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1: 34.9 Graphical Method

§34.9 Graphical Method

►The graphical method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression. …For specific examples of the graphical method of representing sums involving the $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

2: 10.74 Methods of Computation

… ►Methods for obtaining initial approximations to the zeros include asymptotic expansions (§§10.21(vi)-10.21(ix)), graphical intersection of $2D$ graphs in $ℝ$ (e. …

3: 18.40 Methods of Computation

§18.40 Methods of Computation

… ►Usually, however, other methods are more efficient, especially the numerical solution of difference equations (§3.6) and the application of uniform asymptotic expansions (when available) for OP’s of large degree. … ►

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Figure 18.40.1: Histogram approximations to the Repulsive Coulomb–Pollaczek, RCP, weight function integrated over $[-1,x)$, see Figure 18.39.2 for an exact result, for $Z=+1$, shown for $N=12$ and $N=120$. Magnify

… ►Results similar to these appear in Langhoff et al. (1976) in methods developed for physics applications, and which includes treatments of systems with discontinuities in $\mu (x)$, using what is referred to as the Stieltjes derivative which may be traced back to Stieltjes, as discussed by Deltour (1968, Eq. 12). … ►Further, exponential convergence in $N$, 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 $w(x)$ for these OP systems on $x\in [-1,1]$ and $(-\mathrm{\infty },\mathrm{\infty })$ respectively, Reinhardt (2018), and Reinhardt (2021b), Reinhardt (2021a). …

4: 27.17 Other Applications

… ►Congruences are used in constructing perpetual calendars, splicing telephone cables, scheduling round-robin tournaments, devising systematic methods for storing computer files, and generating pseudorandom numbers. … ►Schroeder (2006) describes many of these applications, including the design of concert hall ceilings to scatter sound into broad lateral patterns for improved acoustic quality, precise measurements of delays of radar echoes from Venus and Mercury to confirm one of the relativistic effects predicted by Einstein’s theory of general relativity, and the use of primes in creating artistic graphical designs.

5: Philip J. Davis

… ►Davis joined the Section as part of a distinguished group of researchers studying mathematical methods for exploiting the new computational power. … ►In 1961, Davis hired Frank W. J. Olver, a founding member of the Mathematics Division and Head of the Numerical Methods Section at the National Physical Laboratory, Teddington, U. … ►After receiving an overview of the project and watching a short demo that included a few preliminary colorful, but static, 3D graphs constructed for the first Chapter, “Airy and Related Functions”, written by Olver, Davis expressed the hope that designing a web-based resource would allow the team to incorporate interesting computer graphics, such as function surfaces that could be rotated and examined. This immediately led to discussions among some of the project members about what might be possible, and the discovery that some interactive graphics work had already been done for the NIST Matrix Market, a publicly available repository of test matrices for comparing the effectiveness of numerical linear algebra algorithms. ►Davis’s comments about our uninspired graphs sparked the research and design of techniques for creating interactive 3D visualizations of function surfaces, which grew in sophistication as our knowledge and the technology for developing 3D graphics on the web advanced over the years. …

6: Bibliography K

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Numerical Methods and Software. Prentice Hall, Englewood Cliffs, N.J..

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Notes:

Includes collection of single- and double-precision Fortran subroutines.

External Links:

ISBN 0-13-627258-4, GAMS (package NMS), ZentralBlatt

Referenced by:

NMS (free collection of Fortran subroutines).

Encodings:

BibTeX

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Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.

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External Links:

Document, MathReview (Matti Jutila), ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

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The Rayleigh function: Theory and computational methods. Zh. Vychisl. Mat. Mat. Fiz. 39 (12), pp. 1962–2006.

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Notes:

Translation in Comput. Math. Math. Phys. 39 (1999), No. 12, pp. 1883–1925.

External Links:

ISSN 0044-4669, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.21(xiii).

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An indirect method for evaluating certain infinite integrals. Z. Angew. Math. Phys. 29 (3), pp. 380–386.

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External Links:

ISSN 0044-2275, Document, MathReview, ZentralBlatt

Referenced by:

§10.71(ii).

Encodings:

BibTeX

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Equivalence of two graphical calculi. J. Phys. A 25 (22), pp. 6005–6026.

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External Links:

ISSN 0305-4470, Document, MathReview (Niky Kamran), ZentralBlatt

Referenced by:

§31.17(i).

Encodings:

BibTeX

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7: Mathematical Introduction

… ►In addition, there is a comprehensive account of the great variety of analytical methods that are used for deriving and applying the extremely important asymptotic properties of the special functions, including double asymptotic properties (Chapter 2 and §§10.41(iv), 10.41(v)). … ►

Graphics

►Special functions with one real variable are depicted graphically with conventional two-dimensional (2D) line graphs. … ►All of the special function chapters contain sections that describe available methods for computing the main functions in the chapter, and most also provide references to numerical tables of, and approximations for, these functions. …

8: 9.19 Approximations

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Martín et al. (1992) provides two simple formulas for approximating $\mathrm{Ai}\left(x\right)$ to graphical accuracy, one for , the other for .

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Corless et al. (1992) describe a method of approximation based on subdividing $ℂ$ into a triangular mesh, with values of $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$ stored at the nodes. $\mathrm{Ai}\left(z\right)$ and ${\mathrm{Ai}}^{\prime }\left(z\right)$ are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$ at the node. Similarly for $\mathrm{Bi}\left(z\right)$, ${\mathrm{Bi}}^{\prime }\left(z\right)$.

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9: 3.11 Approximation Techniques

… ►A method for obtaining a sufficiently accurate first approximation is described in the next subsection. … ►For details and examples of these methods, see Clenshaw (1957, 1962) and Miller (1966). … ►The method of the fast Fourier transform (FFT) exploits the structure of the matrix $𝛀$ with elements ${\omega }_n^{jk}$, $j,k=0,1,\mathrm{\dots },n-1$. … ►In computer graphics a special type of spline is used which produces a Bézier curve. …

10: 18.39 Applications in the Physical Sciences

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§18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences

►Shizgal (2015) gives a broad overview of techniques and applications of spectral and pseudo-spectral methods to problems arising in theoretical chemistry, chemical kinetics, transport theory, and astrophysics. … ►Shizgal (2015, Chapter 2), contains a broad-ranged discussion of methods and applications for these, and other, non-classical weight functions. … ►

§18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods

… ►The technique to accomplish this follows the DVR idea, in which methods are based on finding tridiagonal representations of the co-ordinate, $x$. …