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ECIV 301 Programming & Graphics Numerical Methods for Engineers Lecture 10 Roots of Polynomials
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Objective 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
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Polynomial Evaluation Nested Form
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Polynomial Evaluation 6 Multiplications 4 additions 3 Multiplications 3 additions
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Polynomial Evaluation In general n(n+1)/2 Multiplications n Additions n Multiplications n Additions
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Evaluating the Derivative
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Algorithm Store Coefficients in Vector Form Evaluate starting from Most Inner Parenthesis
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Algorithm Continue by adding terms
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Pseudo Code p=0 df=0 DO j=n,0,-1 df = df * x + p p = p * x + a(j) ENDDO Must be evaluated First
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Polynomial Deflation Can also be expressed in factored form
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Polynomial Deflation Divide by any of the factors, e.g. (x+3)
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Deflation – Division by monomial (x-t) Store Coefficients in Vector Form
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Deflation – Division by monomial (x-t) Do i=n-1, 0, -1 EndDo
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Polynomial Division
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Muller’s Method
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Muller’s - Algorithm Define Initial Guesses x o, x 1, x 2 (1) (2) For Each Iteration
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Muller’s - Algorithm (3) (4)
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Muller’s - Algorithm (5) (6) (7)
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Muller’s - Algorithm (8) Compute New Estimate Two Roots: Choose sign that agrees with sign of b
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Muller’s - Algorithm (9) Compute Error If converged FINISH
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Muller’s - Algorithm (10) Assign new guesses (A) For Real Roots ONLY Choose the two points that are closest to x 3 (B) For ALL Roots NEXT ITERATION (11)
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