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Chapter 4 Roots of Polynomials

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Chapter 4

Roots of Polynomials

Objectives

- Understand the importance of finding polynomial roots in engineering applications
- Know the conventional method concept
- Know the Muller’s method concept
- Know the Bairstow’s method

Content

- Polynomials in engineering and science
- Conventional method
- Muller’s method
- Bairstow’s method
- Conclusions

General solutions of linear ODE

Solve for general solution

Change to characteristic equations:

The results can be :-

General solutions of linear ODE

- Problem :

- Follow these rules:
- For an nth order equation, there are n real or complex roots.
- If n is odd, there is at least one real root.
- If complex roots exist, they will be in conjugate pairs (that is, l+mi and l-mi), where i=sqrt(-1).

- Only real roots exist
However,

- Finding good initial guesses complicates both the open and bracketing methods, also the open methods could be susceptible to divergence.

- Real and complex roots of polynomials – Müller and Bairstow methods.

- Like Secant, Müller’s method obtains a root estimate by projecting a parabola to the x axis through three function values.

Secant method

(linear approximation)

Müller method

(Parabola or 2nd order

approximation)

Must use three points to approximate function

Müller methodology derivation

- Write the equation in a convenient form at point x2:

- We then have three eqns now (from x0, x1, and x2)

- Step III Reduce to two eqns

Right now u can solve for a and b from

When u know a, b, c you are ready to

estimate root from

- Step IV Here the new estimated root is

Two roots, but which one ?

- Error can be derived from

- Summary of algorithm

Start with 3 points [x0,f(x0)] [x1,f(x1)] and [x2,f(x2)]

Calculate a, b, and c from

- Summary of algorithm (cont’d)

Calculate new root from

Calculate error

Check whether

new xi-1 = old xi

- An iterative approach loosely related to both Müller and Newton Raphson methods.
- Based on dividing a polynomial by a factor x-t:

Start with

Dividing with x-t yields

and a remainder R=b0

The coefficients of polynomial are

- To permit the evaluation of complex roots, Bairstow’s method divides the polynomial by a quadratic factor x2-rx-s:

For the remainder to be zero, boand b1 must be zero. However, it is unlikely that our initial guesses at the values of r and s will lead to this result, so we do this…

Using a similar approach to Newton Raphson method, both bo and b1can be expanded as function of both r and s in Taylor series.

Neglect higher-order terms

We estimate Δr and Δs from

How can we find

these partial

derivatives ???

Partial derivatives can be obtained by a synthetic division of the b’s in a similar fashion the b’s themselves are derived:

where

then

At each step, the error can be estimated as

The roots can be determined from

- At this point three possibilities exist:
- The quotient is a third-order polynomial or greater. The previous values of r and s serve as initial guesses and Bairstow’s method is applied to the quotient to evaluate new r and s values.
- The quotient is quadratic. The remaining two roots are evaluated directly, using
- The quotient is a 1st order polynomial. The remaining single root can be evaluated simply as x=-s/r.