pseudo spectral methods

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1: 18.38 Mathematical Applications

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Differential Equations: Spectral Methods

►Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a method originated by Clenshaw (1957). This process has been generalized to spectral methods for solving partial differential equations. … ►

Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

… ►These methods have become known as pseudo-spectral, and are overviewed in Cerjan (1993), and Shizgal (2015). …

2: 27.19 Methods of Computation: Factorization

§27.19 Methods of Computation: Factorization

… ►Type I probabilistic algorithms include the Brent–Pollard rho algorithm (also called Monte Carlo method), the Pollard $p-1$ algorithm, and the Elliptic Curve Method (ecm). …As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard $p-1$, and a 67-digit prime for ecm. ►Type II probabilistic algorithms for factoring $n$ rely on finding a pseudo-random pair of integers $(x,y)$ that satisfy $x^2\equiv y^2(modn)$. …

3: Brian D. Sleeman

… ►Sleeman published numerous papers in applied analysis, multiparameter spectral theory, direct and inverse scattering theory, and mathematical medicine. He is author of the book Multiparameter spectral theory in Hilbert space, published by Pitman in 1978, and coauthor (with D. … ►Sleeman was elected a Fellow of the Royal Society of Edinburgh in 1976 and is the founding editor of the journal Computational and Mathematical Methods in Medicine. …

4: Daniel W. Lozier

… ►Then he transferred to NIST (then known as the National Bureau of Standards), where he collaborated for several years with the Building and Fire Research Laboratory developing and applying finite-difference and spectral methods to differential equation models of fire growth. … ►In 2008 he was named an Honorary Fellow of the European Society of Computational Methods in Sciences and Engineering, and in 2017 was named a Fellow of the Washington Academy of Sciences.

5: 31.17 Physical Applications

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§31.17(i) Addition of Three Quantum Spins

►The problem of adding three quantum spins $𝐬$, $𝐭$, and $𝐮$ can be solved by the method of separation of variables, and the solution is given in terms of a product of two Heun functions. … ►Consider the following spectral problem on the sphere $S_2$: ${𝐱}^2=x_s^2+x_t^2+x_u^2=R^2$. … ►For more details about the method of separation of variables and relation to special functions see Olevskiĭ (1950), Kalnins et al. (1976), Miller (1977), and Kalnins (1986). … ►For applications of Heun’s equation and functions in astrophysics see Debosscher (1998) where different spectral problems for Heun’s equation are also considered. …

6: 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, $x$. …

7: 23.5 Special Lattices

… ► $e_1$ and $g_3$ have the same sign unless $2{\omega }_3=(1+\mathrm{i}){\omega }_1$ when both are zero: the pseudo-lemniscatic case. As a function of $\mathrm{\Im }{e}_3$ the root $e_1$ is increasing. …

8: Bibliography R

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Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, Self-Adjointness. Academic Press, New York.

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External Links:

ISBN 978-0-12-585002-5, MathReview Entry

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§1.18(x).

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Methods of Modern Mathematical Physics, Vol. 4, Analysis of Operators. Academic Press, New York.

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External Links:

ISBN 0-12-585004-2(vol.4), MathReview (P. R. Chernoff)

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§1.18(x).

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Methods of Modern Mathematical Physics, Vol. 3, Scattering Theory. Academic Press, New York.

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External Links:

ISBN 0-12-585003-4 (vol.3), MathReview (P. R. Chernoff)

Referenced by:

§1.18(x).

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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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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

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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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External Links:

ISSN 1521-9615, Document, Link

Referenced by:

§18.39(iv), §18.40(ii).

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9: 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 an account of this method see Brink and Satchler (1993, Chapter VII). 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).

10: Bibliography O

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Asymptotic Enumeration Methods. In Handbook of Combinatorics, Vol. 2, L. Lovász, R. L. Graham, and M. Grötschel (Eds.), pp. 1063–1229.

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External Links:

ISBN 0-444-82351-4, Pdf, MathReview (Konrad Engel), ZentralBlatt

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Ch.2.

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Computing $\pi (x)$: The combinatorial method. Revista do DETUA 4 (6), pp. 759–768.

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Notes:

Online fulltext contains postpublication annotations.

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FullText

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§27.18.

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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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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

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§3.11(ii).

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A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations. Proc. Cambridge Philos. Soc. 46 (4), pp. 570–580.

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External Links:

Document, MathReview (J. Todd), ZentralBlatt

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§10.21(ii).

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Spectral decomposition of tent maps using symmetry considerations. J. Statist. Phys. 84 (1-2), pp. 269–276.

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External Links:

ISSN 0022-4715, Document, MathReview (Makoto Mori), ZentralBlatt

Referenced by:

§24.18.

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