Maria Fernanda Vergara Mendoza Petroleum Engineering UIS-COLOMBIA
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.
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)
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.
Write the parabolic equation in this form:
The coefficients a, b, and c can be evaluated by substituting each of the three points to give:
Two of the terms of are zero, it can be solved for c=f(x i+1 ).
Using algebraic manipulations, we solve the remaining coefficients:
These can be substituted 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 :
The error can be 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
If you have as initial values respectively, find the root of the equation: FIRST: Evalue the equation in its initial values
SECOND: This values are used to calculate: THIRD: Find the a, b, c coefficients:
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
Is an iterative approach 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).
The synthetic division can be extended to quadratic factors:
When you multiply and match factors have:
The idea is to 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)
Because both b o 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:
“ If the partial 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.
Then the system of equations can be written as:
When both of these error estimates fall below a stopping criterion, the values of the roots can be determined by:
Employ Bairstow’s method to determine the roots of the polynomial
Use initial guesses of r=s=-1 and iterate to a level of tolerance of 1%
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 :
Which can be solved for Δr=0.3558 and Δs=1.1381.
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
CHAPRA, Steven C. “Numerical methods for engineers”, Fifth edition. Mc Graw Hill.