Significant Digit Error Control

• [Play Video]  4½ minutes: Algebraic View of Limits. Example involving sums and quotients.
• [Play Video]  5½ minutes: Limits and Error Control for Linear Expressions
• [Play Video]  2¾ minutes: Error Control to N decimal Places, say 5 or 10. 
• [Play Video]  3¼ minutes:  Limits as Error Control for an unlimited number of decimal places. 

The question of relative error is related to the unrestricted control of the number of significant digits in computations: For every n is there an m such that

|x-a| |a| < 1 2 1 10m     implies     |f(x)-f(a)| |f(a)| < 1 2 1 10n     (?)

This question can only be answered when division by zero is avoided. In numerical calculations, circumstances may suggest what is more important (more precisely what is feasible): absolute error control or relative error control.

Various error control (or continuity) questions can be based on different measures of closeness for x and f(x), that is, different measures of closeness on the domain and range of a function f. For example, the question of relative error on the domain can also be posed as follows: for every n is there an m such that

|x-a| < 1 2 1 10m     implies     |f(x)-f(a)| |f(a)| < 1 2 1 10n     (?)

For addition and subtraction, absolute error control (the first type introduced in this chapter) is more appropriate than relative error or significant digit control. For multiplication and division, relative error and significant digit error control is more appropriate. When there is a mixture of addition or subtraction with multiplication or division, no simple advice can be offered. A course on numerical methods could discuss this topic further.