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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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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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
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