absolute error
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1: 19.38 Approximations
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►Minimax polynomial approximations (§3.11(i)) for and in terms of with can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸.
Approximations of the same type for and for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸.
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2: 3.1 Arithmetics and Error Measures
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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 is an approximation to a real or complex number , then the absolute error is
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3.1.8
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3.1.9
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3: 8.27 Approximations
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DiDonato (1978) gives a simple approximation for the function (which is related to the incomplete gamma function by a change of variables) for real and large positive . This takes the form , approximately, where and is shown to produce an absolute error as .
4: 18.40 Methods of Computation
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5: 7.13 Zeros
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has a simple zero at , and in the first quadrant of there is an infinite set of zeros , , arranged in order of increasing absolute value.
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►In the sector , has an infinite set of zeros , , arranged in order of increasing absolute value.
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6: 2.3 Integrals of a Real Variable
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►is finite and bounded for , then the th error term (that is, the difference between the integral and th partial sum in (2.3.2)) is bounded in absolute value by when exceeds both and .
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►In both cases the th error term is bounded in absolute value by , where the variational
operator
is defined by
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7: 18.15 Asymptotic Approximations
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►When , the error term in (18.15.1) is less than twice the first neglected term in absolute value, in which one has to take .
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8: 10.40 Asymptotic Expansions for Large Argument
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§10.40(ii) Error Bounds for Real Argument and Order
… ►Then the remainder term does not exceed the first neglected term in absolute value and has the same sign provided that . ►For the error term in (10.40.1) see §10.40(iii). ►§10.40(iii) Error Bounds for Complex Argument and Order
…9: 10.17 Asymptotic Expansions for Large Argument
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§10.17(iii) Error Bounds for Real Argument and Order
… ►If these expansions are terminated when , then the remainder term is bounded in absolute value by the first neglected term, provided that . ►§10.17(iv) Error Bounds for Complex Argument and Order
… ►Corresponding error bounds for (10.17.3) and (10.17.4) are obtainable by combining (10.17.13) and (10.17.14) with (10.4.4). … ►For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).10: 2.7 Differential Equations
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2.7.25
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