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32 COMPUTE! September/October, 198O. Issue 6
Solving
Equations With
A Computer
Marvin L De Jong
Department of Mathematics-Physics
The School of the Ozarks
Pt. Lookout, MO 65726
INTRODUCTION
There is a large body of knowledge, known as
"Numerical Analysis," that is used to solve pro
blems that would be either loo difficult or too ineffi
cient to solve with either hand calculations or an
electronic calculator. The problems generally attack
ed with numerical analysis techniques require either
a computer or a programmable electronic calculator.
The purpose of this article is to show how a few
techniques from numerical analysis can be used to
solve difficult equations. These techniques do not re
quire any extraordinary mathematical skills; a first
course in high-school algebra will suffice.
To begin, we will assume you can solve equations
of the type,
2x + 5 = -3 (1)
This type of equation is solved using the rules:
RULE (1) The same number (or algebraic expression)
can be added or subtracted from both sides of an
equation.
RULE (2) Both sides of an equation can be multi
plied or divided by any non-zero number (or algebraic
expression).
Thus, in Equation (1), we would first subtract
five from both sides of the equation and next both
sides of the equation would be divided by two, giving
x = -4 as the answer. Any equation of the form
Ax + B = C (2)
has a solution x = (C - B)/A, which is very easy
to program in BASIC or FORTRAN. The program
in Listing 1 does this.Listing t program to solve a
linear equation.
10 INPUT A. B, C
20 X = (C - B)/A
30 PRINT X
40 END
Clearly in this case the problem could just as well
have been done with pencil and paper. We are inter
ested in more difficult problems, but RULES (1)
and (2) above describe how equations may be modified
to get the unknown "x" by itself on one side of the
equation, and we will need these rules in what follows.
To these rules we add a third, namely
RULE (3) In certain cases both sides of an
equation may be operated on by the same function
and the results arc still equal.
To illustrate, if x^ = 9, then we may operate on
both sides of this equation with the square root func
tion (SQR in BASIC) to get x = 3. Note that this
technique misses the answer x = -3, but it illustrated
the fact that taking the square root of both sides of
an equation (usually) yields a valid result. Likewise,
one can lake the logarithm (LOG in BASIC) of both
sides of an equation provided we arc dealing with
positive numbers, and we can take the exponential
function (EXP in BASIC) of both sides of an equa
tion, using RULE (3).
The type of equations that are of interest in the
present context can best be illustrated by some ex
amples. How would you solve for X in the following
equations:
x^ = cos(x) (3)
ex - 4x = 0 (4)
log(x) - cos(x) = 0 (5)
These so-called non-linear equations cannot be
solved by a simple application of the rules given so
far. In fact, you may be disappointed to know that
no single technique will solve all possible non-linear
equations. Many people like mathematics because it
seems to follow simple, hard-and-fast rules that lead
to answers that are either right or wrong. On the
contrary, mathematics requires creativity and the
ability to view a problem from many angles. Further
more, more often than not, the answers are only ap
proximately correct rather than absolutely correct. In
any case, let us examine two techniques that may be
used to solve these difficult looking equations.
The Method Of Successive Substitutions
The method of successive substitutions is one of the
Simplest techniques used to solve these equations. It
comes with no guarantee that it will work, but because
it is simple it is frequently worth trying.
The first step is to take the equation to be solved
and using the three rules given in the Introduction.
put the equation in a form with x on the left-hand
side and everything else on the right-hand side of the
equation. For example, the equation x^ = cos(x),
Equation (3) above, becomes either x = (cos(x))/x or
x = V eos (x). It is typical to find several possible
forms. This step is usually described in textbooks by
telling you to put your equation in the form
x = f(x) (6)
In our example, f(x) is either cos(x)/x or \Jc'os(x)
depending on whether we arc using x = (cos(x))/x or
x = Ncos(x). In any case, the equation is modified
so that x is all by itself on one side of the equation
and everything else is on the other side.
The second step is to guess at a value of x that
will satisfy the equation. This is an important step