Polynomials

U n i v e r s i t y


S o f t w a r e





POLYNOMIALS

















SINCLAIR


ZX SPECTRUM (16K-48K)


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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.


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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.


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U S POLYNOMIALS


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UNIVERSITY


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Library of


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