### Error propagation and generation

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 generated error is more sensitive to the actual ordering of the computational steps. It is possible to be more precise, as described below.

### Absolute error

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.

### Relative error

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

### Error propagation

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.

### Error generation

Often (for example, in a computer) an operation × is approximated by an operation $×*$, 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.

### Example

Evaluate (as accurately as possible) the expressions:

1. 3.45+4.87-5.16
2. 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 1S ), 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..

### Checkpoint

1. What distinguishes propagated and generated errors?

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

### EXERCISES

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

1. 8.24 + 5.33.
2. 124.53 - 124.52.
3. 4.27 x 3.13.
4. 9.48 x 0.513 - 6.72.
5. 0.25 x 2.84/0.64.
6. 1.73 - 2.16 + 0.08 + 1.00 - 2.23 - 0.97 + 3.02.

Answers