In this article we give a general methodology to build and work with functional networks, a network-based alternative to the neural networks paradigm. In functional networks, neural functions are allowed to be not only multivariate but also truly multiargument and different for all neurons. Thus neural functions instead of weights are learned. In addition, outputs coming from different neurons can be connected, that is, forced to output the same values. The topology and neuron functions of functional networks can be selected based on data, domain knowledge, or a combination of the two. Functional equations play an important role in functional networks, since the preceding types of connections lead to functional equations that impose a substantial reduction in the degrees of freedom of the initial neural functions. Some methods are given to obtain equivalent functional and differential equations, and they are applied to approximating the solutions of differential equations problems. The examples of an associative operator, a cantilever beam, and a mass supported by two springs and a viscous damper are given to illustrate the methods and show their power.

In this paper we discuss the power of a pivoting transformation introduced by Castillo, Cobo, Jubete, and Pruneda [Orthogonal Sets and Polar Methods in Linear Algebra: Applications to Matrix Calculations, Systems of Equations and Inequalities, and Linear Programming, John Wiley, New York, 1999] and its multiple applications. The meaning of each sequential tableau appearing during the pivoting process is interpreted. It is shown that each tableau of the process corresponds to the inverse of a row modified matrix and contains the generators of the linear subspace orthogonal to a set of vectors and its complement. This transformation, which is based on the orthogonality concept, allows us to solve many problems of linear algebra, such as calculating the inverse and the determinant of a matrix, updating the inverse or the determinant of a matrix after changing a row (column), determining the rank of a matrix, determining whether or not a set of vectors is linearly independent, obtaining the intersection of tw...