eigenvalue methods

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1: 3.8 Nonlinear Equations

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Eigenvalue Methods

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2: 28.34 Methods of Computation

… ►Methods for computing the eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$, and ${\lambda }_{\nu }\left(q\right)$, defined in §§28.2(v) and 28.12(i), include: … ►

Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

… ►Also, once the eigenvalues $a_n\left(q\right)$, $b_n\left(q\right)$, and ${\lambda }_{\nu }\left(q\right)$ have been computed the following methods are applicable: …

3: 29.20 Methods of Computation

… ►The eigenvalues $a_{\nu }^m\left(k^2\right)$, $b_{\nu }^m\left(k^2\right)$, and the Lamé functions ${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$, ${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$, can be calculated by direct numerical methods applied to the differential equation (29.2.1); see §3.7. … ►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). … ►The eigenvalues corresponding to Lamé polynomials are computed from eigenvalues of the finite tridiagonal matrices $𝐌$ given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998). …

4: Bibliography D

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Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.

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External Links:

ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt

Referenced by:

§31.17(ii).

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5: 3.2 Linear Algebra

… ►Many methods are available for computing eigenvalues; see Golub and Van Loan (1996, Chapters 7, 8), Trefethen and Bau (1997, Chapter 5), and Wilkinson (1988, Chapters 8, 9).

6: 30.16 Methods of Computation

… ►The eigenvalues of $𝐀$ can be computed by methods indicated in §§3.2(vi), 3.2(vii). …

7: 12.16 Mathematical Applications

… ►Sleeman (1968b) considers certain orthogonality properties of the PCFs and corresponding eigenvalues. In Brazel et al. (1992) exponential asymptotics are considered in connection with an eigenvalue problem involving PCFs. …

8: 3.7 Ordinary Differential Equations

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§3.7(ii) Taylor-Series Method: Initial-Value Problems

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§3.7(iii) Taylor-Series Method: Boundary-Value Problems

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§3.7(iv) Sturm–Liouville Eigenvalue Problems

… ►The Sturm–Liouville eigenvalue problem is the construction of a nontrivial solution of the system … ►

§3.7(v) Runge–Kutta Method

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9: 18.39 Applications in the Physical Sciences

… ►By Table 18.3.1#12 the normalized stationary states and corresponding eigenvalues are … ►The orthonormal stationary states and corresponding eigenvalues are then of the form …and the corresponding eigenvalues are … ►with eigenvalues … ►

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

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10: Bibliography S

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A global Newton method for the zeros of cylinder functions. Numer. Algorithms 18 (3-4), pp. 259–276.

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ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

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The computation of eigenvalues and solutions of Mathieu’s differential equation for noninteger order. ACM Trans. Math. Software 19 (3), pp. 377–390.

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ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

item (e).

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Practical Extrapolation Methods: Theory and Applications. Cambridge Monographs on Applied and Computational Mathematics, Vol. 10, Cambridge University Press, Cambridge.

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ISBN 0-521-66159-5, MathReview (Thomas A. Atchison), ZentralBlatt

Referenced by:

§3.9(vi).

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Large orders and summability of eigenvalue perturbation theory: A mathematical overview. Int. J. Quantum Chem. 21, pp. 3–25.

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Document

Referenced by:

§22.19(ii).

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Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§31.13.

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