U n i v e r s i t y

                       S o f t w a r e



                         POLYNOMIALS











                          SINCLAIR

                    ZX SPECTRUM (16K-48K)

_____________________________________________________________



                  Programs constituting the



            U n i v e r s i t y   S o f t w a r e



 LIBRARY OF ADVANCED MATH/STAT/ECON have been carefully

 prepared by a team of PhD students from the University

 of London. Although the programs are designed to handle

 the most complex problems, instructions printed on the

 covers intend to introduce the non-specialist to the

 fundamentals of the theory.

 All programs start executing automatically once they are

 loaded and all that is required is to follow the instruc-

 tions displayed on the screen. But, the used is not denied

 the right of having access to the program listings. REM

 statements give the user an idea about how they work. If

 shorter execution times or more memory space is required,

 they may be excluded from programs.







© Copyright 1983



            U n i v e r s i t y   S o f t w a r e



             29 St. Peters Street, London N1 8JP



No part of this recording, its program listings or the

contents of this cassette inlay can be reproduced without

written permission. Although due care has been taken in

preparing these programs, the publisher is not responsible

for any errors or liable for any damage arising from their

use.

_____________________________________________________________



Side A : REAL ROOTS OF POLYNOMIALS



A polynomial is a function of the form :

             n        n-1          2

   F(x) = c x  + c   x    + . . c x + c x + c

           n      n-1            2     1     0



where cn......c0 are called the coefficients and n the

degree of polynomial. The values of x which satisfy F(x)=0

are called the roots of the polynomial.



To calculate the roots the present program employs three

different methods.



(i) Quadratic Equations are in fact second degree poly-

nomials. If the degree of a polynomial is entered as 2, the

roots are calculated by the formula :

                             ____________

                          _ / 2

                      -c  +/c   - 4c c

                      __1_V__1______2_0___

                               2c

                                 0



(ii) Newton-Raphson Method is applied to higher degree (3 or

more) polynomials. The basic step of the method is as

follows : starting with an arbitrarily chosen initial value

of x (say x0), another x (say x1) is produced which is closer

to any one of the roots of the function F(x). x1 is produced

by the formula :



                                F(x )

                     x  = x  - ____0__

                      1    0    F'(x )

                                    0



where F(x0) is the value of the first derivative of F(x) for

x=x0. Of course, we assume that F(x) is differentiable. This

step is repeated until the difference between the new x

and the old x is less than a desired degree of accuracy. In

this program this accuracy level is set as 10^-8. If F(x) does

not have any real roots, iteration is carried out 100 times.

The option of iterating another 100 times is available.



If one root is found, other roots can be searched for by

entering different initial values. Although Newton-Raphson

method of approximation works much faster than the half-

interval search method, it is difficult to guess which initial

value approximates to which root.



(iii) Half-interval Search Method looks for a change of sign

of F(x) within a given interval. First the interval is divided

into 10 equal parts and the sign change is searched for

these increments. If a sign change is found the limits of

the current increment is defined as a new interval and in

turn divided into 10 increments. This procedure is repeated

until the increment is less than 10^-8. If no sign change is

found in first 10 iterations, the original interval is divided

into 100 increments and search is repeated. If still no sign

change is found the absence of real roots within given

interval is reported.



Side B : PLOT OF POLYNOMIALS



This program plots the polynomials between given limits,

prints the extremum values and the lower and upper limits

on the axes. If a root is found x axis adjusts its location

according to the values of polynomial and the approximate

value of the root is printed at the intersection point.



It is advisable to use this program first to have an idea

about the behavious of the polynomial and then use Side A

to determine the precise values of roots.

_____________________________________________________________



 U S                     POLYNOMIALS

_____________________________________________________________



 UNIVERSITY

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Library of

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POLYNOMIALS



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