relative error
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1—10 of 18 matching pages
1: 33.25 Approximations
2: 3.1 Arithmetics and Error Measures
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►Then rounding by chopping or rounding down of gives , with maximum relative error
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Symmetric rounding or rounding to nearest of gives or , whichever is nearer to , with maximum relative error equal to the machine precision .
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►Also in this arithmetic generalized precision can be defined, which includes absolute error and relative precision (§3.1(v)) as special cases.
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►If , the relative error is
…The relative precision is
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3: 25.20 Approximations
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4: DLMF Project News
error generating summary5: 11.6 Asymptotic Expansions
6: 19.36 Methods of Computation
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►If the iteration of (19.36.6) and (19.36.12) is stopped when ( and being approximated by and , and the infinite series being truncated), then the relative error in and is less than if we neglect terms of order .
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7: 7.24 Approximations
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Cody (1969) provides minimax rational approximations for and . The maximum relative precision is about 20S.
Cody et al. (1970) gives minimax rational approximations to Dawson’s integral (maximum relative precision 20S–22S).
8: 3.3 Interpolation
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►With an error term the Lagrange interpolation formula for is given by
…If , (), and the nodes are real, and is continuous on the smallest closed interval containing , then the error can be expressed
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►The divided differences of
relative to a sequence of distinct points are defined by
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►The interpolation error
is as in §3.3(i).
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9: Bibliography W
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Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions.
In Algorithms for Approximation, A. Iske and J. Levesley (Eds.),
pp. 331–348.
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Asymptotic monotonicity of the relative extrema of Jacobi polynomials.
Canad. J. Math. 46 (6), pp. 1318–1337.
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On the relative extrema of the Jacobi polynomials
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SIAM J. Math. Anal. 25 (2), pp. 776–811.
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Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials.
Anal. Appl. (Singap.) 1 (2), pp. 213–241.
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Error bounds for asymptotic expansions of Hankel transforms.
SIAM J. Math. Anal. 7 (6), pp. 799–808.
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10: 3.6 Linear Difference Equations
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►Unless exact arithmetic is being used, however, each step of the calculation introduces rounding errors.
These errors have the effect of perturbing the solution by unwanted small multiples of and of an independent solution , say.
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►For further information on Miller’s algorithm, including examples, convergence proofs, and error analyses, see Wimp (1984, Chapter 4), Gautschi (1967, 1997b), and Olver (1964a).
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►Suppose again that , is given, and we wish to calculate to a prescribed relative accuracy for a given value of .
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►For further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6).
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