Roots of polynomials

Roots of polynomials

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  Maria Fernanda VergaraMendoza Petroleum Engineering UIS-COLOMBIA
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  In this chapter, you will learn some methods to find the roots of polynomial equations of the general form: Where n= the order of the polynomial; a= constant coefficients. RULES: For an n th-order equation, there are n real or complex roots. If n is odd, there is at least one real root The complex roots exsist in conjugate pairs (a+bi and a-bi), i=√(-1)
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  The Muller’s method,is like the secant method, just that this one projects a parabola through three points unlike secant method, who projects a straight line. This method consists of deriving the coefficients of the parabola that goes through the three points.
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  Write the parabolicequation in this form: The coefficients a, b, and c can be evaluated by substituting each of the three points to give:
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  Two of theterms of are zero, it can be solved for c=f(x i+1 ). Using algebraic manipulations, we solve the remaining coefficients:
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  These can besubstituted to give: The results can be summarized as Once you know the approximate coefficients you have to find the approximated root using the quadratic equation :
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  The error canbe calculated as: There is a problem with equation, this equation yields two roots, in this method the sign is chosen with this strategies: 1. If only real roots are being located, we choose the two original points that are nearest the new root estimate, x i+2 . If both real and complex roots are being evaluated, a sequential approach is employed. That means: x i , x i+1 , x i+2 take the place of x i-1 , x i , x i+1
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  If you haveas initial values respectively, find the root of the equation: FIRST: Evalue the equation in its initial values
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  SECOND: This valuesare used to calculate: THIRD: Find the a, b, c coefficients:
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  The error is:This is a huge error, so its necesary to do other iterations: Repeat the calculations and get a low percent of error: Iteration Xr Ea% 0 5 -- 1 3.976487 25.74 2 4.00105 0.6139 3 4 0.0262 4 4 0.0000119
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  Is an iterativeapproach related loosely to both the Muller and Newton Raphson methods. It is based on the idea of synthetic division of the given polynomial by a quadratic function and can be used to find all the roots of a polynomial. The idea is to do a synthetic division of the polynomial P n (x) by the quadratic factor (x 2 - rx - s).
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  The synthetic divisioncan be extended to quadratic factors: When you multiply and match factors have:
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  The idea isto find values of r and s, making b 1 and b 0 zero. The method works taking an initial approach (r 0, s 0 ) and getting better approaches (r k , s k ), this is an iterative procedure, the process ends when the residue of dividing the polynomial by (x 2 - r k x - s k ) its zero. B 1 =f(s, r) B 0 =g(s, r)
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  Because both bo and b 1 are functions of both r and s, they can be expanded using a Taylor series: The changes, Δr and Δs, can be estimated by setting the expansion equal to zero:
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  “ If thepartial derivatives of the b’s can be determined, these are a system of two equations that can be solved simultaneously for the two unknowns, Δr and Δs.” According to Bairstow, the partial derivatives can be obtained by a synthetic division of the b’s.
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  Then the systemof equations can be written as:
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  APPROXIMATED ERROR Whenboth of these error estimates fall below a stopping criterion, the values of the roots can be determined by:
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  Employ Bairstow’s methodto determine the roots of the polynomial Use initial guesses of r=s=-1 and iterate to a level of tolerance of 1% SOLUTION: b 5 =1 b 4 =-4.5 b 3 =6.25 b 2 =0.375 b 1 =-10.5 b 0 =11.375 c 5 =1 c 4 =-5.5 c 3 =10.75 c 2 =-4.875 c 1 =-16.375 Thus, the simultaneous equations to solve Δr and Δs are :
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  Which can besolved for Δr=0.3558 and Δs=1.1381. r=-0.6442 S=0.1381 And the approximate errors are: The computation can be continued with the result that after four iterations the metod converges on velues of r=-0.5 and s=0.5
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  CHAPRA, Steven C.“Numerical methods for engineers”, Fifth edition. Mc Graw Hill. CARRILLO, Eduardo. “Raices de polinomios”. PPT.