Page 190 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 8 ROOTS OF EQUATIONS I67

Examination of the process of synthetic division reveals that there is a
correspondence between the coefficients of the two preceding forms of the
polynomial :
bn = an (8-1 1)

bn-1 = an-1 -pbn (8-12)

bn-2 = ~ - Fpbn-1 - qbn (8-13)
2

bn-k = Un-k -pbn-k+l - qb,+k+2 (k = 2, 3, . . . , n-1) (8-14)
R = al -pb2 - qb3 (8-15)

S = a0 - qb2 (8-16)
If the polynomial has been normalized so that an = 1, then the equations are
simplified somewhat.
The trial quadratic will be a factor of the polynomial if the remainder is zero,
that is, R = S = 0. Since R and S are functions ofp and q:

(8-17)
s = S@, 4) (8-18)

we need to find the values of p and q that make R and S equal to zero. We will
do this by means of a two-dimensional analog of the Newton-Raphson method.
If p* and q* are the desired solution, then the solution can be expressed as a
Taylor series

(8-19)

and

(8-20)

where (8-2 18)

and (8-22)
ignoring terms other that the first, since as we approach the correct answer the
higher terms become negligible. The preceding result in two equations in two
unknowns, which can be solved to obtain