Page 192 - Excel for Scientists and Engineers: Numerical Methods

CHAPTER 8 ROOTS OF EQUATIONS 169

Equations 8-25 to 8-29 can be written in the form
cn =o (8-30)

cn-1 = b,, - PCn (8-3 1)

(8-32)

cn-k = bn-k - Pcn-k+l - qcn-k+2 (8-33)
co = -4c2 (8-34)

The simultaneous equations to be solved are
+
C~AP C3Aq = -b, (8-35)
C~AP + CZAq = -bo (8-36)

Using Cramer's rule, we obtain

(8-3 7)

(8-38)

The procedure for calculating the roots therefore is as follows: with initial
estimates of p and q (zero or one can be used), calculate the values of b, and c,.
Use these values to calculate Ap and Aq, and correct the initial values. Continue
until convergence is reached. Obtain the two roots by use of the quadratic
formula. Use the result of synthetic division of the polynomial as the new
polynomial, and repeat the process. Continue until the polynomial is of order
one or zero.
The VBA code is shown in Figure 8-28. The portion of the code that
performs the Bairstow calculation is based on code found in Shoup, T. E.,
Numerical Methods for the Personal Computer, Prentice-Hall, 1983.