Multiquadric collocation method is highly efficient for solving partial differential equations due to its exponential error convergence rate. More amazingly, there are two ways to reduce the error: the traditional way of refining the grid, and the unexpected way of simply increasing the value of shape constant $c$ contained in the multiquadric basis function, $sqrt{r^2 + c^2}$. The latter is accomplished without increasing computational cost. It has been speculated that in a numerical solution without roundoff error, infinite accuracy can be achieved by letting $c
ightarrow infty$. The ability to obtain infinitely accurate solution is limited only by the roundoff error induced instability of matrix solution with large condition number. Using the arbitrary precision computation capability of {it Mathematica}, this paper tests the above conjecture. A sharper error estimate than previously obtained is presented in this paper. A formula for a finite, optimal $c$ value that minimizes the solution error for a given grid size is obtained. Using residual errors, constants in error estimate and optimal $c$ formula can be obtained. These results are supported by numerical examples.
| Identifer | oai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0614106-030140 |
| Date | 14 June 2006 |
| Creators | Lee, Cheng-Feng |
| Contributors | Chien-Sen Huang, Zi-Cai Li, Leevan Ling, ALLEN T. L. HORNG, Tzon-Tzer Lu |
| Publisher | NSYSU |
| Source Sets | NSYSU Electronic Thesis and Dissertation Archive |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.lib.nsysu.edu.tw/ETD-db/ETD-search/view_etd?URN=etd-0614106-030140 |
| Rights | campus_withheld, Copyright information available at source archive |
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