Remez second algorithm

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1: Bibliography R

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Rational Chebyshev approximation by Remes’ algorithms. Numer. Math. 7 (4), pp. 322–330.

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ISSN 0029-599X, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§3.11(i).

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Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.

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ISSN 0098-3500, Document

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1st item.

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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.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(i), §19.36(ii).

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General Computation Methods of Chebyshev Approximation. The Problems with Linear Real Parameters. Publishing House of the Academy of Science of the Ukrainian SSR, Kiev.

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English translation, AEC-TR-4491, United States Atomic Energy Commission

External Links:

MathReview (A. S. Householder)

Referenced by:

§3.11(iii).

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Precision controlled trigonometric algorithms. Appl. Math. Comput. 2 (4), pp. 335–352.

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Document, MathReview (Luciano Biasini), ZentralBlatt

Referenced by:

§4.45(i).

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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 $z=a_j$, $j=1,2,\mathrm{\dots },N$, and at $\mathrm{\infty }$, 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

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Evaluation of $q$-gamma function and $q$-analogues by iterative algorithms. Numer. Algorithms 49 (1-4), pp. 159–168.

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ISSN 1017-1398, Document, FullText, MathReview (Necdet Batir), ZentralBlatt

Referenced by:

§17.18, §17.18, §5.21, §5.21.

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A continued fraction algorithm for the computation of higher transcendental functions in the complex plane. Math. Comp. 21 (97), pp. 18–29.

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FullText, MathReview (D. C. Gilles), ZentralBlatt

Referenced by:

§10.74(v), §3.10(ii).

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Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.

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ISSN 0001-0782, Document

Referenced by:

§33.26(ii).

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Algorithm 471: Exponential integrals. Comm. ACM 16 (12), pp. 761–763.

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Notes:

Includes Algol60 program, maximum accuracy 10S.

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ISSN 0001-0782, Document

Referenced by:

§6.21(ii), §8.28(vi).

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Algorithm 542: Incomplete gamma functions. ACM Trans. Math. Software 5 (4), pp. 482–489.

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Notes:

Single-precision Fortran.

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ISSN 0098-3500, Document, GAMS (module 542; package TOMS), ZentralBlatt

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2nd item.

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6: Bibliography T

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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.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.74(vi), §10.77(x).

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A set of algorithms for the incomplete gamma functions. Probab. Engrg. Inform. Sci. 8, pp. 291–307.

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Notes:

Includes Pascal program

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FullText

Referenced by:

§5.24(ii), §7.25(ii), §8.28(ii).

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Numerical algorithms for uniform Airy-type asymptotic expansions. Numer. Algorithms 15 (2), pp. 207–225.

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ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§10.19(iii), §10.20(i), §10.74(i).

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Algorithm 165: Complete elliptic integrals. Comm. ACM 6 (4), pp. 163–164.

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Notes:

Includes Algol program, maximum accuracy 10D. For certification see same journal 6 (1963), pp. 166–167

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ISSN 0001-0782, Document

Referenced by:

§19.39(ii).

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Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.

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Document, Code, ZentralBlatt

Referenced by:

§33.23(vi), I. J. Thompson and A. R. Barnett (1985).

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7: Bibliography Z

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Algorithm 916: computing the Faddeyeva and Voigt functions. ACM Trans. Math. Software 38 (2), pp. Art. 15, 22.

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ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

2nd item, 5th item.

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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.

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Notes:

Includes MATLAB and Fortran programs claiming 6S or 13S accuracy for single and double precision

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ISSN 0098-3500, Link, Document, MathReview Entry

Referenced by:

3rd item, 6th item, §7.25(iii), §7.25(vi).

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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.

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ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

4th item, §7.25(iii).

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Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.

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Notes:

Includes hypergeometric and $q$-hypergeometric summation, solution of difference equations (for example, by Petkovšek’s algorithm), combinatorial algorithms, and (package LUC) evaluation of zonal polynomials.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, L. Lapointe and L. Vinet (1996).

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A fast Fourier-Bessel transform algorithm. Zh. Vychisl. Mat. i Mat. Fiz. 35 (7), pp. 1128–1133 (Russian).

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Notes:

English translation in Comput. Math. Math. Phys. 35(7), pp. 901–905

External Links:

ISSN 0044-4669, MathReview (Vítĕzslav Veselý), ZentralBlatt

Referenced by:

§10.74(vii).

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8: Bibliography O

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Hyperasymptotic solutions of second-order linear differential equations. I. Methods Appl. Anal. 2 (2), pp. 173–197.

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ISSN 1073-2772, FullText, MathReview (A. D. Wood), ZentralBlatt

Referenced by:

§10.17(v), §10.40(iv), §13.29(i), §13.7(iii), §2.11(v), §9.7(v).

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Note on backward recurrence algorithms. Math. Comp. 26 (120), pp. 941–947.

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FullText, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.74(iv), §3.6(v).

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Error analysis of Miller’s recurrence algorithm. Math. Comp. 18 (85), pp. 65–74.

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FullText, MathReview (J.C.P. Miller), ZentralBlatt

Referenced by:

§3.6(iii).

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Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.

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External Links:

FullText, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v).

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Algorithm 22: Riccati-Bessel functions of first and second kind. Comm. ACM 3 (11), pp. 600–601.

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Notes:

Includes Algol code, minimum accuracy 9S. For certification see same journal 13(1970), p. 448.

External Links:

ISSN 0001-0782, Document

Referenced by:

§10.77(ii).

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9: 3.6 Linear Difference Equations

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§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 $J_N\left(1\right)$ and $J_{N+1}\left(1\right)$ or by use of Miller’s algorithm (§3.6(iii)) or Olver’s algorithm (§3.6(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 $K\left(k\right)$ and $E\left(k\right)$, with $0\le k^2(=m)\le 1$. … ►Unless otherwise stated, the variables are real, and the functions are $F(\varphi ,k)$ and $E(\varphi ,k)$. ►For research software see Bulirsch (1965b, function $\mathrm{el2}$), Bulirsch (1969b, function $\mathrm{el3}$), Jefferson (1961), and Neuman (1969a, functions $E(\varphi ,k)$ and $\mathrm{\Pi }(\varphi ,k^2,k)$). …