Simpson%20rule
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1: 3.5 Quadrature
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§3.5(ii) Simpson’s Rule
… ►Then the elementary Simpson’s rule is … ►Then the composite Simpson’s rule is … ► ►Rules of closed type include the Newton–Cotes formulas such as the trapezoidal rules and Simpson’s rule. …2: 3.8 Nonlinear Equations
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§3.8(ii) Newton’s Rule
… ► … ►Newton’s rule is given by … ►Another iterative method is Halley’s rule: …The rule converges locally and is cubically convergent. …3: 18.40 Methods of Computation
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►Results of low ( to decimal digits) precision for are easily obtained for to .
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Derivative Rule Approach
►An alternate, and highly efficient, approach follows from the derivative rule conjecture, see Yamani and Reinhardt (1975), and references therein, namely that … ► ►Further, exponential convergence in , via the Derivative Rule, rather than the power-law convergence of the histogram methods, is found for the inversion of Gegenbauer, Attractive, as well as Repulsive, Coulomb–Pollaczek, and Hermite weights and zeros to approximate for these OP systems on and respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …4: 20 Theta Functions
Chapter 20 Theta Functions
…5: Bibliography R
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On the definition and properties of generalized - symbols.
J. Math. Phys. 20 (12), pp. 2398–2415.
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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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6: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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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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7: 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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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Calculation of Gauss quadrature rules.
Math. Comp. 23 (106), pp. 221–230.
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8: 1.11 Zeros of Polynomials
9: 3.4 Differentiation
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►The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2).
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►As explained in §§3.5(i) and 3.5(ix) the composite trapezoidal rule can be very efficient for computing integrals with analytic periodic integrands.
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10: 8 Incomplete Gamma and Related
Functions
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