Lecture+18--+Polynomial+root+_+Gauss+ellimination

Lecture 18 Polynom - Root of Polynomials • Bisection false-position Newton-Raphson secant methods cannot be easily used to determine all roots of

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Unformatted text preview: Root of Polynomials • Bisection, false-position, Newton-Raphson, secant methods cannot be easily used to determine all roots of higher-order polynomials • MATLAB function: roots for polynomials; recast root finding in terms of an eigenvalue problem; MATLAB Function: roots • Zeros of a nth-order polynomial c = poly(r) -coefficients x = roots(c) - roots [ ] 1 2 1 n n 1 2 2 1 n 1 n n n c c c c c c vector t coefficien c x c x c x c x c x p , , , , , ) ( --- = + + + + + = Roots of Polynomial • Consider the 6th-order polynomial >> c = [1 -6 14 10 -111 56 156]; >> r = roots(c) r = 2.0000 + 3.0000i 2.0000 - 3.0000i 3.0000 2.0000 -2.0000 -1.0000 >> polyval(c, r), format long g ans = 1.0e-011 * 0.204636307898909 + 0.025579538487364i 0.204636307898909 - 0.025579538487364i 0.090949470177293 0 0.090949470177293 -0.008526512829121 (4 real and 2 complex roots) Verify f ( xr ) =0 i 3 2 3 2 2 1 x 156 x 56 x 111 x 10 x 14 x 6 x x f r 2 3 4 5 6 ±-- = = + +- + +- = , , , , ) ( f ( x ) = x5 11x4 + 46x3 90x2 + 81x 27 = ( x 1 ) 2 ( x 3 ) 3 >> c = [1 -11 46 -90 81 -27]; r = roots(c) r = 3.00003015641971 2.99998492179015 + 2.61163002529051e-005i 2.99998492179015 2....
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This note was uploaded on 11/07/2011 for the course EGM 3344 taught by Professor Raphaelhaftka during the Spring '09 term at University of Florida.

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Lecture 18 Polynom - Root of Polynomials • Bisection false-position Newton-Raphson secant methods cannot be easily used to determine all roots of

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