backward%20recursion%20method
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1: 5.21 Methods of Computation
§5.21 Methods of Computation
►An effective way of computing in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (5.11.3). …For the left half-plane we can continue the backward recurrence or make use of the reflection formula (5.5.3). …2: 3.6 Linear Difference Equations
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►A “trial solution” is then computed by backward recursion, in the course of which the original components of the unwanted solution die away.
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►If, as , the wanted solution grows (decays) in magnitude at least as fast as any solution of the corresponding homogeneous equation, then forward (backward) recursion is stable.
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►Then is generated by backward recursion from
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►Within this framework forward and backward recursion may be regarded as the special cases and , respectively.
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3: 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.
4: 16.25 Methods of Computation
§16.25 Methods of Computation
►Methods for computing the functions of the present chapter include power series, asymptotic expansions, integral representations, differential equations, and recurrence relations. …In these cases integration, or recurrence, in either a forward or a backward direction is unstable. …5: 7.22 Methods of Computation
§7.22 Methods of Computation
… ►The methods available for computing the main functions in this chapter are analogous to those described in §§6.18(i)–6.18(iv) for the exponential integral and sine and cosine integrals, and similar comments apply. … ►The recursion scheme given by (7.18.1) and (7.18.7) can be used for computing . See Gautschi (1977a), where forward and backward recursions are used; see also Gautschi (1961). … ►For a comprehensive survey of computational methods for the functions treated in this chapter, see van der Laan and Temme (1984, Ch. V).6: 29.20 Methods of Computation
§29.20 Methods of Computation
… ►Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. … ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). …The numerical computations described in Jansen (1977) are based in part upon this method. ►A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. …7: 11.13 Methods of Computation
§11.13 Methods of Computation
… ►For complex variables the methods described in §§3.5(viii) and 3.5(ix) are available. … ►The solution needs to be integrated backwards for small , and either forwards or backwards for large depending whether or not exceeds . For both forward and backward integration are unstable, and boundary-value methods are required (§3.7(iii)). … ►In consequence forward recurrence, backward recurrence, or boundary-value methods may be necessary. …8: 20 Theta Functions
Chapter 20 Theta Functions
…9: Bibliography O
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Asymptotic Enumeration Methods.
In Handbook of Combinatorics, Vol. 2, L. Lovász, R. L. Graham, and M. Grötschel (Eds.),
pp. 1063–1229.
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Computing : The combinatorial method.
Revista do DETUA 4 (6), pp. 759–768.
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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series.
J. Inst. Math. Appl. 20 (3), pp. 379–391.
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Note on backward recurrence algorithms.
Math. Comp. 26 (120), pp. 941–947.
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A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations.
Proc. Cambridge Philos. Soc. 46 (4), pp. 570–580.
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