computation%20by%20recursion
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6 matching pages
1: 6.20 Approximations
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Cody and Thacher (1968) provides minimax rational approximations for , with accuracies up to 20S.
Cody and Thacher (1969) provides minimax rational approximations for , with accuracies up to 20S.
MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions and , with accuracies up to 20S.
Luke (1969b, p. 25) gives a Chebyshev expansion near infinity for the confluent hypergeometric -function (§13.2(i)) from which Chebyshev expansions near infinity for , , and follow by using (6.11.2) and (6.11.3). Luke also includes a recursion scheme for computing the coefficients in the expansions of the functions. If the scheme can be used in backward direction.
2: Bibliography G
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Recursive computation of the repeated integrals of the error function.
Math. Comp. 15 (75), pp. 227–232.
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A note on the recursive calculation of incomplete gamma functions.
ACM Trans. Math. Software 25 (1), pp. 101–107.
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The ABC of hyper recursions.
J. Comput. Appl. Math. 190 (1-2), pp. 270–286.
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Numerically satisfactory solutions of hypergeometric recursions.
Math. Comp. 76 (259), pp. 1449–1468.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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3: Bibliography M
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Computation of inhomogeneous Airy functions.
J. Comput. Appl. Math. 53 (1), pp. 109–116.
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The efficient computation of some generalised exponential integrals.
J. Comput. Appl. Math. 148 (2), pp. 363–374.
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The computation of elementary functions in radix
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Computing 53 (3-4), pp. 219–232.
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Recursion relations for the - symbols.
Nuclear Physics A 113 (1), pp. 215–220.
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The -analogue of the Laguerre polynomials.
J. Math. Anal. Appl. 81 (1), pp. 20–47.
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4: Bibliography F
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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Faster computation of Bernoulli numbers.
J. Algorithms 13 (3), pp. 431–445.
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On weighted polynomial approximation on the whole real axis.
Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
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On the coefficients in the recursion formulae of orthogonal polynomials.
Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.
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5: 18.40 Methods of Computation
§18.40 Methods of Computation
►§18.40(i) Computation of Polynomials
… ►A numerical approach to the recursion coefficients and quadrature abscissas and weights
… ►A simple set of choices is spelled out in Gordon (1968) which gives a numerically stable algorithm for direct computation of the recursion coefficients in terms of the moments, followed by construction of the J-matrix and quadrature weights and abscissas, and we will follow this approach: Let be a positive integer and define …and these can be used for the recursion coefficients …6: 18.39 Applications in the Physical Sciences
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►The discrete variable representations (DVR) analysis is simplest when based on the classical OP’s with their analytically known recursion coefficients (Table 3.5.17_5), or those non-classical OP’s which have analytically known recursion coefficients, making stable computation of the and , from the J-matrix as in §3.5(vi), straightforward.
…Table 18.39.1 lists typical non-classical weight functions, many related to the non-classical Freud weights of §18.32, and §32.15, all of which require numerical computation of the recursion coefficients (i.
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►Following the method of Schwartz (1961), Yamani and Reinhardt (1975), Bank and Ismail (1985), and Ismail (2009, §5.8) have shown this is equivalent to determination of such that in the recursion scheme
…The recursion of (18.39.46) is that for the type 2 Pollaczek polynomials of (18.35.2), with , , and , and terminates for being a zero of the polynomial of order .
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►As this follows from the three term recursion of (18.39.46) it is referred to as the J-Matrix approach, see (3.5.31), to single and multi-channel scattering numerics.
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