1550078481-Ordinary_Differential_Equations__Roberts_

The Initial Value Problem y' = J(x, y); y(c) = d 75

a finite set of rational numbers can be represented exactly and they

are not equ ally spaced throughout the range ofrepresentable values. When at-

tempting to p erform an arithmetic operation whose result would be a number

whose magnitude is larger than the largest number representable on the com-

puter overflow occurs and most computers terminate execution immediately.

Likewise, underflow occurs and execution is terminated when attempting to

perform a n arithmetic operation whose result would be a number whose mag-

nitude is less than the smallest nonzero number representable. In addition,

contrary to your experience with real numbers, the floating-point operations

of addition and multiplication are not commutative and the distributive law

fails to hold.

When a number which the computer cannot represent exactly is entered

into the computer or is calculated within the computer , the computer selects

the nearest number in its representable set by rounding-off or chopping-off

the number. The error created by rounding or chopping after the final digit

is called the round-off error. For example, suppose we have a calculator

which uses base 10 and has four digits accuracy. If our calculator rounds,

it represents 2/3 as .6667; whereas, if our calculator chops, it represents 2/3

as .6666. Now suppose that our calculator actually rounds and we use it to

compute

x^2 - -^4

--2-^9

x--

3

for x = .6666. Calculating, we get

(.6666)^2 - .4444

. 6666 - .6667

.4444 - .4444 = 0.

-.0001

Factoring and cancelling before using our calculator , we find

x^2 - -^4

--2-^9

x--

3

2 2

(x-3)(x+3) 2

---~ 2 --= x + 3·

x-3

Then using our calculator to evaluate this last expression with x = .6666, we
obtain .6666 + .6667 = 1.333. This example illustrates that round-off error
can create some very serious computing problems. Round-off error depends
upon the computer and the coding of the algorithm. In general, round-off
error is difficult to analyze and, indeed, somewhat unpredictable. In order to
minimize round-off error, it is a good idea to use high precision arithmetic
and to reduce the number of computations as much as possible.
Throughout this section, we will let (x) denote the unique, explicit solution
of the initial value problem (1) y' = j(x, y); y(xo) = Yo on the interval I
which contains x 0. By the fundamental existence and uniqueness theorem

and the continuation theorem requiring that f and f y be continuous in some