(4)ROOTS_NON_LINEAR_EQNS.pdf

8/10/2019 (4)ROOTS_NON_LINEAR_EQNS.pdf

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Numerical Methods Applied to Mechatronics

Lecture No 4

Escuela de Ingeniera Mecatrnica

Universidad Nacional de Trujillo

NON-LINEAR EQUATIONS ROOTS

Dr. Jorge A. Olortegui Yume Ph.D.


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Non-Linear Equations Roots Dr. Jorge A. Olortegui Yume Ph.D.2

ROOTS PROBLEMS

Functionf can be written as:f(x)=0

particular x satisfying equality is a root

These problems often occur when a design

problem presents an implicit equation for a

required parameter.


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GRAPHICAL METHODS

Plot of the function and observewhere it crosses thex-axis.

Graphing can also indicate whereroots may be and where some root-

finding methods may fail:

a) Same sign, no roots

b) Different sign, one root

c) Same sign, two roots

d) Different sign, three roots


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BRACKETING METHODS

Make 2 initial guesses that bracket the root

Brackets: 2 guessesxlandxufulfillingf(xl)f(xu) < 0

[ ]xl xu

f(xu)

f(xl)


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Incremental Search Methods

Disadvantages:

BRACKETING METHODS

If spacing are too far

apart, brackets may

be missed due tocapturing an even

number of roots

within two points.

Cannot find bracketscontaining even-

multiplicity roots

Tests the value of function at evenly spaced intervals to

find brackets by function sign change.

Good Practice: Plot the function along with the algorithm


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Incremental Search MethodsBRACKETING METHODS

Example: Matlab program


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The Bisection Method

BRACKETING METHODSIncremental Search Methods

a variation of the incremental

search methodthe interval

is always divided in half.

If a function changes sign over

an interval, the function value

at the midpoint is evaluated.

The location of the root is then

determined as lying within the

subinterval where the sign

change occurs.

The absolute error is reduced

by a factor of 2 for each

iteration.


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Bisection Method Errors

BRACKETING METHODSIncremental Search Methods

The absolute error of the bisection method is solelydependent on the absolute error at the start of the

process (the space between the two guesses) and the

number of iterations:

The required number of iterations to obtain a particular

absolute error can be calculated based on the initialguesses:

Ean x

0

2n

nlog2x 0

Ea,d


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BRACKETING METHODSIncremental Search MethodsThe Bisection Method

Example: Matlab program


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BRACKETING METHODSThe False Position Method

It determines the nextguess not by splitting the

bracket in half but byconnecting the endpointswith a straight line and

determining the locationof the intercept of thestraight line (xr).

The value ofxrthenreplaces whichever of the

two initial guesses yieldsa function value with thesame sign asf(xr).

xr xu f(xu)(xlxu)

f(xl) f(xu)


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BRACKETING METHODS

Bisection vs False Position

Bisection does not take intoaccount the shape of the

function; this can be good

or bad depending on thefunction!

Bad: f(x)x10 1


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BRACKETING METHODSThe False Position Method

Homework: Write a general purpose Matlab program for the

False position method


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OPEN METHODS

Open methodsdiffer from bracketing methods, in

that open methods require:

1 starting value or

2 starting values (not necessarily bracketing a

root).

Open methods may diverge as the computation

progresses, butwhen they do converge, they

usually do so much faster than bracketing methods.


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OPEN METHODS

GRAPHICAL COMPARISON OF METHODSa) Bracketing method

b) Diverging open method

c) Converging open method

Note speed!


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OPEN METHODS

SIMPLE FIXED-POINT METHOD

Rearrangef(x)=0 to the form

Use g(x)to predict a new value ofx- that is,

Approximate error : a xi1x i

xi1100%

xgx

ii

xgx 1


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OPEN METHODSSIMPLE FIXED-POINT METHODExample: Solve for the roots of f(x)=e-x-x

Solution:

1. Rearrange

2. Start with a guess for x0. For this case x0 =0

3. Use the iteration formula

4. Continue until some tolerance is reached

xgx x=e-x

i xi g(xi)=e-xi |a| %

0 0.0000 1.0000

1 1.0000 0.3679 100.000

2 0.3679 0.6922 171.828

3 0.6922 0.5005 46.854

4 0.5005 0.6062 38.309

5 0.6062 0.5454 17.436

ii xgx 1

a xi1x i

xi1100%


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Solution:

xgx

567143.0x

00x


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Solution:

00x

10 xg

101

xgx

3679.01 xg

3679.012

xgx

6922.02 xg


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OPEN METHODSSIMPLE FIXED-POINT METHODConvergence:

Requires :

a) Convergent, 0 g(x) 1

d) Divergent, g(x)


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OPEN METHODSSIMPLE FIXED-POINT METHODConvergence:

Requires :

a) Convergent, 0 g(x) 1

d) Divergent, g(x)


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OPEN METHODS

NEWTON-RAPHSON METHOD

Based on forming the tangent line to thef(x) curve at

some guessx, then following the tangent line to where it

crosses thex-axis.

f'(x i) f(xi)0

x ix i1

x i1x i

f(x i)

f'(x i)


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OPEN METHODS

NEWTON-RAPHSON METHOD

Pros and Cons

Pro: The error of the i+1thiteration is

roughly proportional to the square of

the error of the ithiteration - this is

called quadratic convergence

Con: Some functions show slow or

poor convergence


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OPEN METHODSNEWTON-RAPHSON METHOD Matlab Code

O O S


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OPEN METHODS

SECANT METHOD

Newton-Raphson method problem:

Evaluation of the derivative - may be difficult or

inconvenient to evaluate.

Then, the derivative can be approximated by abackward finite divided difference:

f'

(x i) f(x

i1) f(x

i

)

x i1xi

OPEN METHODS


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OPEN METHODS

SECANT METHOD (contd)

Substitution of this approximation for thederivative to the Newton-Raphson method

equation gives:

Note - this method requires twoinitial estimatesofxbut does notrequire an analytical expression

of the derivative.

xi1xi f(xi) xi1x i

f(xi1) f(xi)

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(4)ROOTS_NON_LINEAR_EQNS.pdf