W8_Finding the Roots

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Centre for Computer Technology

ICT114ICT114Mathematics forMathematics for

ComputingComputing

Week 8Week 8

Finding the Roots of f(x)Finding the Roots of f(x)


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March 20, 2012March 20, 2012 Copyright Box Hill Institute

ObjectivesObjectives

Review week 7Review week 7

Convergence CriteriaConvergence Criteria

Errors in ComputingErrors in ComputingFixed Point IterationFixed Point Iteration

Bisection MethodBisection Method

SummarySummary


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Gauss Jordan MethodGauss Jordan Method

For the above matrix,For the above matrix,perform a series of rowperform a series of rowoperations to obtain theoperations to obtain thematrix in the formmatrix in the form

The matrix B is theThe matrix B is theinverse of the matrixinverse of the matrix


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Gauss Elimination MethodGauss Elimination Method

A x = bA x = b

aa1111 aa1212 aa1313 xx11 bb11

aa2121 aa2222 aa2323 xx22 = b= b22

aa3131 aa3232 aa3333 xx33 bb33

The solution of the above system ofThe solution of the above system ofequations isequations is x = Ax = A-1-1bb


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Finding the Roots of f(x)Finding the Roots of f(x)


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IntroductionIntroduction

We will discuss methods to find the rootsWe will discuss methods to find the roots

of a polynomialof a polynomial

Fixed Point IterationFixed Point IterationBisection MethodBisection Method

Newton's MethodNewton's Method

Secant MethodSecant Method


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Convergence CriteriaConvergence Criteria

Convergence is a procedure to monitor progress

toward the root and stop when current guess is

close enough to the desired root.

Convergence checking

will avoid searching to unnecessary accuracy.

whether two successive approximations to the

root are close enough to be considered equal. f(x) is sufficiently close to zero at the current

guess.


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Fixed Point IterationFixed Point Iteration


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ItIt involves evaluating a formulainvolves evaluating a formula that takesthat takes

a guess at a root as inputa guess at a root as input andand returns anreturns an

updated guess at the root as outputupdated guess at the root as output. The. Thesuccess of this method depends on thesuccess of this method depends on the

choice of the formula that is iteratedchoice of the formula that is iterated


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Algorithm : Fixed Point IterationAlgorithm : Fixed Point Iteration

To solve

f(x) = 0

rewrite as

xnew = g(xold)

initialize:x0 = . . .

fork= 1, 2, . . .xk= g(xk-1)

if converged, stop

end


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Given anGiven an initial bracket for a rootinitial bracket for a root, the, thesystematic halving of the of the bracketsystematic halving of the of the bracketaround the rootaround the root is called the bisectionis called the bisectionmethod. Though it does it slowly,method. Though it does it slowly, it alwaysit alwaysconverges.converges.

note:note: when a root is suspected to lie inwhen a root is suspected to lie inthe rangethe range xxleftleft x x x xrightright, the pair (x, the pair (xleftleft,,xxrightright ) is) is

referred to as a bracket.referred to as a bracket.


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March 20, 2012March 20, 2012

Copyright Box Hill Institute



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March 20, 2012March 20, 2012




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March 20, 2012March 20, 2012


Algorithm : Bisection MethodAlgorithm : Bisection Method

initialize: a = . . ., b = . . .

fork= 1, 2, . . .xm= (a + b)/2

iff(xm)


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March 20, 2012March 20, 2012Copyright Box Hill Institute


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SummarySummary

Fixed Point Iteration - It involves evaluating aFixed Point Iteration - It involves evaluating a

formula that takes a guess at a root as input andformula that takes a guess at a root as input and

returns an updated guess at the root as output.returns an updated guess at the root as output.

The success of this method depends on theThe success of this method depends on thechoice of the formula that is iteratedchoice of the formula that is iterated

Bisection Method - Given an initial bracket for aBisection Method - Given an initial bracket for a

root, the systematic halving of the of the bracketroot, the systematic halving of the of the bracketaround the root is called the bisection method.around the root is called the bisection method.

Though it does it slowly, it always converges.Though it does it slowly, it always converges.


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ReferencesReferences

Gerald W. Recktenwald, Numerical MethodsGerald W. Recktenwald, Numerical Methodswith MATLAB, Implementation and Application,with MATLAB, Implementation and Application,Prentice HallPrentice Hall

H L Verma and C W Gross : Introduction toH L Verma and C W Gross : Introduction toQuantitative Methods,John WileyQuantitative Methods,John Wiley

JB Scarborough : Numerical MathematicalJB Scarborough : Numerical MathematicalAnalysis, Jon Hopkins Hall, New JerseyAnalysis, Jon Hopkins Hall, New Jersey

Finding the Roots off(x) = 0, Gerald W.Recktenwald, Department of MechanicalEngineering, Portland State University

W8_Finding the Roots

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