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ICT114ICT114Mathematics forMathematics for
ComputingComputing
Week 8Week 8
Finding the Roots of f(x)Finding the Roots of f(x)
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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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Fixed Point IterationFixed Point Iteration
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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Fixed Point IterationFixed Point Iteration
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Bisection MethodBisection Method
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Bisection MethodBisection Method
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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Bisection MethodBisection Method
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Bisection MethodBisection Method
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Algorithm : Bisection MethodAlgorithm : Bisection Method
initialize: a = . . ., b = . . .
fork= 1, 2, . . .xm= (a + b)/2
iff(xm)
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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
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