Numerical Methods - PART 2 (Chapter 7)

7/24/2019 Numerical Methods - PART 2 (Chapter 7)

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ROOTS OF EQUATIONS


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ROOTS OF POLYNOMIALS


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In this chapter, we consider methods to findthe roots of polynomial equations of thegeneral form:

where n= order of the polynomial and the asare constant coefficients

The roots of such polynomials follow theserules: For an nthorder equation, there are nreal or

complex roots (not necessarily distinct) If nis odd, there is at least one real root. If complex roots exist, they exist in conjugate pairs.


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Mullers method is a generalization of the secant

method, in the sense that it does not require thederivative of the function. It is an iterative method that requires three

starting points (p0, f (p0)), (p1, f (p1)), and (p2, f(p2)).

A parabola is constructed that passes throughthe three points; then the quadratic formula isused to find a root of the quadratic for the nextapproximation.

It has been proved that near a simple root

Mullers method converges faster than the secantmethod and almost as fast as Newtons method. The method can be used to find real or complex

zeros of a function and can be programmed touse complex arithmetic.


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acbb

c

xx4

2

223

In Mullers method,the sign is chosen toagree with the sign ofb

01

01

hha

11 ahb 2xfc

010 xxh

121 xxh

01

01

0

xx

xfxf

12

12

1

xx

xfxf


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Example:

Use Mullers method with guesses of x0=

4.5, x1=5.5 and x3=5 to determine a root ofthe equation f(x)=x313x-12

h0= 1 0= 62.25

h1=-.5 1= 69.75

f(x0) = 20.625

f(x1) = 82.875

f(x2) = 48


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x3= 3.976487

New iteration: x0= 5.5x1= 5x2 = 3.976487

n

r

1 5.000000

2 3.976487

3 4.001050

4 4.000000

5 4.000000


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Bairstowsmethod is an iterative approachrelated loosely to both the Muller andNewton-Raphson methods

Approach: Guess a value for the root at x= t

Divide the polynomial by the factor x t

Determine whether there is a remainder

If not, the guess was perfect and the root is equal to t

If there is a remainder, the guess can be systematicallyand the procedure repeated until the remainderdisappears and a root is located.


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To permit the evaluation of complexroots, Bairstowsmethod divides thepolynomial by a quadratic factor

If this is done, the result is a newpolynomial = 3 3

with a remainder =


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Adjust rand sso that the divisionyields a zero remainder

sss

rrr

i

i

11

11


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Identify the acoefficients Determine the bcoefficients

= = = + + = 2 0

Determine the ccoefficients

= = = + + = 2 1

Solve the system: 3 =

Determine the next rand s


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By: synthetic division:

na 1na 2na 3na 2a 1a 0a

1r nbr1 11 nbr 21 nbr

1s nbs1 11 nbs

nb 1nb 2nb 3nb 2b 1b 0b

ncr1 11 ncr

ncs1

nc 1nc 2nc .. 2c 1c 0c

Example: Determine the roots of the polynomialf(x) = x5-3.5x4+2.125x2-3.875x+1.25 with initialguesses r=s=-1.00000

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Numerical Methods - PART 2 (Chapter 7)