numerical evaluation

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1: 3 Numerical Methods

Chapter 3 Numerical Methods

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2: 36.15 Methods of Computation

… ►Direct numerical evaluation can be carried out along a contour that runs along the segment of the real $t$-axis containing all real critical points of $\mathrm{\Phi }$ and is deformed outside this range so as to reach infinity along the asymptotic valleys of $\exp \left(\mathrm{i}\mathrm{\Phi }\right)$. … ►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\mathrm{\Phi }$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …

3: 3.10 Continued Fractions

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§3.10(iii) Numerical Evaluation of Continued Fractions

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Forward Recurrence Algorithm

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Backward Recurrence Algorithm

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Forward Series Recurrence Algorithm

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4: Bibliography N

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Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes. ACM Trans. Math. Software 18 (3), pp. 345–349.

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Notes:

Double-precision Fortran, minimum accuracy 9S, maximum accuracy 13S.

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ISSN 0098-3500, Document, GAMS (module 707; package TOMS), ZentralBlatt

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1st item.

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Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes. J. Comput. Appl. Math. 39 (2), pp. 193–200.

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Notes:

Extended-precision Fortran evaluator, minimum accuracy 9S, maximum accuracy 13S.

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ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§13.32(iii).

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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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ISSN 0010-4655, Document, ZentralBlatt

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§25.18(i).

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5: Bibliography T

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On the numerical evaluation of the modified Bessel function of the third kind. J. Comput. Phys. 19 (3), pp. 324–337.

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Notes:

Algol program.

External Links:

Document, MathReview (Carl C. Grosjean), ZentralBlatt

Referenced by:

§10.74(i), §10.74(iv), §10.77(iii).

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On the numerical evaluation of the ordinary Bessel function of the second kind. J. Computational Phys. 21 (3), pp. 343–350.

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Notes:

Algol program

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ISSN 0021-9991, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.77(iii).

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Numerical evaluation of exponential integral: Theis well function approximation. Journal of Hydrology 205 (1-2), pp. 38–51.

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Document

Referenced by:

§6.18(i).

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6: 20.14 Methods of Computation

… ►In practice a value with, say, $\mathrm{\Im }\tau \ge 1/2$, $\left|q\right|\le 0.2$, is found quickly and is satisfactory for numerical evaluation.

7: Bibliography C

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Numerical evaluation of the Lambert $W$ function and application to generation of generalized Gaussian noise with exponent 1/2. IEEE Trans. Signal Process. 50 (9), pp. 2160–2165.

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ISSN 1053-587X, Document, MathReview (J. Borwein), ZentralBlatt

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§4.44, §4.45(iii).

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Numerical evaluation of the Fermi-Dirac integrals. The Astrophysical Journal Supplement Series 71, pp. 677–699.

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Notes:

Double-precision Fortran, maximum precision 12D.

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ISSN 0067-0049, Document, Code

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§25.12(iii), §25.18(i), 3rd item, 3rd item.

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A differential equation method for the numerical evaluation of the Airy, Pearcey and swallowtail canonical integrals and their derivatives. Molecular Phys. 48 (6), pp. 1305–1330.

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ISSN 0026-8976, Document, MathReview (Peter Giblin)

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§36.10(i), §36.10(ii), §36.15(v), §36.8.

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A method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes: Application to Pearcey’s integral and its derivatives. J. Phys. A 15 (4), pp. 1179–1190.

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ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§36.15(iii), §36.8.

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Numerical evaluation of Airy functions with complex arguments. J. Comput. Phys. 99 (1), pp. 106–114.

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ISSN 0021-9991, Document, MathReview, ZentralBlatt

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4th item, 1st item.

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8: Bibliography L

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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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ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§19.36(iii), §19.5.

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Numerical evaluation of integrals containing a spherical Bessel function by product integration. J. Math. Phys. 22 (7), pp. 1399–1413.

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ISSN 0022-2488, Document, MathReview (C. W. Clenshaw), ZentralBlatt

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§10.74(vii).

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Numerical Evaluation of Special Functions. In Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics (Vancouver, BC, 1993), Proc. Sympos. Appl. Math., Vol. 48, pp. 79–125.

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Notes:

An updated version exists online.

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FullText, MathReview (John P. Coleman), ZentralBlatt

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Ch.3.

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9: 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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External Links:

FullText, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§3.5(viii).

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Numerical evaluation of integrals of the form $I={\int }_{x_1}^{x_2}f(x)e^{i\varphi (x)}𝑑x$ and the tabulation of the function $\mathrm{Gi}(z)=(1/\pi ){\int }_0^{\mathrm{\infty }}\sin (uz+\frac{1}{3}{u}^3)𝑑u$ . Quart. J. Mech. Appl. Math. 3 (1), pp. 107–112.

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Document, MathReview (S. C. van Veen), ZentralBlatt

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1st item.

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Numerical evaluation of the Hankel transform. Comput. Phys. Comm. 116 (2-3), pp. 278–294.

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External Links:

ISSN 0010-4655, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii).

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Evaluation of associated Legendre functions off the cut and parabolic cylinder functions. Electron. Trans. Numer. Anal. 9, pp. 137–146.

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Notes:

Orthogonal polynomials: numerical and symbolic algorithms (Leganés, 1998)

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ISSN 1068-9613, FullText, MathReview (Adam Marlewski), ZentralBlatt

Referenced by:

§14.30(i), 3rd item.

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Numerical evaluation of spherical Bessel transforms via fast Fourier transforms. J. Comput. Phys. 100 (2), pp. 294–296.

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ISSN 0021-9991, Document, MathReview, ZentralBlatt

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§10.74(vii).

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10: Bibliography P

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Evaluation of Fermi-Dirac Integral. In Nonlinear Numerical Methods and Rational Approximation (Wilrijk, 1987), A. Cuyt (Ed.), Mathematics and Its Applications, Vol. 43, pp. 435–444.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§25.18(i).

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A numerical evaluator for the generalized hypergeometric series. Comput. Phys. Comm. 77 (2), pp. 249–254.

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Notes:

Extended-precision Fortran, maximum accuracy 12D.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

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