rhombus rule
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11: 29.20 Methods of Computation
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►A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8.
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12: Bibliography R
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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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13: 34.7 Basic Properties: Symbol
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►This equation is the sum rule.
It constitutes an addition theorem for the symbol.
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14: 1.4 Calculus of One Variable
15: 1.11 Zeros of Polynomials
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Descartes’ Rule of Signs
…16: 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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17: 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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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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18: 3.7 Ordinary Differential Equations
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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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19: 6.18 Methods of Computation
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►Zeros of and can be computed to high precision by Newton’s rule (§3.8(ii)), using values supplied by the asymptotic expansion (6.13.2) as initial approximations.
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