Comments on: “A comparison of global methods for linear two-point boundary value problems” (Math. Comp. 29 (1975), no. 132, 1007–1019) by R. D. Russell and J. M. Varah

de Boor, Carl; Swartz, Blair

doi:10.1090/S0025-5718-1977-0501939-4

Comments on: “A comparison of global methods for linear two-point boundary value problems” (Math. Comp. 29 (1975), no. 132, 1007–1019) by R. D. Russell and J. M. Varah
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by Carl de Boor and Blair Swartz PDF

Math. Comp. 31 (1977), 916-921 Request permission

Abstract:

A more careful count of the operations involved in solving the linear system associated with collocation of a two-point boundary value problem using rough splines reverses results recently reported by others in this journal. In addition, it is observed that the use of the technique of "condensation of parameters" can decrease the computer storage required. Furthermore, the use of a particular highly localized basis can also reduce the setup time when the mesh is irregular. Finally, operation counts are roughly estimated for the solution of certain linear systems associated with two competing collocation methods; namely, collocation with smooth splines and collocation of the equivalent first order system with continuous piecewise polynomials.

References

R. D. Russell and J. M. Varah, A comparison of global methods for linear two-point boundary value problems, Math. Comput. 29 (1975), no. 132, 1007–1019. MR 0388788, DOI 10.1090/S0025-5718-1975-0388788-3
Carl de Boor and Blâir Swartz, Collocation at Gaussian points, SIAM J. Numer. Anal. 10 (1973), 582–606. MR 373328, DOI 10.1137/0710052

SOLVEBLOK

A Package for Solving Almost Block Diagonal Linear Systems

Robert D. Russell, Collocation for systems of boundary value problems, Numer. Math. 23 (1974), 119–133. MR 416074, DOI 10.1007/BF01459946
Richard Weiss, The application of implicit Runge-Kutta and collection methods to boundary-value problems, Math. Comp. 28 (1974), 449–464. MR 341881, DOI 10.1090/S0025-5718-1974-0341881-2
R. Russell, Efficiencies of $B$-spline methods for solving differential equations, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) Congressus Numerantium, No. XVI, Utilitas Math. Publ., Winnipeg, Man., 1976, pp. 599–617. MR 0405871
Robert D. Russell, A comparison of collocation and finite differences for two-point boundary value problems, SIAM J. Numer. Anal. 14 (1977), no. 1, 19–39. MR 451745, DOI 10.1137/0714003

Collocation for Systems of Ordinary Differential Equations

BIT