Interactive solvers for large, dense matrices

Eowyn Wilhelmina Cenek, University of Southern Mississippi

Abstract

Stochastic Interpolation (SI) uses a continuous, centrally symmetric probability distribution function to interpolate a given set of data points, and splits the interpolation operator into a discrete deconvolution followed by a discrete convolution of the data. The method is particularly effective for large data sets, as it does not suffer from the problem of oversampling, where too many data points cause the interpolating function to oscillate wildly. Rather, the interpolation improves every time more data points are added. The method relies on the inversion of relatively large, dense matrices to solve A nn x = b for x . Based on the probability distribution function chosen, the matrix Ann may have specific properties that make the problem of solving for x tractable. The iterative Shulz Jones Mayer (SJM) method relies on an initial guess, which is iterated to approximate A nn -1 nn . We present initial guesses that are guaranteed to converge quadratically for several classes of matrices, including diagonally and tri-diagonally dominant matrices and the structured matrices we encounter in the implementation of SI. We improve the method, creating the Polynomial Shulz Jones Mayer method, and take advantage of the more efficient matrix operations possible for Toeplitz matrices. We calculate error bounds and use those to improve the method's accuracy, resulting in a method requiring [Special characters omitted.] O ( nlogn ) operations that returns x with double precision. The use of SI and PSJM is illustrated in interpolating functions and images in grey scale and color.