STEP 1

ERRORS 1

Sources of error

The main sources of error in obtaining numerical solutions to mathematical problems are:

1. the model - its construction usually involves simplifications and omissions;
   
2. the data - there may be errors in measuring or estimating values;
   
3. the numerical method - generally based on some approximation;
   
4. the representation of numbers, for example, p cannot be represented exactly by a finite number of digits;
   
5. The arithmetic - frequently errors are introduced in carrying out operations such as addition (+) and multiplication (*).

Onee can pass responsibility for 1. onto the applied mathematician, but the others are not so easily dismissed. Thus, if the errors in the data are known to lie within certain bounds, one should be able to estimate the consequential errors in the results. Similarly, given the characteristics of a computer, one should he able to account for the effects of 4. and 5. As for 3., when a numerical method is devised, it is customary to investigate its error properties.

1. Example

   

   In order to illustrate the ways in which the above errors arise,1et us take the example of the simple pendulum (Fig. 1). After various physical assumptions, including air resistance and friction at the pivot to be negligible, one obtains the simple (non-linear) differential equation

   

   In introductory courses of Mechanics, the customary next step is to use the approximation

   

   to arrive at the even simpler linear differential equation

   

   This has the analytical solution

   

   where A and B are suitable constants.

   One can then deduce that the period of the simple pendulum (i.e., the smallest positive value of T such that

   

   is

   

   Up to this point, only errors of Type 1 have been encountered; other errors are introduced when one tries to obtain a numerical value for T in a particular case. Thus, both 5. and 7. will be subject to measurement errors; p must be represented as a finite decimal number, the square root must be computed (usually by an iterative process) after dividing l by g (which may involve a rounding error), and finally the square root must be multiplied by 2p.

   Checkpoint

   1. What sources of error are of concern to the numerical analyst?
   2. Which types of error depend upon the computer used?

   EXERCISES

   When carrying out the following calculations, note all the points at which errors of one kind or another arise.

   1. Calculate the period of a simple pendulum of length 75 cm, given that g is 981 cm/s².
   2. The rate of flow of a liquid through a circular hole of diameter d is given by

      

      where C is a coefficient of discharge and H is the head of liquid causing the flow. Calculate R for a head of 650 cm, given that d = 15 cm and the coefficient of discharge C is estimated to be 0.028.

      Answers

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