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11—15 of 15 matching pages
11: 1.11 Zeros of Polynomials
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1.11.7
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►A monic polynomial of even degree with real coefficients has at least two zeros of opposite signs when the constant term is negative.
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►The number of positive zeros of a polynomial with real coefficients cannot exceed the number of times the coefficients change sign, and the two numbers have same parity.
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§1.11(iii) Polynomials of Degrees Two, Three, and Four
… ►Resolvent cubic is with roots , , , and , , . …12: Bibliography O
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Summing one- and two-dimensional series related to the Euler series.
J. Comput. Appl. Math. 98 (2), pp. 245–271.
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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series.
J. Inst. Math. Appl. 20 (3), pp. 379–391.
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Second-order linear differential equations with two turning points.
Philos. Trans. Roy. Soc. London Ser. A 278, pp. 137–174.
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Improved error bounds for second-order differential equations with two turning points.
J. Res. Nat. Bur. Standards Sect. B 80B (4), pp. 437–440.
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Iterative Solution of Nonlinear Equations in Several Variables.
Academic Press, New York.
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13: 19.36 Methods of Computation
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►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18).
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19.36.2
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►Complex values of the variables are allowed, with some restrictions in the case of that are sufficient but not always necessary.
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►The incomplete integrals and can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to , accompanied by two quadratically convergent series in the case of ; compare Carlson (1965, §§5,6).
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►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20).
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14: Bibliography G
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The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals.
Ann. of Math. (2) 48 (4), pp. 785–826.
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Algorithm 819: AIZ, BIZ: two Fortran 77 routines for the computation of complex Airy functions.
ACM Trans. Math. Software 28 (3), pp. 325–336.
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Algorithm 822: GIZ, HIZ: two Fortran 77 routines for the computation of complex Scorer functions.
ACM Trans. Math. Software 28 (4), pp. 436–447.
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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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Mutual integrability, quadratic algebras, and dynamical symmetry.
Ann. Phys. 217 (1), pp. 1–20.
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