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By John F. Monahan

ISBN-10: 0521139511

ISBN-13: 9780521139519

ISBN-10: 0521191580

ISBN-13: 9780521191586

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6023). 6022) and will bias numbers downward. 6023) is preferred, but still a detail remains if z is in the middle. Knuth (1997b, p. 237) discussed the merits of rounding to the nearest even or odd number, with a minor edge attributed to even. The rounding versus chopping issue has important consequences in the analysis of accuracy. 17). Here U is called the machine unit, which takes the value B chopping and half that for rounding to nearest. One expected implication is that the machine unit U bounds the relative error |(fl(z) − z)/z| for the floating point representation.

For an interesting application that computes the digits of π, see Bailey (1988). The third and final approach for unusual problems are methods for doing rational arithmetic. Here, the results are entirely exact and the computations are all performed using integers. Different representations can be used. The simplest one, and the only one recommended for amateurs, is to represent a number using its prime factorization by the list of exponents: 33/34 = 2−131 5 0 7 011113017−1 = (−1, 1, 0, 0, 1, 0, −1).

1000), as will e x . Consequently, both log(1 + x) and e x − 1 will be zero when the result (to first order) should actually be x for both. 2) e x − 1 ≈ x + x 2/2! + x 3/3! + x 4/4! + · · · . 3) Thankfully, when such rescues are required the series will usually converge very quickly. 2) also includes some cancellation. For most computations, a certain level of cancellation is quite acceptable. Here, for small x, any cancellation would be in a smaller order of magnitude. 5, two decimal digits were lost in cancellation, magnifying the relative importance of the rounding error in the intermediate expressions into the second significant digit.

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Numerical Methods of Statistics by John F. Monahan

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