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Nodal spline error analysis and applications in quadrature
Thesis (Ph. D.) -- University of Stellenbosch, 1996.
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| Format: | Thesis |
| Language: | English |
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Stellenbosch : Stellenbosch University
2012
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| _version_ | 1867613910068101120 |
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| access_status_str | Open Access |
| author | De Swardt, Susan Antoinette |
| author2 | De Villiers, J. M. |
| author_browse | De Swardt, Susan Antoinette De Villiers, J. M. |
| author_facet | De Villiers, J. M. De Swardt, Susan Antoinette |
| author_sort | De Swardt, Susan Antoinette |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (Ph. D.) -- University of Stellenbosch, 1996. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/55190 |
| institution | Stellenbosch University (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:43:38.086Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2012 |
| publishDateRange | 2012 |
| publishDateSort | 2012 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/55190 Nodal spline error analysis and applications in quadrature De Swardt, Susan Antoinette De Villiers, J. M. Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Interpolation Spline theory Quadratic fields Dissertations -- Mathematics Thesis (Ph. D.) -- University of Stellenbosch, 1996. In this thesis we consider the approximation technique of nodal spline interpolation, as first introduced in the quadratic case on the real line by Rohwer [29], after which, in a sequence of five papers[13, 14, 15, 16, 17], and as summarized in our Chapter 1, de Villiers and Rohwer further extended and developed the subject. In Chapter 2, we specialize to the case of quadratic nodal splines, and proceed by means of a Peano kernel technique to estimate, for arbitrary knot sequences on a bounded interval [a,b], the interpolation error explicitly in terms of the mesh norm H and local mesh ratio R. The resulting Jackson-type error estimates in Theorem 2.11 are a clear improvement on those in[17], where a method based on sharp estimates for the Lebesgue constant (i.e. the operator norm) of the quadratic nodal spline operator was used instead. In [16], de Villiers showed that, in the case of uniform knots, the definite integral over [a,b] of the nodal spline interpolant yields precisely the Gregory rule of even order, for which he then proceeded to deduce, in particular for the order two case, i.e. the Lacroix rule, non-optimal order error estimates. Here, in Chapter 3, we define a generalized Gregory rule as the definite integral over [a,b] of the nodal spline interpolant on an arbitrary knot sequence, yielding an integration rule with arbitrary knot spacing. We then show how the results of Chapter 2 can immediately be employed to deduce, for the quadratic case, and in particular (if the knots are uniform) for the Lacroix rule, error estimates explicitly in terms of the mesh parameters H and R. Next, we show how an application of a Peano kernel theorem for quadrature error analysis yields error bounds for the quadratic nodal spline quadrature rule which, for values of R which are not too large, and in particular including the uniform knot case R = 1, are an improvement on the abovementioned ones (as obtained from Chapter 2). The resulting non-optimal order error estimates in Corolly 3.12 for the Lacroix rule can now be shown to compare favourably with the analogous estimates for the Simpson rule, as derived by Stroud in [33]. As a numerical illustration of the above quadrature error bounds, we introduce, in Section 3.6, examples of partitions of [a,b], and proceed to demonstrate that the theoretical convergence orders are indeed achieved for the integrands f which we choose. As a further application of nodal spline interpolation in quadrature, we define, in Chapter 4, an arbitrary order nodal spline quadrature rule for product integration. The error analysis of Chapter 2 is then immediately used to establish the expected convergence orders. Next, in Section 4.4, following Alaylioglu, Lubinksy and Eyre[1], we introduce and analyze a quadrature rule for product integration based on the socalled "not-a-knot" cubic spline interpolant. Specifically the quadratic nodal spline quadrature rule for product integrals is then compared with this "not-a-knot" rule, followed by some numerical illustrations. Finally, we consider the case of product integration in the presence of either jump discontinuities or an infinite singularity, for which convergence results have recently been established by Rabinowitz [25,27]. In particular, for the case of an infinite singularity, we define, following [25], a modified nodal spline quadrature rule for product integration, after which we state and prove (as was done independently in [27]) sufficient conditions for convergence. Numerical examples are then provided. Doctoral 2012-08-27T11:36:56Z 2012-08-27T11:36:56Z 1996 Thesis http://hdl.handle.net/10019.1/55190 en Stellenbosch University 124 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Interpolation Spline theory Quadratic fields Dissertations -- Mathematics De Swardt, Susan Antoinette Nodal spline error analysis and applications in quadrature |
| title | Nodal spline error analysis and applications in quadrature |
| title_full | Nodal spline error analysis and applications in quadrature |
| title_fullStr | Nodal spline error analysis and applications in quadrature |
| title_full_unstemmed | Nodal spline error analysis and applications in quadrature |
| title_short | Nodal spline error analysis and applications in quadrature |
| title_sort | nodal spline error analysis and applications in quadrature |
| topic | Interpolation Spline theory Quadratic fields Dissertations -- Mathematics |
| url | http://hdl.handle.net/10019.1/55190 |
| work_keys_str_mv | AT deswardtsusanantoinette nodalsplineerroranalysisandapplicationsinquadrature |