Ch1 - 1 NUMERICAL ANALYSIS AAOC C341 Second Semester...

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Unformatted text preview: 1 NUMERICAL ANALYSIS AAOC C341 Second Semester 2008-2009 BITS-Pilani, Goa Campus 2 Numerical Analysis ¡ Introduction: Numerical Analysis is very important for mathematicians, Engineers and the related peoples because it provides the desired accurate solution to mathematical, engineering or any related problem when there is no analytical solution. These days the use of computers make it very easy to use the very complicated numerical techniques. 3 CHAPTER 1 Finite Digit Arithmetic and Errors ¡ Floating point Arithmetic ¡ Propagated Error ¡ Generated Error ¡ Error in Evaluation of a Function 4 Floating point Arithmetic Calculations are usually carried out in floating point arithmetic by computers. An n-digit floating point number in base has the form , where the number is mantissa with base number and e is an integer called the exponent . The mantissa lies between [.1, 1) or we can write [.1/ β , 1), in case of normal floating point form. β e n d d d x β β ) . ( 2 1 … ± = n d d d … 2 1 . β 5 Floating point Arithmetic continued … The number is called normalized floating point number if If d 1 and d n are non zero then we say x has n significant digits . There are two ways of translating a real number into n- β digit floating point number fl(x) . Where n is the number of significant digits and β is the base. One is rounding and the other is chopping . e n d d d x β β ) . ( 2 1 … ± = . 1 ≠ d 6 Floating point Arithmetic continued … Rounding off and Chopping Suppose we want to round off the following number up to k decimal places where k<n. Then the new number will be given by Where if d k+1 < 5, then d k * is not changed. If then d k * is increased by one. Chopping of the number up to k decimal places is the number n n d d d a a a x … … 2 1 2 1 . = * 2 1 2 1 . * k n d d d a a a x … … = 9 5 1 ≤ ≤ + k d k n d d d a a a x … … 2 1 2 1 . * = 7 Floating point Arithmetic continued … In rounding fl(x) is chosen as the normalized floating point number nearest x . In chopping fl(x) is chosen as the normalized floating point number between x and . fl(x) may not be equal to x i.e. Therefore error in the value of x , denoted by e x and is given by The absolute error in the value of x is . . ) ( ≠ − x fl x x e ). ( x fl x e x − = 8 Floating point Arithmetic continued … Relative error: e x is not a proper measure of the error because it depends on the value of x , therefore we consider the relative error. The relative error in x , denoted by r x is which is considered as the best measure of the error in x . , ) ( x x fl x x e r x x − = = 9 Floating point Arithmetic continued … Theorem: Let fl(x) be n- β floating point representation of a real number x then 10) decimal if 2, binary if base, the is ( used. is chopping if , r (ii) used is rounding if , 2 1 | | ) ( 1 x 1 = = < ≤ < + − + − β β β β β n n x r i 10 Floating point Arithmetic continued … Proof of the Theorem: (i) Let . When rounding is used we get...
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This note was uploaded on 02/23/2011 for the course MATH 122 taught by Professor Ritadubey during the Spring '11 term at Amity University.

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Ch1 - 1 NUMERICAL ANALYSIS AAOC C341 Second Semester...

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