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1: 3.8 Nonlinear Equations
§3.8 Nonlinear Equations
… ►Because the method requires only one function evaluation per iteration, its numerical efficiency is ultimately higher than that of Newton’s method. … ►Consider and . We have and . … ►Corresponding numerical factors in this example for other zeros and other values of are obtained in Gautschi (1984, §4). …2: Bibliography G
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Questions of Numerical Condition Related to Polynomials.
In Studies in Numerical Analysis, G. H. Golub (Ed.),
pp. 140–177.
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Computing complex Airy functions by numerical quadrature.
Numer. Algorithms 30 (1), pp. 11–23.
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Numerically satisfactory solutions of hypergeometric recursions.
Math. Comp. 76 (259), pp. 1449–1468.
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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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Computing the zeros and turning points of solutions of second order homogeneous linear ODEs.
SIAM J. Numer. Anal. 41 (3), pp. 827–855.
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3: Bibliography S
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Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms.
Math. Tables Aids Comput. 9 (52), pp. 164–177.
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Numerical evaluation of the Hankel transform.
Comput. Phys. Comm. 116 (2-3), pp. 278–294.
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Uniform asymptotic forms of modified Mathieu functions.
Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.
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On numerical Bessel transformation.
Comput. Phys. Comm. 16 (3), pp. 383–387.
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Numerical Methods Based on Sinc and Analytic Functions.
Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.
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4: 27.15 Chinese Remainder Theorem
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►The Chinese remainder theorem states that a system of congruences , always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod ), where is the product of the moduli.
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►Their product has 20 digits, twice the number of digits in the data.
…These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result , which is correct to 20 digits.
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►Details of a machine program describing the method together with typical numerical results can be found in Newman (1967).
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5: Bibliography K
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Numerical Methods and Software.
Prentice Hall, Englewood Cliffs, N.J..
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The second Painlevé equation in electric probe theory. Some numerical solutions.
Zh. Vychisl. Mat. Mat. Fiz. 38 (6), pp. 992–1000 (Russian).
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The second Painlevé equation in the electrostatic probe theory: Numerical solutions for the partial absorption of charged particles by the surface.
Technical Physics 49 (1), pp. 1–7.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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On Nielsen’s generalized polylogarithms and their numerical calculation.
Nordisk Tidskr. Informationsbehandling (BIT) 10, pp. 38–73.
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6: Bibliography O
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On the asymptotic and numerical solution of linear ordinary differential equations.
SIAM Rev. 40 (3), pp. 463–495.
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Error bounds for asymptotic solutions of second-order differential equations having an irregular singularity of arbitrary rank.
J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 244–249.
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On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions.
J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 225–243.
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Numerical solution of second-order linear difference equations.
J. Res. Nat. Bur. Standards Sect. B 71B, pp. 111–129.
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Numerical solution of Riemann-Hilbert problems: Painlevé II.
Found. Comput. Math. 11 (2), pp. 153–179.
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7: Bibliography L
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Asymptotic and numeric study of eigenvalues of the double confluent Heun equation.
J. Phys. A 31 (42), pp. 8521–8531.
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Application of theta functions for numerical evaluation of complete elliptic integrals of the first and second kinds.
Comput. Phys. Comm. 60 (3), pp. 319–327.
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An asymptotic estimate for the Bernoulli and Euler numbers.
Canad. Math. Bull. 20 (1), pp. 109–111.
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Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions.
Numer. Math. 116 (2), pp. 269–289.
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Numerical Solution of Linear Difference Equations.
NBSIR
Technical Report 80-1976, National Bureau of Standards, Gaithersburg, MD 20899.
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8: Bibliography V
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Integral representations for products of Lamé functions by use of fundamental solutions.
SIAM J. Math. Anal. 15 (3), pp. 559–569.
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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation.
Constr. Approx. 20 (1), pp. 39–54.
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On the rational solutions of the second Painlevé equation.
Differ. Uravn. 1 (1), pp. 79–81 (Russian).
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RFSFNS: A portable package for the numerical determination of the number and the calculation of roots of Bessel functions.
Comput. Phys. Comm. 92 (2-3), pp. 252–266.
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The topological degree theory for the localization and computation of complex zeros of Bessel functions.
Numer. Funct. Anal. Optim. 18 (1-2), pp. 227–234.
9: Bibliography B
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Pionic atoms.
Annual Review of Nuclear and Particle Science 20, pp. 467–508.
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Numerical calculation of a generalized complete elliptic integral.
Rev. Mod. Phys. 10, pp. 264–269.
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Some solutions of the problem of forced convection.
Philos. Mag. Series 7 20, pp. 322–343.
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Numerical Methods for Least Squares Problems.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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Numerical evaluation of continued fractions.
SIAM Rev. 6 (4), pp. 383–421.
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10: Bibliography N
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On the numerical evaluation of the generalised Fermi-Dirac integrals.
Comput. Phys. Comm. 76 (1), pp. 48–50.
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On an integral transform involving a class of Mathieu functions.
SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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Reduction and evaluation of elliptic integrals.
Math. Comp. 20 (94), pp. 223–231.
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A table of integrals of the error functions.
J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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Bisection hardly ever converges linearly.
Numer. Math. 70 (1), pp. 111–118.
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