DLMF: Untitled Document

… ►Numerical methods and issues for solution of (1.2.61) appear in §§3.2(i) to 3.2(iii). … ►

Eigenvectors and Eigenvalues of Square Matrices

… ►Eigenvalues are the roots of the polynomial equation …Numerical methods and issues for solution of (1.2.72) appear in §§3.2(iv) to 3.2(vii). … ►

Relation of Eigenvalues to the Determinant and Trace

…

… ► ►►►►

	Open Source	With Book			Commercial
…
28.36(ii) Exponents, Eigenvalues		✓		✓	✓	✓	✓	a
…
30.18(ii) Eigenvalues $\lambda^{m}_{n}\left(\gamma^{2}\right)$		✓	✓	✓			✓
…

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Research Software.

This is software of narrow scope developed as a byproduct of a research project and subsequently made available at no cost to the public. The software is often meant to demonstrate new numerical methods or software engineering strategies which were the subject of a research project. When developed, the software typically contains capabilities unavailable elsewhere. While the software may be quite capable, it is typically not professionally packaged and its use may require some expertise. The software is typically provided as source code or via a web-based service, and no support is provided.

…

… ►

J. C. Lagarias, V. S. Miller, and A. M. Odlyzko (1985) Computing

\pi(x)

: The Meissel-Lehmer method. Math. Comp. 44 (170), pp. 537–560.

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External Links:: ISSN 0025-5718, FullText, MathReview (Kenneth A. Jukes), ZentralBlatt
Referenced by:: §27.18.
Encodings:: BibTeX

… ►

T. M. Larsen, D. Erricolo, and P. L. E. Uslenghi (2009) New method to obtain small parameter power series expansions of Mathieu radial and angular functions. Math. Comp. 78 (265), pp. 255–274.

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External Links:: ISSN 0025-5718, Document, FullText, MathReview (Junesang Choi), ZentralBlatt
Referenced by:: §28.24, §28.24.
Encodings:: BibTeX

… ►

W. Lay, K. Bay, and S. Yu. Slavyanov (1998) Asymptotic and numeric study of eigenvalues of the double confluent Heun equation. J. Phys. A 31 (42), pp. 8521–8531.

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External Links:: ISSN 0305-4470, Document, MathReview, ZentralBlatt
Referenced by:: §31.13, §31.18.
Encodings:: BibTeX

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D. Le (1985) An efficient derivative-free method for solving nonlinear equations. ACM Trans. Math. Software 11 (3), pp. 250–262.

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Notes:: Corrigendum, ibid. 15(3) (1989), 287
External Links:: ISSN 0098-3500, Document, MathReview (T. Feagin), ZentralBlatt
Referenced by:: §3.8(iii).
Encodings:: BibTeX

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I. M. Longman (1956) Note on a method for computing infinite integrals of oscillatory functions. Proc. Cambridge Philos. Soc. 52 (4), pp. 764–768.

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External Links:: Document, MathReview (D. J. Hofsommer), ZentralBlatt
Referenced by:: §3.5(vii).
Encodings:: BibTeX

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E. A. Bender (1974) Asymptotic methods in enumeration. SIAM Rev. 16 (4), pp. 485–515.

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External Links:: Document, MathReview (Bernard Harris), ZentralBlatt
Referenced by:: Ch.2.
Encodings:: BibTeX

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M. V. Berry and J. P. Keating (1999) The Riemann zeros and eigenvalue asymptotics. SIAM Rev. 41 (2), pp. 236–266.

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External Links:: ISSN 1095-7200, Document, MathReview (Richard B. Paris), ZentralBlatt
Referenced by:: §25.17.
Encodings:: BibTeX

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Å. Björck (1996) Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:: ISBN 0-89871-360-9, MathReview (Hongyuan Zha), ZentralBlatt
Referenced by:: §3.11(v).
Encodings:: BibTeX

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G. Blanch (1966) Numerical aspects of Mathieu eigenvalues. Rend. Circ. Mat. Palermo (2) 15, pp. 51–97.

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External Links:: Document, MathReview (J. Meixner), ZentralBlatt
Referenced by:: item (e).
Encodings:: BibTeX

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W. G. C. Boyd (1995) Approximations for the late coefficients in asymptotic expansions arising in the method of steepest descents. Methods Appl. Anal. 2 (4), pp. 475–489.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Dan Polis̆evski), ZentralBlatt
Referenced by:: §2.4(iv).
Encodings:: BibTeX

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… ►A convenient way of constructing the coefficients, together with the eigenvalues, is as follows. Equations (29.6.4), with

p=1,2,\dots,n

, (29.6.3), and

A_{2n+2}=0

can be cast as an algebraic eigenvalue problem in the following way. …Let the eigenvalues of

\mathbf{M}

be

H_{p}

with … ►

29.15.7

a^{2m}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2}),

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Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : real parameter, $\nu$ : real parameter and $H_{p}$ : eigenvalues
Permalink:: http://dlmf.nist.gov/29.15.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

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29.15.22

a^{2m}_{\nu}\left(k^{2}\right)=\tfrac{1}{2}(H_{m}+\nu(\nu+1)k^{2}),

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Symbols:: $a^{\NVar{n}}_{\NVar{\nu}}\left(\NVar{k^{2}}\right)$ : eigenvalues of Lamé’s equation, $m$ : nonnegative integer, $k$ : real parameter, $\nu$ : real parameter and $H_{p}$ : eigenvalues
Permalink:: http://dlmf.nist.gov/29.15.E22
Encodings:: pMML, png, TeX
See also:: Annotations for §29.15(i), §29.15(i), §29.15 and Ch.29

…

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35.7.3

{{}_{2}F_{1}}\left({a,b\atop c};\begin{bmatrix}t_{1}&0\\ 0&t_{2}\end{bmatrix}\right)=\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{% \left(c-a\right)_{k}}{\left(b\right)_{k}}{\left(c-b\right)_{k}}}{k!\,{\left(c% \right)_{2k}}{\left(c-\tfrac{1}{2}\right)_{k}}}\*(t_{1}t_{2})^{k}{{}_{2}F_{1}}% \left({a+k,b+k\atop c+2k};t_{1}+t_{2}-t_{1}t_{2}\right).

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Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $a$ : complex variable, $t_{j}$ : eigenvalues of $\mathbf{T}$ , $b$ : complex variable, $c$ : complex variable and $k$ : nonnegative integer
Referenced by:: Erratum (V1.0.25) for Equation (35.7.3)
Permalink:: http://dlmf.nist.gov/35.7.E3
Encodings:: pMML, png, TeX
Notational change (effective with 1.0.25):: Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${{}_{2}F_{1}}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

… ►Let

f:{\boldsymbol{\Omega}}\to\mathbb{C}

(a) be orthogonally invariant, so that

f(\mathbf{T})

is a symmetric function of

t_{1},\dots,t_{m}

, the eigenvalues of the matrix argument

\mathbf{T}\in{\boldsymbol{\Omega}}

; (b) be analytic in

t_{1},\dots,t_{m}

in a neighborhood of

\mathbf{T}=\boldsymbol{{0}}

; (c) satisfy

f(\boldsymbol{{0}})=1

. … ►

35.7.9

t_{j}(1-t_{j})\frac{{\partial}^{2}F}{{\partial t_{j}}^{2}}-\frac{1}{2}\sum_{% \begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{m}\frac{t_{k}(1-t_{k})}{t_{j}-t_{k}}\frac{\partial F}{% \partial t_{k}}+\left({c-\tfrac{1}{2}(m-1)-\left(a+b-\tfrac{1}{2}(m-3)\right)t% _{j}}+\frac{1}{2}\sum_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{m}\frac{t_{j}(1-t_{j})}{t_{j}-t_{k}}\right)\frac{% \partial F}{\partial t_{j}}=abF,

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Symbols:: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $a$ : complex variable, $t_{j}$ : eigenvalues of $\mathbf{T}$ , $b$ : complex variable, $c$ : complex variable, $j$ : nonnegative integer, $k$ : nonnegative integer and $m$ : positive integer
Referenced by:: §35.7(iii)
Permalink:: http://dlmf.nist.gov/35.7.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(iii), §35.7 and Ch.35

… ►Butler and Wood (2002) applies Laplace’s method (§2.3(iii)) to (35.7.5) to derive uniform asymptotic approximations for the functions …

… ►

M. Reed and B. Simon (1975) Methods of Modern Mathematical Physics, Vol. 2, Fourier Analysis, Self-Adjointness. Academic Press, New York.

ⓘ

External Links:: ISBN 978-0-12-585002-5, MathReview Entry
Referenced by:: §1.18(x).
Encodings:: BibTeX

►

M. Reed and B. Simon (1978) 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)
Referenced by:: §1.18(x).
Encodings:: BibTeX

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M. Reed and B. Simon (1979) 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).
Encodings:: BibTeX

►

M. Reed and B. Simon (1980) Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis. Elsevier, New York.

ⓘ

Notes:: revised and expanded Edition from 1972,
External Links:: ISBN 10: 0-12-585050-6(vol. 1), MathReview Entry
Referenced by:: §1.18(x).
Encodings:: BibTeX

… ►

S. Ritter (1998) On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.

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External Links:: ISSN 0044-2267, Document, MathReview Entry, ZentralBlatt
Referenced by:: §29.20(ii).
Encodings:: BibTeX

…

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L. Infeld and T. E. Hull (1951) The factorization method. Rev. Modern Phys. 23 (1), pp. 21–68.

ⓘ

Referenced by:: §18.38(iii).
Encodings:: BibTeX

… ►

M. E. H. Ismail and E. Koelink (2011) The

J

-matrix method. Adv. in Appl. Math. 46 (1-4), pp. 379–395.

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External Links:: ISSN 0196-8858, Document, Link, MathReview (Boris Petrovich Osilenker)
Referenced by:: §18.39(iv).
Encodings:: BibTeX

… ►

A. R. Its, A. S. Fokas, and A. A. Kapaev (1994) On the asymptotic analysis of the Painlevé equations via the isomonodromy method. Nonlinearity 7 (5), pp. 1291–1325.

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External Links:: ISSN 0951-7715, Document, MathReview (Alexander Vladimirovich Kitaev), ZentralBlatt
Referenced by:: §32.11(iii).
Encodings:: BibTeX

… ►

A. R. Its and V. Yu. Novokshënov (1986) The Isomonodromic Deformation Method in the Theory of Painlevé Equations. Lecture Notes in Mathematics, Vol. 1191, Springer-Verlag, Berlin.

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External Links:: ISBN 3-540-16483-9, MathReview (Alexander A. Pankov), ZentralBlatt
Referenced by:: §32.11(iv), §32.4(i), Profile Alexander A. Its.
Encodings:: BibTeX

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C. Itzykson and J. Drouffe (1989) Statistical Field Theory: Strong Coupling, Monte Carlo Methods, Conformal Field Theory, and Random Systems. Vol. 2, Cambridge University Press, Cambridge.

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External Links:: ISBN 0-521-37012-4; 0-521-40806-7, MathReview (C. A. Hurst), ZentralBlatt
Referenced by:: 2nd item.
Encodings:: BibTeX

…

… ►

G. Valent (1986) An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

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External Links:: ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt
Referenced by:: §31.10(i).
Encodings:: BibTeX

… ►

H. Volkmer (1998) On the growth of convergence radii for the eigenvalues of the Mathieu equation. Math. Nachr. 192, pp. 239–253.

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External Links:: ISSN 0025-584X, Document, MathReview (M. E. Muldoon), ZentralBlatt
Referenced by:: §28.6(i).
Encodings:: BibTeX

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H. Volkmer (2004a) 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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External Links:: ISSN 0176-4276, Document, MathReview, ZentralBlatt
Referenced by:: §30.16(i), §30.16(ii), §30.16(ii).
Encodings:: BibTeX

►

H. Volkmer (2004b) Four remarks on eigenvalues of Lamé’s equation. Anal. Appl. (Singap.) 2 (2), pp. 161–175.

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External Links:: ISSN 0219-5305, Document, MathReview, ZentralBlatt
Referenced by:: §29.3(ii), §29.3(vii), §29.5, §29.7(i), §29.7(i).
Encodings:: BibTeX

►

H. Volkmer (2008) Approximation of eigenvalues of some differential equations by zeros of orthogonal polynomials. J. Comput. Appl. Math. 213 (2), pp. 488–500.

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External Links:: ISSN 0377-0427, Document, FullText, MathReview (Javier Segura), ZentralBlatt
Referenced by:: item (f), §28.34(ii), §29.20(i).
Encodings:: BibTeX

…

… ►

P. L. Walker (2009) The distribution of the zeros of Jacobian elliptic functions with respect to the parameter

k

. Comput. Methods Funct. Theory 9 (2), pp. 579–591.

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External Links:: ISSN 1617-9447, MathReview (M. K. Jain), ZentralBlatt
Referenced by:: §22.4(i).
Encodings:: BibTeX

… ►

J. A. Wheeler (1937) Wave functions for large arguments by the amplitude-phase method. Phys. Rev. 52, pp. 1123–1127.

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External Links:: Document, ZentralBlatt
Referenced by:: §33.5(ii).
Encodings:: BibTeX

… ►

J. H. Wilkinson (1988) The Algebraic Eigenvalue Problem. Monographs on Numerical Analysis. Oxford Science Publications, The Clarendon Press, Oxford University Press, Oxford.

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Notes:: Paperback reprint of original edition, published in 1965 by Clarendon Press, Oxford.
External Links:: ISBN 0-19-853418-3, MathReview, ZentralBlatt
Referenced by:: §3.2(v), §3.2(vi), §3.2(vii).
Encodings:: BibTeX

… ►

R. Wong and M. Wyman (1974) The method of Darboux. J. Approximation Theory 10 (2), pp. 159–171.

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External Links:: Document, MathReview (M. E. Noble), ZentralBlatt
Referenced by:: §2.10(iv).
Encodings:: BibTeX

… ►

R. Wong and Y. Zhao (2005) On a uniform treatment of Darboux’s method. Constr. Approx. 21 (2), pp. 225–255.

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External Links:: ISSN 0176-4276, Document, MathReview Entry, ZentralBlatt
Referenced by:: §2.10(iv).
Encodings:: BibTeX

…

eigenvalue methods

11—20 of 21 matching pages

11: 1.2 Elementary Algebra

Eigenvectors and Eigenvalues of Square Matrices

Relation of Eigenvalues to the Determinant and Trace

12: Software Index

13: Bibliography L

14: Bibliography B

15: 29.15 Fourier Series and Chebyshev Series

16: 35.7 Gaussian Hypergeometric Function of Matrix Argument

17: Bibliography R

18: Bibliography I

19: Bibliography V

20: Bibliography W