Sources of Errors

Following are the broad sources of errors in numerical analysis:

(1) Input errors.The input information is rarely exact since it comes from the experiments and any experiment can give results of only limited accuracy. Moreover, the quantity used can be represented in a computer for only a limited number of digits.

(2) Algorithmic errors.If direct algorithms based on a finite sequence of operations are used, errors due to limited steps don’t amplify the existing errors, but if infinite algorithms are used, exact results are expected only after an infinite number of steps. As this cannot be done in practice, the algorithm has to be stopped after a finite number of steps and the results are not exact.

(3) Computational errors.Even when elementary operations such as multiplication and division are used, the number of digits increases greatly so that the results cannot be held fully in a register available in a given computer. In such cases, a certain number of digits must be discarded. Furthermore, the errors here accumulate one after another from operation to operation, changing during the process and producing new errors.

The following diagram gives a schematic sequence for solving a problem using a digital computer pointing out the sources of errors.

Our effort will be to minimize these errors so as to get the best possible results.

We begin by explaining the various kinds of errors and approximations that may occur in a problem and derive some results on error propagation in numerical calculations.

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