eciv 301 programming & graphics numerical methods for engineers lecture 10 roots of polynomials
Post on 22-Dec-2015
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Objective
nnn xaxaxaaxf 2
210
Calculate Roots of a polynomial
• There are n complex or real roots
• If n odd, at least one real root
• Complex roots exist in conjugate pairs
Polynomial Evaluation
33
2210 xaxaxaaxfn
xaaxaxaxfn 3210
6 Multiplications 4 additions
3 Multiplications 3 additions
Polynomial EvaluationIn general
nnn xaxaxaaxf 2
210
xaaxaxaxf nnn 110
n(n+1)/2 Multiplications n Additions
n Multiplications n Additions
Algorithm
xaaxaxaxf nnn 110
nn aaaaa 110 Store Coefficients in Vector Form
Evaluate starting from Most Inner Parenthesis
xaaxf nnnn 1
Polynomial Deflation
2x5x4x1xxf4
Divide by any of the factors, e.g. (x+3)
432 xx10x27x240
2x3x5x4x1xxf5
Muller’s - Algorithm
(8)
ac4bb
c2xx
223
Compute New Estimate
Two Roots: Choose sign that agrees with sign of b
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