Conditioning

Even when calculations are performed exactly, there are situations for which it can be difficult or impossible to obtain a correct answer. Chaotic systems are extremely sensitive to small changes in the start conditions. A well-known example of such a system is weather forecasting. The chaotic nature of the system imposes a limit on the period at which the weather can be forecasted with some degree of confidentiality (this period is about 15 days!).
The sensitivity of a system to changes in the start conditions is expressed in terms of the condition number, which is defined as

.

The condition number determines whether a system is sensitive to changes in the start conditions. A large condition number makes a system ill-conditioned and sensitive to small changes. A small condition number on the other hand defines a well-conditioned system where the system behaves well to small changes.

Condition number of linear functions

Assume that the approximated result for follows from using the approximated value of , for some linear function . (Whenever is not a linear function take the first order Taylor expansion.) It was shown in the previous section that for any number the computed or approximated value can be written as

.

Applying this to gives

,             (1)

where is the approximated value to . The assumption that follows from the approximated value for can be written as

.             (2)

Combining (1) and (2) leads to

  (3)

The fraction is a measure for the sensitivity of this system. If the derivative of exists in the following is true

.

Together with (3) this gives

,

as . This result gives us a measure for the relative sensitivity of for small variations in . By taking the absolute value the condition number of for is obtained:

.

Take for example the function . The relative derivative is given by . The condition number is therefore . This implies that for   the function is sensitive to small changes in its arguments. For on the other hand, small variations in the arguments will have only small effects.

REQUIRED
IV.2a Give the condition number for an argument for the function . Also discuss when the function is sensitive to small changes in its arguments.