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Remez second algorithm

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1: Bibliography R
  • A. Ralston (1965) Rational Chebyshev approximation by Remes’ algorithms. Numer. Math. 7 (4), pp. 322–330.
  • M. Razaz and J. L. Schonfelder (1981) Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.
  • K. Reinsch and W. Raab (2000) Elliptic Integrals of the First and Second Kind – Comparison of Bulirsch’s and Carlson’s Algorithms for Numerical Calculation. In Special Functions (Hong Kong, 1999), C. Dunkl, M. Ismail, and R. Wong (Eds.), pp. 293–308.
  • E. Ya. Remez (1957) General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.
  • P. A. Rosenberg and L. P. McNamee (1976) Precision controlled trigonometric algorithms. Appl. Math. Comput. 2 (4), pp. 335–352.
  • 2: 3.11 Approximation Techniques
    A widely implemented and used algorithm for calculating the coefficients p j and q j in (3.11.16) is Remez’s second algorithm. See Remez (1957), Werner et al. (1967), and Johnson and Blair (1973). … is of fundamental importance in the FFT algorithm. …For further details and algorithms, see Van Loan (1992). …
    3: 31.14 General Fuchsian Equation
    The general second-order Fuchsian equation with N + 1 regular singularities at z = a j , j = 1 , 2 , , N , and at , is given by …
    §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).
    4: 27.19 Methods of Computation: Factorization
    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 p - 1 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). … 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). …
    5: Bibliography G
  • B. Gabutti and G. Allasia (2008) Evaluation of q -gamma function and q -analogues by iterative algorithms. Numer. Algorithms 49 (1-4), pp. 159–168.
  • I. Gargantini and P. Henrici (1967) A continued fraction algorithm for the computation of higher transcendental functions in the complex plane. Math. Comp. 21 (97), pp. 18–29.
  • W. Gautschi (1966) Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.
  • W. Gautschi (1973) Algorithm 471: Exponential integrals. Comm. ACM 16 (12), pp. 761–763.
  • W. Gautschi (1979a) Algorithm 542: Incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 482–489.
  • 6: Bibliography Z
  • M. R. Zaghloul and A. N. Ali (2011) Algorithm 916: computing the Faddeyeva and Voigt functions. ACM Trans. Math. Software 38 (2), pp. Art. 15, 22.
  • M. R. Zaghloul (2016) Remark on “Algorithm 916: computing the Faddeyeva and Voigt functions”: efficiency improvements and Fortran translation. ACM Trans. Math. Softw. 42 (3), pp. 26:1–26:9.
  • M. R. Zaghloul (2017) Algorithm 985: Simple, Efficient, and Relatively Accurate Approximation for the Evaluation of the Faddeyeva Function. ACM Trans. Math. Softw. 44 (2), pp. 22:1–22:9.
  • Zeilberger (website) Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.
  • Ya. M. Zhileĭkin and A. B. Kukarkin (1995) A fast Fourier-Bessel transform algorithm. Zh. Vychisl. Mat. i Mat. Fiz. 35 (7), pp. 1128–1133 (Russian).
  • 7: Bibliography O
  • A. B. Olde Daalhuis and F. W. J. Olver (1995a) Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.
  • F. W. J. Olver and D. J. Sookne (1972) Note on backward recurrence algorithms. Math. Comp. 26 (120), pp. 941–947.
  • F. W. J. Olver (1964a) Error analysis of Miller’s recurrence algorithm. Math. Comp. 18 (85), pp. 65–74.
  • F. W. J. Olver (1977c) Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.
  • H. Oser (1960) Algorithm 22: Riccati-Bessel functions of first and second kind. Comm. ACM 3 (11), pp. 600–601.
  • 8: Bibliography T
  • N. M. Temme (1979a) An algorithm with ALGOL 60 program for the computation of the zeros of ordinary Bessel functions and those of their derivatives. J. Comput. Phys. 32 (2), pp. 270–279.
  • N. M. Temme (1994a) A set of algorithms for the incomplete gamma functions. Probab. Engrg. Inform. Sci. 8, pp. 291–307.
  • N. M. Temme (1997) Numerical algorithms for uniform Airy-type asymptotic expansions. Numer. Algorithms 15 (2), pp. 207–225.
  • H. C. Thacher Jr. (1963) Algorithm 165: Complete elliptic integrals. Comm. ACM 6 (4), pp. 163–164.
  • I. J. Thompson (2004) Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.
  • 9: 3.6 Linear Difference Equations
    §3.6(iii) Miller’s Algorithm
    For further information on Miller’s algorithm, including examples, convergence proofs, and error analyses, see Wimp (1984, Chapter 4), Gautschi (1967, 1997b), and Olver (1964a). …
    §3.6(v) Olver’s Algorithm
    The backward recursion can be carried out using independently computed values of J N ( 1 ) and J N + 1 ( 1 ) or by use of Miller’s algorithm3.6(iii)) or Olver’s algorithm3.6(v)). …
    10: 19.39 Software
    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. … Unless otherwise stated, the functions are K ( k ) and E ( k ) , with 0 k 2 ( = m ) 1 . … Unless otherwise stated, the variables are real, and the functions are F ( ϕ , k ) and E ( ϕ , k ) . For research software see Bulirsch (1965b, function el2 ), Bulirsch (1969b, function el3 ), Jefferson (1961), and Neuman (1969a, functions E ( ϕ , k ) and Π ( ϕ , k 2 , k ) ). …