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
SOFTWARE
Library of
Advanced
Math/
Stat/
Econ:
MATRIX
OPERATIONS
POLYNOMIALS
INTEGRATION
REGRESSION
LINEAR
PROGRAMMING