It has been noted that a number is to be
represented by a finite number of digits, and hence often by an
approximation. It is to be expected that the result of any
arithmetic procedure (any **algorithm**) involving a set of numbers will have an** ****implicit
error** relating to the error of
the original numbers. One says that the initial errors **propagate** through the computation. In addition, errors may he **generated** at each step in an algorithm,
and one speaks of the **total
cumulative error **at any step as
the ** accumulated error**.

Since one ains to produce results within some
chosen limit of error, it is useful to consider **error propagation**. Roughly speaking, based on experience, the **propagated error** depends on the **mathematical
algorithm** chosen, whereas the g**enerated error** is more sensitive to the actual ordering of the
computational steps. It is possible to be more precise, as
described below.

The absolute error is the absolute difference
between the exact number *x* and the approximate number *X*,
i.e.,

A number correct to *n* decimal places
has

We expect that the absolute error involved in any approximate number is no more than five units at the first neglected digit.

This error is the ratio of the absolute error to the absolute exact number, i.e.,

(Note that the upper bound follows from the triangle inequality; thus

A decimal number correct to *n*
significant digits has

Consider two numbers

Under the operations of **addition
or subtraction**,

The magnitude of the propagated error is therefore not more than the sum of the initial absolute enors; of course, it may be zero.

Under the operation of **multiplication**:

The maximum relative error propagated is approximately the sum of the initial relative errors. The same result is obtained when the operation is division.

Often (for example, in a computer) an operation
× is approximated by an operation $\times *$, say.
Consequently, *x×y* is represented by *x*^{*}*×y*^{*}.
Indeed, one has

so that the accumulated enor does not exceed the sum of the propagated and generated errors. Examples may be found in Step 4.

Evaluate (as accurately as possible) the expressions:

- 3.45+4.87-5.16

- 3.55 x 2.73

There are two methods which the student may
consider: The first is to invoke the concepts of absolute and
relative error as defined above. Thus, the result for **1.**
is 3.16 +/- 0.015, since the **maximum
absolute error****
**is 0.005 + 0.005 + 0.005 = 0.015.
We conclude that the answer is 3 (to 1*S* ), for the number
certainly lies between 3.145 and 3.175. In **2.**, the product
9.6915 is subject to the **maximum relative error****:**

whence the maximum (absolute) error ~ (2.73 + 3.55) x 0.005 ~ 0.03, so that the answer is 9.7.

A second approach is **interval
arithmetic**. Thus, the approximate
number 3.45 represents a number in the interval (3.445, 3.455),
etc. Consequently, the result for **1.** lies in the interval
bounded below by

and above by

Similarly, in **2.**, the result lies in the
interval bounded below by

and above by

,

whence once again the approximate numbers 3 and
9.7 correctly represent the respective results to **1.** and **2.**.

1. What distinguishes **propagated and generated errors**?

2. How to determine the **propagated error** for the operations addition (subtraction) and
multiplication (division)?

Evaluate the following operations as accurately as possible, assuming all values to the number of digits given:

- 8.24 + 5.33.

- 124.53 - 124.52.

- 4.27 x 3.13.

- 9.48 x 0.513 - 6.72.

- 0.25 x 2.84/0.64.

1.73 - 2.16 + 0.08 + 1.00 - 2.23 - 0.97 + 3.02.