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2 Machine Arithmetic The arithmetic used on computers unfortunately does not respect the laws of ordinary arithmetic. Each elementary floating-point operation, in general, generates a small error that may then propagate through subsequent machine operations. As a rule, this error propagation is harmless, except in the case of subtraction, where cancellation effects may seriously compromise the accuracy of the results. 1 A Model of Machine Arithmetic Any of the four basic arithmetic operations, when applied to two machine numbers, may produce a result no longer representable on the computer.

If a fellow came along who was too tall to fit into his beds, he cut off his feet. If the fellow was too short, he stretched him. In the same way, if a real number comes along that is too long, its tail end (not the head) is cutoff; if it is too short, it is padded by zeros at the end. 6 1 Machine Arithmetic and Related Matters More specifically, let x 2 R; x D ˙ 1 X ! t; s/; x D ˙ t X ! 8) kD1 the rounded number. One then distinguishes between two methods of rounding, the first being Procrustes’ method.

Subject to a nonzero rounding error), overflows, or underflows. y=4// 5. The Matlab “machine precision” eps is twice the unit roundoff (2 2 t , t D 53; cf. Sect. 3). It can be computed by the following Matlab program (attributed to CLEVE MOLER): 32 1 Machine Arithmetic and Related Matters %EI_5 Matlab machine precision % a=4/3; b=a-1; c=b+b+b; eps0=abs(c-1) Run the program and prove its validity. 6. 12). 7. z; y/ (triangle inequality). J. x; y/ D j ln jxj ln jyj j. "/ as " ! 0. 8. Assume that x1 , x2 are approximations to x1 , x2 with relative errors E1 and E2 , respectively, and that jEi j Ä E, i D 1; 2.