Roots of Equations - Numerical Methods in Engineering with MATLAB - page 174

Deflation of Polynomials

Afteraroot r of P n ( x ) = 0has been computed, it is desirable to factor the polynomial

as follows:

P n ( x ) = ( x − r ) P n − 1 ( x ) (4.12)

This procedure, known as deflation or synthetic division , involves nothing morethan
computing the coefficients of P n − 1 ( x ) . Since the remaining zeros of P n ( x ) are also the

zeros of P n − 1 ( x ) , the root-finding procedurecan nowbe applied to P n − 1 ( x ) rather than

Deflation thus makes it progressively easier to find successive roots, because
the degree of the polynomial is reduced every time aroot isfound. Moreover, by
eliminating the roots thathave already been found, the chances of computing the
same root morethan once areeliminated.
If we let

P n ( x ) .

b 1 x n − 1 b 2 x n − 2

P n − 1 ( x ) = + +···+ b n − 1 x + b n

thenEq. (4.12) becomes

a 1 x n a 2 x n − 1

+ +···+ +

a n x a n + 1

r )( b 1 x n − 1 b 2 x n − 2 = ( x − + +···+ b n − 1 x + b n )

Equating the coefficients of like powersof x , weobtain

b 1 = a 1 b 2 = a 2 + rb 1 ··· b n = a n + rb n − 1 (4.13)

which leadsto Horner'sdeflation algorithm :

b(1) = a(1);
fori=2:n
b(i) = a(i) + r*b(i-1);

end

Laguerre's Method

Laguerre'sformulas are not easilyderived forageneral polynomial P n ( x ) However, the

derivationis greatly simplifiedif weconsider the specialcase where the polynomial
has a zero at x

.

= r and ( n − 1)zeros at x = q . If the zeros wereknown,this polynomial

can be writtenas

q ) n − 1 P n ( x ) = ( x − r )( x − (a)

Our problemis now this: given the polynomial in Eq. (a) in the form

a 1 x n a 2 x n − 1 P n ( x ) = + +···+ a n x + a n + 1