tailieunhanh - numerical mathematics and scientific computation volume 1 Episode 4

Tham khảo tài liệu 'numerical mathematics and scientific computation volume 1 episode 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 100 Chapter 2. How to Obtain and Estimate Accuracy numerical quadrature. Another example is the case in the numerical solution of initial value problems for ordinary differential equations. Note that in this application the terms have to be added in the order in which they are generated. Compensated is based on the possibility to simulate double precision floating point addition in single precision arithmetic. To illustrate the basic idea we take as in Example a 10 b 104 so that s fl a b 104 Suppose we form c fl fl b s a 10 10 4567000 10 3. Note that the variable c is computed without error and picks up the information that was lost in the operation fl a b . The following algorithm uses this idea to accurately computing xi Algorithm Compensated Summation. 8 Xi c 0 for i 2 n y c Xi t 8 y c s t y s t end It can be proved see Goldberg 16 1991 that on binary machines with a guard digit the computed sum satisfies 8 1 fi 2u O nu2 . i l This formulation is a typical example of a backward error analysis see Sec. . The single precision term in the error bound is independent of n. Standard Error The bounds for the accumulated rounding error we have derived so far are estimates of the maximal error. These bounds are often much too pessimistic when the number of variables is large. As a complement one can use the standard error. The theory of standard error is based on probability theory and will not be treated in detail here. The standard error of an estimate of a given quantity is the same as the standard deviation of its sampling distribution. If in a sum y Xi each Xi has error Aj Ô then the maximum error bound for y is nổ. Thus the maximal error grows proportionally to n. If . Accuracy and Rounding Errors 101 n is large for example n 1000 then it is in fact highly improbable that the real error will be anywhere near nd since that bound is attained only when every A has the same sign and