Block Gaussian Decomposition-Arbitrary Precision Arithmetic Applied To Accurate Solutions of Ill Conditioned Elliptic Partial Differential Equations

Continuously differentiable radial basis functions (C∞-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill-conditioned. Similarly, the Hilbert and Vandermonde systems involve full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C∞-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C∞-RBFs may vary are very sensitive to small changes in the location of data and evaluation centers, shape parameters. affect the condition number and solution accuracy. This study demonstrates that a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors.