Descartes’ rule of signs (for polynomials)
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11—20 of 340 matching pages
11: Bibliography R
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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials.
Canad. J. Math. 33 (4), pp. 915–928.
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Universality properties of Gaussian quadrature, the derivative rule, and a novel approach to Stieltjes inversion.
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Erratum to:Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging.
Computing in Science and Engineering 23 (4), pp. 91.
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Relationships between the zeros, weights, and weight functions of orthogonal polynomials: Derivative rule approach to Stieltjes and spectral imaging.
Computing in Science and Engineering 23 (3), pp. 56–64.
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Finite-sum rules for Macdonald’s functions and Hankel’s symbols.
Integral Transform. Spec. Funct. 10 (2), pp. 115–124.
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12: Bibliography
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Numerical computation of Tricomi’s psi function by the trapezoidal rule.
Computing 39 (3), pp. 271–279.
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Numerical evaluation of the Kummer function with complex argument by the trapezoidal rule.
Rend. Sem. Mat. Univ. Politec. Torino 49 (3), pp. 315–327.
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Numerical calculation of incomplete gamma functions by the trapezoidal rule.
Numer. Math. 50 (4), pp. 419–428.
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Numerical Calculation of the Riemann Zeta Function and Generalizations by Means of the Trapezoidal Rule.
In Numerical and Applied Mathematics, Part II (Paris, 1988), C. Brezinski (Ed.),
IMACS Ann. Comput. Appl. Math., Vol. 1, pp. 467–472.
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Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials.
Mem. Amer. Math. Soc. 54 (319), pp. iv+55.
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13: 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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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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The non-symmetric Wilson polynomials are the Bannai-Ito polynomials.
Proc. Amer. Math. Soc. 144 (12), pp. 5217–5226.
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Computing special functions by using quadrature rules.
Numer. Algorithms 33 (1-4), pp. 265–275.
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Calculation of Gauss quadrature rules.
Math. Comp. 23 (106), pp. 221–230.
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14: 1.4 Calculus of One Variable
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►If the
sign is replaced by , then is increasing (also called strictly
increasing) on .
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Chain Rule
… ►L’Hôpital’s Rule
…15: 1.2 Elementary Algebra
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►Let be distinct constants, and be a polynomial of degree less than .
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►If are positive integers and , then there exist polynomials
, , such that
…To find the polynomials
, , multiply both sides by the denominator of the left-hand side and equate coefficients.
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►This is the row times column rule.
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►Eigenvalues are the roots of the polynomial equation
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16: 3.7 Ordinary Differential Equations
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►This converts the problem into a tridiagonal matrix problem in which the elements of the matrix are polynomials in ; compare §3.2(vi).
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►The method consists of a set of rules each of which is equivalent to a truncated Taylor-series expansion, but the rules avoid the need for analytic differentiations of the differential equation.
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►For the standard fourth-order rule reads
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►For the standard fourth-order rule reads
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17: 8.25 Methods of Computation
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►See Allasia and Besenghi (1987b) for the numerical computation of from (8.6.4) by means of the trapezoidal rule.
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►A numerical inversion procedure is also given for calculating the value of (with 10S accuracy), when and are specified, based on Newton’s rule (§3.8(ii)).
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18: 9.17 Methods of Computation
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►The trapezoidal rule (§3.5(i)) is then applied.
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►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations.
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19: Bibliography T
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Zonal Polynomials.
Institute of Mathematical Statistics Lecture Notes—Monograph
Series, 4, Institute of Mathematical Statistics, Hayward, CA.
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Laguerre polynomials: Asymptotics for large degree.
Technical report
Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.
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Asymptotic estimates for Laguerre polynomials.
Z. Angew. Math. Phys. 41 (1), pp. 114–126.
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Bernoulli polynomials old and new: Generalizations and asymptotics.
CWI Quarterly 8 (1), pp. 47–66.
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The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions.
Report TW 183/78
Mathematisch Centrum, Amsterdam, Afdeling Toegepaste
Wiskunde.
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