# 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: …
• (f)

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 $a^{m}_{\nu}\left(k^{2}\right)$, $b^{m}_{\nu}\left(k^{2}\right)$, and the Lamé functions $\mathit{Ec}^{m}_{\nu}\left(z,k^{2}\right)$, $\mathit{Es}^{m}_{\nu}\left(z,k^{2}\right)$, 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 $\mathbf{M}$ given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998). …
##### 4: Bibliography D
• A. Debosscher (1998) Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.
• ##### 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 $\mathbf{A}$ 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
###### §3.7(iv) Sturm–Liouville Eigenvalue Problems
The Sturm–Liouville eigenvalue problem is the construction of a nontrivial solution of the system …
##### 9: Bibliography S
• J. Segura (1998) A global Newton method for the zeros of cylinder functions. Numer. Algorithms 18 (3-4), pp. 259–276.
• R. B. Shirts (1993a) The computation of eigenvalues and solutions of Mathieu’s differential equation for noninteger order. ACM Trans. Math. Software 19 (3), pp. 377–390.
• A. Sidi (2003) Practical Extrapolation Methods: Theory and Applications. Cambridge Monographs on Applied and Computational Mathematics, Vol. 10, Cambridge University Press, Cambridge.
• B. Simon (1982) Large orders and summability of eigenvalue perturbation theory: A mathematical overview. Int. J. Quantum Chem. 21, pp. 3–25.
• S. Yu. Slavyanov and N. A. Veshev (1997) Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.
• ##### 10: Software Index
 Open Source With Book Commercial … 28.36(ii) Exponents, Eigenvalues ✓ ✓ ✓ ✓ ✓ a … 30.18(ii) Eigenvalues $\lambda^{m}_{n}\left(\gamma^{2}\right)$ ✓ ✓ ✓ ✓ …
• 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.