On the Condition of the Linear Systems Associated with Discretized BVPs of ODEs

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Previous article Next article On the Condition of the Linear Systems Associated with Discretized BVPs of ODEsC. de Boor and H.-O. KreissC. de Boor and H.-O. Kreisshttps://doi.org/10.1137/0723061PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThe way in which improved conditioning can result when one rewrites an mth order ordinary differential equation as a first order vector system for numerical solution is explained.[A] U. Ascher, Collocation for two-point boundary value problems revisited, Technical Report, 84-17, Dept. Computer Science, Univ. British Columbia, Vancouver Google Scholar[APR] U. Ascher, , S. Pruess and , R. Russell, On spline basis selection for solving differential equations, SIAM J. Numer. Anal., 20 (1983), 121–142 10.1137/0720009 84h:65075 0525.65060 LinkISIGoogle Scholar[GG] John H. George and , Robert W. Gunderson, Conditioning of linear boundary value problems, Nordisk Tidskr. Informationsbehandling (BIT), 12 (1972), 172–181 46:8427 0243.65045 Google Scholar[K] Herbert B. Keller, Numerical solution of two point boundary value problems, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976viii+61, CBMS Regional Conference Series Applied Mathematics, 24 55:6868 LinkGoogle Scholar[O2A] Michael R. Osborne, On the numerical solution of boundary value problems for ordinary differential equationsInformation processing 74 (Proc. IFIP Congress, Stockholm, 1974), North-Holland, Amsterdam, 1974, 673–677 55:4693 0303.65074 Google Scholar[PR] John Paine and , Robert D. Russell, Conditioning of collocation matrices and discrete Green's functions, SIAM J. Numer. Anal., 23 (1986), 376–392 10.1137/0723025 88a:65086 0631.65077 LinkISIGoogle Scholar[O1] M. R. Osborne, J. J. H. Miller, Collocation, difference equations, and stitched function representationsTopics in numerical analysis, II (Proc. Roy. Irish Acad. Conf., Univ. College, Dublin, 1974), Academic Press, London, 1975, 121–132 53:14921 0357.65057 CrossrefGoogle Scholar[R] J. R. Rice, Numerical Methods, Software, and Analysis, McGraw-Hill, New York, 1983 Google Scholar[W] J. H. Wilkinson, Rounding errors in algebraic processes, Prentice-Hall Inc., Englewood Cliffs, N.J., 1963vi+161 28:4661 01024452 Google ScholarKeywordscondition numberfinite differencesODEBVP Previous article Next article FiguresRelatedReferencesCited byDetails An overview of the method of fundamental solutions—Solvability, uniqueness, convergence, and stabilityEngineering Analysis with Boundary Elements, Vol. 120 Cross Ref Efficient Block Preconditioning for a $C^1$ Finite Element Discretization of the Dirichlet Biharmonic ProblemJ. Pestana, R. Muddle, M. Heil, F. Tisseur, and M. Mihajlović28 January 2016 | SIAM Journal on Scientific Computing, Vol. 38, No. 1AbstractPDF (1572 KB)Partial eigenvalue assignment of high order systems with time delayLinear Algebra and its Applications, Vol. 438, No. 5 Cross Ref Numerical simulation of two-dimensional combustion using mesh-free methodsEngineering Analysis with Boundary Elements, Vol. 33, No. 7 Cross Ref Transformation of high order linear differential-algebraic systems to first order28 September 2006 | Numerical Algorithms, Vol. 42, No. 3-4 Cross Ref Matrix enlarging methods and their applicationBIT Numerical Mathematics, Vol. 37, No. 3 Cross Ref The conditioning of some numerical methods for first kind boundary integral equationsJournal of Computational and Applied Mathematics, Vol. 67, No. 1 Cross Ref A Calculus of Difference Schemes for the Solution of Boundary Value Problems on Irregular MeshesThomas A. Manteuffel and Andrew B. White, Jr.14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 29, No. 5AbstractPDF (1771 KB)A note on conditioning, stability and collocation matricesApplied Mathematics and Computation, Vol. 31 Cross Ref On the Conditioning of Multipoint and Integral Boundary Value ProblemsF. R. De Hoog and R. M. M. Mattheij17 February 2012 | SIAM Journal on Mathematical Analysis, Vol. 20, No. 1AbstractPDF (1049 KB)Effectively Well-Conditioned Linear SystemsTony F. Chan and David E. Foulser13 July 2006 | SIAM Journal on Scientific and Statistical Computing, Vol. 9, No. 6AbstractPDF (761 KB)Conditioning CollocationBlair Swartz14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 25, No. 1AbstractPDF (3114 KB) Volume 23, Issue 5| 1986SIAM Journal on Numerical Analysis History Submitted:25 July 1985Accepted:15 November 1985Published online:14 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsKeywordscondition numberfinite differencesODEBVPMSC codes65L1065F35PDF Download Article & Publication DataArticle DOI:10.1137/0723061Article page range:pp. 936-939ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics