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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
§28.34(iii) Floquet Solutions
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 denote a computed solution of the system (3.2.1), with 𝐫 = 𝐛 𝐀 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 ρ 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 ( γ 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.