computation of solutions

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1: 31.18 Methods of Computation

§31.18 Methods of Computation

►Independent solutions of (31.2.1) can be computed in the neighborhoods of singularities from their Fuchs–Frobenius expansions (§31.3), and elsewhere by numerical integration of (31.2.1). …

2: 28.34 Methods of Computation

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§28.34(iii) Floquet Solutions

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3: 30.12 Generalized and Coulomb Spheroidal Functions

… ►For the theory and computation of solutions of (30.12.1) see Falloon (2001), Judd (1975), Leaver (1986), and Komarov et al. (1976). …

4: 3.2 Linear Algebra

… ►When the factorization (3.2.5) is available, the accuracy of the computed solution $𝐱$ can be improved with little extra computation. … ►Let ${𝐱}^{\ast }$ denote a computed solution of the system (3.2.1), with $𝐫=𝐛-𝐀{𝐱}^{\ast }$ again denoting the residual. …

5: Bibliography G

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Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments. ACM Trans. Math. Software 30 (2), pp. 145–158.

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External Links:

ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(viii).

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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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External Links:

ISSN 0036-1429, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vi), §3.8(v).

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BibTeX

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6: 33.23 Methods of Computation

… ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii $\rho$ and $r$, respectively, and may be used to compute the regular and irregular solutions. … ►Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. …

7: 3.6 Linear Difference Equations

… ►In this situation the unwanted multiples of $g_n$ grow more rapidly than the wanted solution, and the computations are unstable. … ►A “trial solution” is then computed by backward recursion, in the course of which the original components of the unwanted solution $g_n$ die away. … ►See also Gautschi (1967) and Gil et al. (2007a, Chapter 4) for the computation of recessive solutions via continued fractions. … ►It is applicable equally to the computation of the recessive solution of the homogeneous equation (3.6.3) or the computation of any solution $w_n$ of the inhomogeneous equation (3.6.1) for which the conditions of §3.6(iv) are satisfied. … ►We first compute, by forward recurrence, the solution $p_n$ of the homogeneous equation (3.6.3) with initial values $p_0=0$, $p_1=1$. …

8: 3.8 Nonlinear Equations

§3.8 Nonlinear Equations

… ►Corresponding numerical factors in this example for other zeros and other values of $j$ are obtained in Gautschi (1984, §4). …

9: 30.16 Methods of Computation

… ►The coefficients $a_{n,r}^m({\gamma }^2)$ are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5). …

10: Bibliography S

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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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External Links:

ISSN 0098-3500, Document, ZentralBlatt

Referenced by:

item (e).

Encodings:

BibTeX

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Algorithm 721: MTIEU1 and MTIEU2: Two subroutines to compute eigenvalues and solutions to Mathieu’s differential equation for noninteger and integer order. ACM Trans. Math. Software 19 (3), pp. 391–406.

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Notes:

Double-precision Fortran, maximum accuracy 18D

External Links:

ISSN 0098-3500, Document, GAMS (module 721; package TOMS), ZentralBlatt

Referenced by:

3rd item, 5th item.

Encodings:

BibTeX

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