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Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method. (English) Zbl 1195.65176
Summary: Multiquadric (MQ) collocation method is highly efficient for solving partial differential equations due to its exponential error convergence rate. A special feature of the method is that error can be reduced by increasing the value of shape constant $$c$$ in the MQ basis function, without refining the grid. It is believed that in a numerical solution without roundoff error, infinite accuracy can be achieved by letting $$c\rightarrow \infty$$. Using the arbitrary precision computation, this paper tests the above conjecture. A sharper error estimate than previously obtained is presented. 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.

##### MSC:
 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
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