Newton's Method of Approximating Real Roots of P(x)=0,

FOCAL8-109

Where the Degree of P(x) is 4 or Less

Jeff Gelpey

Submitted by: Brother John F. O'Connell, St. John's

Preparatory School, Danvers, Massachusetts

With Newton's Method the first approximation is usually

found by -f(0)/f(0). In the program listing this is -G/M.

However, if f(0) = 0, this cannot be used, so the program al-

lows the user to make his own approximation and carry on

from there. Since no provision is made for complex roots the

user should examine the polynomial to determine the possible

number of real roots (by Descartes Rule of Signs). He should

also try to locate possible roots between integers so that he can

arrive at the approximations more quickly. The synthetic

substitution option allows this to be done. Using the synthetic

substitution, the upper and lower bounds can be established,

while the changes in sign of F(x) will locate possible real roots

between consecutive integers. When the output BETTER APP.

repeats its numerical value, a root has been found correct to

three decimal places. The other real roots can be found in a

similar manner.

Minimum Hardware:
4K PDP-8/S

Source Language:
FOCAL-69

Catalog: July 1973