketing methods a. Bisection b. False position

3. Open methodsa. Simple fixed-point iterationb. Newton-Raphsonc. Secant methods

Single starting point or two starting points (not necessary to bracket the roots)

conv

erge

nt

31

OPEN METHODS

dive

rgen

t

● Bracketing methods are “convergent”.

● Fixed-point methods may sometime “diverge”, depending on the starting point (initial guess) and how the function behaves.

Rearrange the function so that x is on the left side of the equation:

32

SIMPLE FIXED-POINT ITERATION

0 ⇒1, 2,… ,

is given or guessed Example: Solve 2,for 0Solution:

2or, 2or, 1

12

i x g(x) 1 1.000 3.000 2 3.000 1.667 3 1.667 2.200 4 2.200 1.909 5 1.909 2.048 6 2.048 1.977 7 1.977 2.012 8 2.012 1.994 9 1.994 2.003

10 2.003 1.999 11 1.999 2.001

● can be expressed as a pair of equations:

1

2 (component equations)

●Plot them separately●Their point of intersection

is the solution.

33

FIXED-POINT-GRAPHICAL EXPLANATION

Fixed-point iteration converges if

1)( xg

34

FIXED-POINT:CONVERGENCE

●Based on Taylor series expansion:

32

1 !2)()()()( xOxxfxxfxfxf iiii

35

NEWTON-RAPHSON METHOD

●The root is the value of when 0

●After rearranging and neglecting the higher order terms:0

36

●Newton-Raphson is A convenient method for functions whose derivatives can be evaluated analytically

Rate of convergence:● , ∝ ,

●It may not always converge●There is no convergence

criteria●Sometimes, it may converge

very slowly

NEWTON-RAPHSON METHOD…

Fixed Point iteration:

Newton-Raphson iteration:

37

NEWTON-RAPHSON METHOD…

38

SECANT METHOD

●A slight variation of Newton-Raphson’s method for functions whose derivatives are difficult to evaluate. ●For these cases, the derivative can be

approximated by a backward finite divided difference

1,2,3, …

●Requires two initial estimates of , e. g, , 1. However, because is not required to change signs between estimates, it is not classified as a “bracketing” method.

●The secant method has the same properties as Newton’s method. Convergence is not guaranteed for all , .

39

SECANT METHOD…

Modified Secant Method

)()()( Secant Modified

formula. R-N original in the )()()('

compute tofraction on perturbati small a Use

1:iii

iiii

i

iiii

xfxxfxxfxx

xxfxxfxf

40

Example: Falling ParachuteFind out ? (using modified secant method)Given that 9.81 m/s2,

0.25 kg/m,36 m/s at 4 s

41

Modified Secant Method

42

Modified Secant Method

Roots of Polynomials

nnon xaxaxaaxf 2

21)(

… will be covered in lab session

43

44

SUMMARY

…

…

…

…

…