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CHAPTER I NUMERICAL ANALYSIS AND ERRORS 0 1.1, Tatroduction. 226 Numerical analysis is concerned with methods which give 128 aumerical solution to mathematical problems by arithmatic opera- 233 | tions on numbers. In Applied Mathematics, Theoretical Physics and 235 . Engineering, the ultimate problem is to compute numerical results 237 . Using Certain data. The school arithmetic is inadequate for handling 239 ) such Problems, The methods sometimes involve development of 4] algorithms, 4 s¢quence of steps, to solve a preblem. Thetools that 80 \ are used in the development of numerical analysis are those of exact classical mathematica] analysis 43 With the advent of modern high speed digital computers the ++ | numerical analysis of today is a very different discipline. In fact the +8 | development of the subject of numerical analysis has received an 49 enormous impetus in the last three decades, 49 Not only the methods employed in numerical analysis are some- . times approximate but the data are also inaccurate. Hence in most 59 of the cases the result of the numerical computation will have some. errors. Bzfore dealing with the methods we first discuss the sources : | of error in numerical solutions. ; 1.2. Sources of Error. , The errors, in general, can be basically of two types. The error which is inherent io a numerical method itself is called truncation error, The other type of error isthe computational error which arises during arithmetic computation due to the finite representation of numbers. Most numbers have infinite decimal representation but for computational purpose they will be represented bya finite number of digits. < ve The trunc.tion error arises due to the replacement of an : nite (©g summation or integration) or infinitesimal (¢ g. differentiation) Process by a finite one, for example. i. (i) in computation of a function by using the ir Taylor series expansion. a ti) in appr Oximation of an integral by st few terms of a finite sum of function 
