Chapter-07 - Roots of Polynomials Chapter 7 The roots of...

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Roots of Polynomials Chapter 7 The roots of polynomials such as n n o n x a x a x a a x f + + + + = K 2 2 1 ) ( Follow these rules: 1. For an n th order equation, there are n real or complex roots. 2. If n is odd, there is at least one real root. 3. If complex root exist in conjugate pairs (that is, λ + μ i and - i ), where i=sqrt(-1).
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Conventional Methods The efficacy of bracketing and open methods depends on whether the problem being solved involves complex roots. If only real roots exist, these methods could be used. However, Finding good initial guesses complicates both the open and bracketing methods, also the open methods could be susceptible to divergence. Special methods have been developed to find the real and complex roots of polynomials – Müller and Bairstow methods.
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Müller Method Müller’s method obtains a root estimate by projecting a parabola to the x axis through three function values. Insert Figure 7.3 The method consists of deriving the coefficients of parabola that goes through the three points: 1. Write the equation in a convenient form: c x x b x x a x f + - + - = 29 ( 29 ( 29 ( 2 2 2 2
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Fig 7.3
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1. The parabola should intersect the three points [x o , f(x o )], [x 1 , f(x 1 )], [x 2 , f(x 2 )]. The coefficients of the polynomial can be estimated by substituting three
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Chapter-07 - Roots of Polynomials Chapter 7 The roots of...

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