# 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

…##### 3: 30.12 Generalized and Coulomb Spheroidal Functions

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►For the theory and computation of solutions of (30.12.1) see Falloon (2001), Judd (1975), Leaver (1986), and Komarov et al. (1976).
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##### 4: 3.2 Linear Algebra

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►When the factorization (3.2.5) is available, the accuracy of the computed solution
$\mathbf{x}$ can be improved with little extra computation.
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►Let ${\mathbf{x}}^{\ast}$ denote a computed solution of the system (3.2.1), with $\mathbf{r}=\mathbf{b}-\mathbf{A}{\mathbf{x}}^{\ast}$ again denoting the residual.
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##### 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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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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##### 6: 33.23 Methods of Computation

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►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.
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►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.
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##### 7: 3.6 Linear Difference Equations

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►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

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►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).
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##### 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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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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