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11: 13.5 Continued Fractions
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►This continued fraction converges to the meromorphic function of on the left-hand side everywhere in .
For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980).
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►This continued fraction converges to the meromorphic function of on the left-hand side throughout the sector .
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12: 13.17 Continued Fractions
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►This continued fraction converges to the meromorphic function of on the left-hand side for all .
For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980).
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►This continued fraction converges to the meromorphic function of on the left-hand side throughout the sector .
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13: 18.40 Methods of Computation
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A numerical approach to the recursion coefficients and quadrature abscissas and weights
… ►See Gautschi (1983) for examples of numerically stable and unstable use of the above recursion relations, and how one can then usefully differentiate between numerical results of low and high precision, as produced thereby. ►Having now directly connected computation of the quadrature abscissas and weights to the moments, what follows uses these for a Stieltjes–Perron inversion to regain . … ►The question is then: how is this possible given only , rather than itself? often converges to smooth results for off the real axis for at a distance greater than the pole spacing of the , this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to and evaluating these on the real axis in regions of higher pole density that those of the approximating function. Results of low ( to decimal digits) precision for are easily obtained for to . …14: 33.21 Asymptotic Approximations for Large
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►We indicate here how to obtain the limiting forms of , , , and as , with and fixed, in the following cases:
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(a)
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(b)
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(c)
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►For asymptotic expansions of and as with and fixed, see Curtis (1964a, §6).
15: 5.4 Special Values and Extrema
16: Bibliography G
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Stable computation of high order Gauss quadrature rules using discretization for measures in radiation transfer.
J. Quant. Spectrosc. Radiat. Transfer 68 (2), pp. 213–223.
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How and how not to check Gaussian quadrature formulae.
BIT 23 (2), pp. 209–216.
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Evaluation of the modified Bessel function of the third kind of imaginary orders.
J. Comput. Phys. 175 (2), pp. 398–411.
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Algorithm 831: Modified Bessel functions of imaginary order and positive argument.
ACM Trans. Math. Software 30 (2), pp. 159–164.
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Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments.
ACM Trans. Math. Software 30 (2), pp. 145–158.
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17: 3.6 Linear Difference Equations
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►In practice, however, problems of severe instability often arise and in §§3.6(ii)–3.6(vii) we show how these difficulties may be overcome.
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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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§3.6(vii) Linear Difference Equations of Other Orders
►Similar considerations apply to the first-order equation …Thus in the inhomogeneous case it may sometimes be necessary to recur backwards to achieve stability. …18: 1.13 Differential Equations
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►(More generally in (1.13.5) for th-order differential equations, is the coefficient multiplying the th-order derivative of the solution divided by the coefficient multiplying the th-order derivative of the solution, see Ince (1926, §5.2).)
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►For extensions of these results to linear homogeneous differential equations of arbitrary order see Spigler (1984).
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►For an extensive collection of solutions of differential equations of the first, second, and higher orders see Kamke (1977).
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►The functions which correspond to these being eigenfunctions.
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Transformation to Liouville normal Form
…19: 3.2 Linear Algebra
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►To solve the system
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►During this reduction process we store the multipliers
that are used in each column to eliminate other elements in that column.
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§3.2(iii) Condition of Linear Systems
… ►where and are the normalized right and left eigenvectors of corresponding to the eigenvalue . … ►Lanczos’ method is related to Gauss quadrature considered in §3.5(v). …20: 19.36 Methods of Computation
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►In the Appendix of the last reference it is shown how to compute without computing more than once.
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►As , , , and converge quadratically to limits , , and , respectively; hence
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►To (19.36.6) add
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►(19.22.20) reduces to
if or , and (19.22.19) reduces to
if or .
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►Quadratic transformations can be applied to compute Bulirsch’s integrals (§19.2(iii)).
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