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

Authors:
Carl de Boor and Blair Swartz

Journal:
Math. Comp. **31** (1977), 916-921

MSC:
Primary 65L10

DOI:
https://doi.org/10.1090/S0025-5718-1977-0501939-4

Original Article:
Math. Comp. **29** (1975), 1007-1019.

MathSciNet review:
0501939

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**R. D. RUSSELL & J. M. VARAH, "A comparison of global methods for linear two-point boundary value problems,"*Math. Comp.*, v. 29, 1975, pp. 1007-1019. MR**52**#9622. MR**0388788 (52:9622)****[2]**C. DE BOOR & B. SWARTZ, "Collocation at Gaussian points,"*SIAM J. Numer. Anal.*, v. 10, 1973, pp. 582-606. MR**51**#9528. MR**0373328 (51:9528)****[3]**C. DE BOOR & R. WEISS,*SOLVEBLOK*:*A Package for Solving Almost Block Diagonal Linear Systems*, MRC Tech. Report #1625, Mathematics Research Center, University of Wisconsin, Madison, Wisconsin, 1976.**[4]**R. D. RUSSELL, "Collocation for systems of boundary value problems,"*Numer. Math.*, v. 23, 1974, pp. 119-133. MR**0416074 (54:4150)****[5]**R. WEISS, "The application of implicit Runge-Kutta and collocation methods to boundary-value problems,"*Math. Comp.*, v. 28, 1974, pp. 449-464. MR**49**#6627. MR**0341881 (49:6627)****[6]**R. RUSSELL, "Efficiences of B-spline methods for solving differential equations,"*Proc.*1975*Manitoba Conference on Num. Math. and Computing*, pp. 599-617. MR**0405871 (53:9663)****[7]**R. D. RUSSELL, "A comparison of collocation and finite differences for two-point boundary value problems,"*SIAM J. Numer. Anal.*, v. 14, 1977, pp. 19-39. MR**0451745 (56:10027)****[8]**J. H. CERUTTI,*Collocation for Systems of Ordinary Differential Equations*, Tech. Report No. 230, Dept. of Computer Sciences, University of Wisconsin, Madison, Wisconsin, 1974; to appear in*BIT*.

Retrieve articles in *Mathematics of Computation*
with MSC:
65L10

Retrieve articles in all journals with MSC: 65L10

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1977-0501939-4

Article copyright:
© Copyright 1977
American Mathematical Society