W9_Finding the Roots

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Centre for Computer Technology

ICT114Mathematics for

Computing

Week 9

Finding the Roots of f(x)


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March 20, 2012 Copyright Box Hill Institute

Objectives

Review week 8

Errors in Computing

Differential Newtons Method

Secant Method

Summary


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Fixed Point Iteration

It involves evaluating a formula that takesa guess at a root as input and returns an

updated guess at the root as output. Thesuccess of this method depends on thechoice of the formula that is iterated


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Algorithm : Fixed Point Iteration

To solvef(x) = 0

rewrite as

xnew = g(xold)

initialize: x0 = . . .for k= 1, 2, . . .

xk= g(xk-1)if converged, stop

end


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Bisection Method

Given an initial bracket for a root, thesystematic halving of the of the bracketaround the root is called the bisectionmethod. Though it does it slowly, it alwaysconverges.

note: when a root is suspected to lie in therange xleft x xright, the pair (xleft,xright ) isreferred to as a bracket.


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Algorithm : Bisection Method

initialize: a= . . ., b= . . .for k= 1, 2, . . .

xm= (a+ b)/2

if f(xm)


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Newtons Method


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Newton's Method

Using an initial guess at the root and the

slope of f(x), Newton's method usesextrapolation to estimate where f(x)crosses the x axis. This method converges

very quickly, but it can diverge if f(x) = 0 isencountered during iterations.

(f(x) is the differential of f(x))


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Newtons Method


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Newtons Method

f(x) is the differential for f(x)


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Algorithm

initialize: x1 = . . .

for k= 2, 3, . . .xk= xk-1- f(xk-1)/f(xk-1)

if converged, stop

end


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Example


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Secant Method


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March 20, 2012Copyright Box Hill Institute

Secant Method

The secant method approximates f(x)from the value of f(x) at two previous

guesses at the root. It is as fast as theNewton's method but can also fail atf(x)=0.


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Secant Method


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Secant Method


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Algorithm

initialize: x1 = . . ., x2 = . . .

for k= 2, 3 .. .

xk+1 = xk- f(xk)(xk- xk-1)/(f(xk) - f(xk-1))If f(xk+1)


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Example


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Summary

Newtons Method - Using an initial guessat the root and the slope of f(x), Newton'smethod uses extrapolation to estimate

where f(x) crosses the x axis. Secant Method - The secant method

approximates f(x) from the value of f(x) at

two previous guesses at the root. It is asfast as the Newton's method but can alsofail at f(x)=0.


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References

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

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

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

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

W9_Finding the Roots

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