By Michael T. Heath
Heath 2/e, offers a vast evaluate of numerical tools for fixing the entire significant difficulties in clinical computing, together with linear and nonlinear equations, least squares, eigenvalues, optimization, interpolation, integration, traditional and partial differential equations, speedy Fourier transforms, and random quantity turbines. The remedy is accomplished but concise, software-oriented but appropriate with various software program programs and programming languages. The booklet positive factors greater than one hundred sixty examples, 500 assessment questions, 240 routines, and 2 hundred computing device difficulties. adjustments for the second one variation contain: extended motivational discussions and examples; formal statements of all significant algorithms; elevated discussions of life, area of expertise, and conditioning for every form of challenge in order that scholars can realize "good" and "bad" challenge formulations and comprehend the corresponding caliber of effects produced; and improved assurance of a number of issues, rather eigenvalues and restricted optimization. The publication incorporates a wealth of fabric and will be utilized in various one- or two-term classes in computing device technological know-how, arithmetic, or engineering. Its comprehensiveness and smooth viewpoint, in addition to the software program guidelines supplied, additionally make it a hugely worthwhile reference for training pros who have to resolve computational difficulties.
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Additional resources for Scientific Computing: An Introductory Survey (2nd Edition)
Example text
Multiplication of two floating-point numbers does not require that their exponents match—the exponents are simply summed and the mantissas multiplied. However, the product of two t-digit mantissas will in general contain up to 2t digits, and thus once again the correct result cannot be represented exactly in the floating-point system and must be rounded. 7 Floating-Point Arithmetic. Consider a floating-point system with β = 10 and t = 6. 93039 × 102 , assuming rounding to nearest. Note that the last two digits of y have no effect on the result.
How does the value obtained using this approximate formula compare with that obtained from the “exact” formula in part b? (d ) Determine which of the previous two answers is more nearly correct by repeating both computations using higher precision, say, sixdigit decimal arithmetic. (e) Explain the results you obtained in parts a–d. (f ) Try this problem on a computer. How small must the change h in radius be for the 30 CHAPTER 1. SCIENTIFIC COMPUTING same phenomenon to occur? Try both single precision and double precision, if available.
D ) For what values of the argument x is this problem highly sensitive? 5 Consider the function f : R2 → R defined by f (x, y) = x − y. Measuring the size of the input (x, y) by |x| + |y|, and assuming that |x| + |y| ≈ 1 and x − y ≈ , show that cond(f ) ≈ 1/ . What can you conclude about the sensitivity of subtraction? 6 The sine function is given by the infinite series sin(x) = x − x3 x5 x7 + − + ···. 3! 5! 7! 0? 0? 7 A floating-point number system is characterized by four integers: the base β, the precision t, and the lower and upper limits L and U of the exponent range.