B. S. Grewal - Numerical Methods in Engineering and Science: C, C++, and MATLAB

Numerical
Methods
in Engineering
and Science
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Methods
in Engineering
and Science C, C++, and MATLAB B. S.

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Original Title and Copyright: Numerical Methods in Engineering and Science 3/E.
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• Introduction
• Accuracy of numbers
• Errors
• Useful rules for estimating errors
• Error propagation
• Error in the approximation of a function
• Error in a series approximation
• Order of approximation
• Growth of error
• Objective type of questions

The input information is rarely exact since it comes from some measurement or the other and the method also introduces further error. As such, the error in the final result may be due to an error in the initial data or in the method or both. Our effort will be to minimize these errors, so as to get the best possible results. We therefore begin by explaining various kinds of approximations and errors which may occur in a problem and derive some results on error propagation in numerical calculations.

1. Approximate numbers. There are two types of numbers: exact and approximate. etc. etc.
   
   

   But there are numbers such as 4/3 ( = 1.33333...), (= 1.414213...) and ( = 3.141592...) which cannot be expressed by a finite number of digits. These may be approximated by numbers 1.3333, 1.4142 and 3.1416, respectively. Such numbers which represent the given numbers to a certain degree of accuracy are called approximate numbers.

2. Significant figures. The digits used to express a number are called significant digits (figures). Thus each of the numbers 7845, 3.589, and 0.4758 contains four significant figures while the numbers 0.00386, 0.000587, and 0.0000296 contain only three significant figures since zeros only help to fix the position of the decimal point. Similarly the numbers 45000 and 7300.00 have two significant figures only.
3. Rounding off. There are numbers with large number of digits, e.g., 22/7 = 3.142857143. In practice, it is desirable to limit such numbers to a manageable number of digits such as 3.14 or 3.143.