Array Algebra Expansion of Matrix and Tensor Calculus: Part 2

Abstract

Part 1 of this paper summarized some extended matrix and tensor operators of the multilinear array algebra and loop inverse estimation. It also introduced the compact shorthand tensor notations of array polynomials for expressing the Taylor expansion of a nonlinear function and its least squares normal equation. These matrix-like operators will be further expanded using the nonlinear tensor transforms of Blaha's Q-surface technique. They enable an analytical derivation of the direct (one hyper iteration) solution of nonlinear systems. New operators are found such as the nonlinear counterpart of the Lm- or general pseudoinverse and neural net operators. The increased pull-in range and rate of convergence of the new technique are applied in some standard examples of nonlinear equations. Some applications of the nonlinear array algebra involving literally billions of parameters are discussed, including the automated stereo mensuration of three-dimensionaldigital imaging and mapping systems.