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set4 - $ Set 4 Basics Part 2 Kyle A Gallivan Department of...

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a39 a38 a36 a37 Set 4: Basics Part 2 Kyle A. Gallivan Department of Mathematics Florida State University Foundations of Computational Math 1 Fall 2011 1
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a39 a38 a36 a37 Finite Precision All discussions so far have assumed we can compute with elements of R and by extension C . Representation of numbers was symbolic and therefore exact, e.g., π , e x , 1 / 3 arithmetic was symbolic and therefore exact. Computers store information using a finite number of digits (bits). Computers do not perform exact arithmetic. 2
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a39 a38 a36 a37 Finite Precision Consequently, The problem solved is not necessarily the problem posed. The computed solution does not necessarily solve the original problem or the problem stored in the computer. The computed solution may solve a problem that is not representable in the computer. 3
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a39 a38 a36 a37 Numerical Analysis and Finite Precision Numerical error analysis addresses the effects of finite precision of representation and computation. Topics: Representation Arithmetic Errors: forms and bounds Conditioning of problem Stability of an algorithm 4
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a39 a38 a36 a37 Sources In addtion to the textbook the following are sources for this presentaion and useful references (see also their citations): What Every Computer Scientist Should Know About Floating-Point Arithmetic David Goldberg, ACM Computing Surveys, March, 1991. http://docs.sun.com/source/806-3568/ncg_goldberg.html Matrix Algorithms, Volume 1: Basic Decompositions , G. W. Stewart, SIAM, 1998. Accuracy and Stability of Numerical Algorithms , N. J. Higham, SIAM, Second Edition, 2002. The Handbook of Mathematical Functions , M. Abramowitz and I. Stegun, Tenth Printing, available on the web at several URLs 5
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a39 a38 a36 a37 Integers – finite number exactly represented Binary unsigned sign excess two’s magnitude three complement 000 0 0 -3 0 001 1 1 -2 1 010 2 2 -1 2 011 3 3 0 3 100 4 -0 1 -4 101 5 -1 2 -3 110 6 -2 3 -2 111 7 -3 4 -1 6
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a39 a38 a36 a37 Real Number Representation Real numbers can be written as decimal expansions: π = 3 . 14159265358979 . . . 1 3 = 0 . 3333333 . . . 98 . 6 = 986 × 10 1 = 0 . 986 × 10 2 = ( 9 10 + 8 100 + 6 1000 ) × 10 2 x R x = ± 10 e summationdisplay k =1 d k × 10 k , 0 d k 9 , d 1 negationslash = 0 , e Z 7
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a39 a38 a36 a37 Floating Point Representation A floating point number system is characterized by four integers β the base or radix t the precision exponent bounds L e U Definition 4.1. A floating point number system F R is a set of real numbers of the form y = ± m × β e t y = ± β e t summationdisplay k =1 d k × β k = ± β e (0 .d 1 d 2 . . . d t ) where m Z , 0 m β t 1 . 8
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a39 a38 a36 a37 Floating Point Representation Definition 4.2. A floating point number system F ⊂ R is normalized if the mantissa, m , ( or significand or fraction) satisfies β t 1 m < β t when y negationslash = 0 . This is the same as d 1 negationslash = 0 . It is assumed that 0 has a special representation.
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