# 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). …
##### 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 $\mathbf{x}$ can be improved with little extra computation. … Let $\mathbf{x}^{*}$ denote a computed solution of the system (3.2.1), with $\mathbf{r}=\mathbf{b}-\mathbf{A}\mathbf{x}^{*}$ again denoting the residual. …
##### 5: Bibliography G
• A. Gil, J. Segura, and N. M. Temme (2004b) Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments. ACM Trans. Math. Software 30 (2), pp. 145–158.
• A. Gil and J. Segura (2003) Computing the zeros and turning points of solutions of second order homogeneous linear ODEs. SIAM J. Numer. Anal. 41 (3), pp. 827–855.
• ##### 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 $a^{m}_{n,r}(\gamma^{2})$ are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5). …
##### 10: Bibliography S
• 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.
• R. B. Shirts (1993b) 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.