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21: 28.34 Methods of Computation

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Solution of the matrix eigenvalue problem for each of the five infinite matrices that correspond to the linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4). See Zhang and Jin (1996, pp. 479–482) and §3.2(iv).

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Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

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22: Bibliography W

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Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94 (1), pp. 147–194.

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External Links:

ISSN 0029-599X, Document, MathReview (Sergey M. Zagorodnyuk), ZentralBlatt

Referenced by:

§2.9(iii).

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Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.

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External Links:

ISSN 0025-5718, Document, MathReview (B. G. Pachpatte), ZentralBlatt

Referenced by:

§2.9(iii).

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Linear Turning Point Theory. Applied Mathematical Sciences No. 54, Springer-Verlag, New York.

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External Links:

ISBN 0-387-96046-5, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

Ch.9.

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Linear and Nonlinear Waves. John Wiley & Sons, New York.

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Notes:

Reprinted in 1999.

External Links:

ISBN 0471359424, MathReview, ZentralBlatt

Referenced by:

§9.16.

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Asymptotics of linear recurrences. Anal. Appl. (Singap.) 12 (4), pp. 463–484.

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External Links:

ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§2.9(iii), §2.9(iii).

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23: 1.1 Special Notation

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$x,y$	real variables.
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$ℒ$	linear operator defined on a manifold $ℳ$
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24: 7.21 Physical Applications

… ►Fried and Conte (1961) mentions the role of $w\left(z\right)$ in the theory of linearized waves or oscillations in a hot plasma; $w\left(z\right)$ is called the plasma dispersion function or Faddeeva (or Faddeyeva) function; see Faddeeva and Terent’ev (1954). …

25: Bibliography Z

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Weighted derangements and the linearization coefficients of orthogonal Sheffer polynomials. Proc. London Math. Soc. (3) 65 (1), pp. 1–22.

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ISSN 0024-6115, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

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Error bounds for asymptotic solutions of second-order linear difference equations. J. Comput. Appl. Math. 71 (2), pp. 191–212.

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ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§2.9(i), §2.9(ii).

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26: 1.3 Determinants, Linear Operators, and Spectral Expansions

§1.3 Determinants, Linear Operators, and Spectral Expansions

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§1.3(iv) Matrices as Linear Operators

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Linear Operators in Finite Dimensional Vector Spaces

►Square matices can be seen as linear operators because $𝐀(\alpha 𝐚+\beta 𝐛)=\alpha 𝐀𝐚+\beta 𝐀𝐛$ for all $\alpha ,\beta \in ℂ$ and $𝐚,𝐛\in {𝐄}_n$, the space of all $n$-dimensional vectors. …

27: 2.9 Difference Equations

… ►Many special functions that depend on parameters satisfy a three-term linear recurrence relation …or equivalently the second-order homogeneous linear difference equation … ►This situation is analogous to second-order homogeneous linear differential equations with an irregular singularity of rank 1 at infinity (§2.7(ii)). … ►For discussions of turning points, transition points, and uniform asymptotic expansions for solutions of linear difference equations of the second order see Wang and Wong (2003, 2005). ►For an introduction to, and references for, the general asymptotic theory of linear difference equations of arbitrary order, see Wimp (1984, Appendix B). …

28: Bibliography I

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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of $J_0(z)-i{J}_1(z)$ and of Bessel functions $J_m(z)$ of any real order $m$ . Linear Algebra Appl. 194, pp. 35–70.

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ISSN 0024-3795, Document, MathReview (Alan L. Andrew), ZentralBlatt

Referenced by:

§10.21(ix).

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Quasi-linear Stokes phenomenon for the second Painlevé transcendent. Nonlinearity 16 (1), pp. 363–386.

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External Links:

ISSN 0951-7715, Document, MathReview (Youmin Lu), ZentralBlatt

Referenced by:

§32.12(ii).

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29: Philip J. Davis

… ►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. …

30: Bibliography G

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The solution of Cauchy’s problem for two totally hyperbolic linear differential equations by means of Riesz integrals. Ann. of Math. (2) 48 (4), pp. 785–826.

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External Links:

FullText, MathReview (E. T. Copson), ZentralBlatt

Referenced by:

§35.2, §35.3(ii).

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The Computation of Special Functions by Linear Difference Equations. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 213–243.

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External Links:

ISBN 90-5699-521-9, MathReview, ZentralBlatt

Referenced by:

§3.6(iii), §3.6(vii).

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Computing the zeros and turning points of solutions of second order homogeneous linear ODEs. SIAM J. Numer. Anal. 41 (3), pp. 827–855.

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ISSN 0036-1429, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vi), §3.8(v).

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Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

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External Links:

ISBN 0-8176-3837-7, MathReview, ZentralBlatt

Referenced by:

§15.17(v).

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Kinetic theory of linear shear flow. Phys. Fluids 1 (3), pp. 215–224.

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Referenced by:

§18.39(iii).

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