Kovacic%20algorithm
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1: 31.14 General Fuchsian Equation
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§31.14(ii) Kovacic’s Algorithm
►An algorithm given in Kovacic (1986) determines if a given (not necessarily Fuchsian) second-order homogeneous linear differential equation with rational coefficients has solutions expressible in finite terms (Liouvillean solutions). The algorithm returns a list of solutions if they exist. ►For applications of Kovacic’s algorithm in spatio-temporal dynamics see Rod and Sleeman (1995).2: 31.8 Solutions via Quadratures
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►For more details see Smirnov (2002).
►The solutions in this section are finite-term Liouvillean solutions which can be constructed via Kovacic’s algorithm; see §31.14(ii).
3: Bibliography K
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Linear convergence and the bisection algorithm.
Amer. Math. Monthly 93 (1), pp. 48–51.
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Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library.
ACM Trans. Math. Software 20 (4), pp. 447–459.
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Algorithm 327: Dilogarithm [S22].
Comm. ACM 11 (4), pp. 270–271.
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An algorithm for solving second order linear homogeneous differential equations.
J. Symbolic Comput. 2 (1), pp. 3–43.
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Algorithm 421. Complex gamma function with error control.
Comm. ACM 15 (4), pp. 271–272.
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4: 20 Theta Functions
Chapter 20 Theta Functions
…5: Bibliography G
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Algorithm 292: Regular Coulomb wave functions.
Comm. ACM 9 (11), pp. 793–795.
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Algorithm 471: Exponential integrals.
Comm. ACM 16 (12), pp. 761–763.
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Algorithm 542: Incomplete gamma functions.
ACM Trans. Math. Software 5 (4), pp. 482–489.
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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules.
ACM Trans. Math. Software 20 (1), pp. 21–62.
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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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6: 27.19 Methods of Computation: Factorization
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►Deterministic algorithms are slow but are guaranteed to find the factorization within a known period of time.
…Fermat’s algorithm is another; see Bressoud (1989, §5.1).
►Type I probabilistic algorithms include the Brent–Pollard rho algorithm (also called Monte Carlo method), the Pollard
algorithm, and the Elliptic Curve Method (ecm).
Descriptions of these algorithms are given in Crandall and Pomerance (2005, §§5.2, 5.4, and 7.4).
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►These algorithms include the Continued Fraction Algorithm (cfrac), the Multiple Polynomial Quadratic Sieve (mpqs), the General
Number Field Sieve (gnfs), and the Special Number Field Sieve (snfs).
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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: Bibliography
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Algorithm 724: Program to calculate F-percentiles.
ACM Trans. Math. Software 19 (4), pp. 481–483.
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Algorithm 39: Areas under the normal curve.
The Computer Journal 12 (2), pp. 197–198.
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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics.
Comput. Phys. Comm. 150 (1), pp. 1–20.
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Algorithms for special integrals of ordinary differential equations.
J. Phys. A 29 (5), pp. 973–991.
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Algorithm 804: Subroutines for the computation of Mathieu functions of integer orders.
ACM Trans. Math. Software 26 (3), pp. 408–414.
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9: 35.12 Software
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►In this section we provide links to the research literature describing the implementation of algorithms in software for the evaluation of functions described in this chapter.
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►For an algorithm to evaluate zonal polynomials, and an implementation of the algorithm in Maple by Zeilberger, see Lapointe and Vinet (1996).
10: Bibliography N
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Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes.
ACM Trans. Math. Software 18 (3), pp. 345–349.
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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Combinatorial Algorithms.
Academic Press, New York.
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