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p103_2_S10

# p103_2_S10 - EE103 Lecture Notes Spring 2010 Prof S...

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EE103 Lecture Notes, Spring 2010, Prof S. Jacobsen Section 2 © Copyright Stephen E Jacobsen, 2010 ....................................................................................................... i SECTION 2: AN INTRODUCTION TO FLOATING POINT ARITHMETIC AND RATES OF CONVERGENCE ...................................................................................................................................... 16 Floating Point Arithmetic ...................................................................................................................... 16 A Bound on the Relative Error .......................................................................................................... 18 Significant Digits: ............................................................................................................................... 21 Sequences, Big “O”, Little “o”, Rates of Convergence ........................................................................ 22 Big O and little o notation: ................................................................................................................. 23 Rates of Convergence ......................................................................................................................... 25 Geometric Sums ...................................................................................................................................... 29 Back to Floating Point Arithmetic ........................................................................................................ 30 © Copyright Stephen E Jacobsen, 2010

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EE103 Lecture Notes, Spring 2010, Prof S. Jacobsen Section 2 16 SECTION 2: AN INTRODUCTION TO FLOATING POINT ARITHMETIC AND RATES OF CONVERGENCE Consider the number 12. Of course, we can think of the number as 10 10 12 1 10 2 10 (12)   Or, consider the number 12.625. We can think of this number as 1 2 3 12.625 1 10 2 10 6 10 2 10 6 10      02 10 10 (12.625) 10 (.12625) 10  On the other hand, we can also think of the number 12 as 32 1 0 0 2 12 1 2 1 2 0 2 0 2 (1100) 2      Similarly, we can think of 12.625 as 32 1 0 1 2 3 12.625 1 2 1 2 0 2 0 2 1 2 0 2 1 2          04 22 (1100.101) 2 (.1100101) 2 Or, we can think of 12.625 as 1 0 2 88 12.625 1 8 4 8 5 8 (14.5) 8 (.145) 8       Floating Point Arithmetic Definition : an n digit floating point number, in base , has the form 123 (. ) e n ddd d  1 where 0,1,2,. .., 1 i d  and is an integer. Unless the number in question is zero, we always write the number so that 1 0 d (i.e., we say the number is "normalized"). The term (. ) n d is called the mantissa; e , the exponent, is an integer. The number n is, of course, finite and is often called the "precision". The size of n depends upon the word length of the computer in question and, of course, this value varies considerably. The exponent, e , is an integer and is limited to a range, denoted by min max ee e 1 Storage of floating point numbers on a real computer is, of course, slightly different.
EE103 Lecture Notes, Spring 2010, Prof S. Jacobsen Section 2 17 For instance, the largest number that can be represented on a ( , ) n machine is max max ( . (1 ) ) ) ) ( 1 ) ee n    What is the smallest positive, normalized, number that can be represented? The representation of decimal fractions, using a (,) n machine, is the source of "round- off" error. Example . 0 ,2 ) n  Let 2/3 x . Then, using "rounding", the floating point representation of 2/3 is 0 10 () ( . 6 7 ) 1 0 fl x and using another method, called "chopping", we have 0 10 . 6 6 0 fl x or, for 838 x  , we have 3 10 ( ) (.84) 10 fl x and 3 10 () ( . 8 3 0 fl x respectively. Definition: The value, using a ( , ) n machine, | ( ) | f lx x is called the round-off error . For 0 x , the value |( ) | || f x is called the relative error (when the relative error is multiplied by 100, we have the percentage error ).

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p103_2_S10 - EE103 Lecture Notes Spring 2010 Prof S...

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