Resolución de sistemas lineales y no lineales

Abstract

[en] The problem of solving systems of equations is one of the oldest and with more applications in a variety of situations. A first example is finding a curve to which belong $n$ fixed points $(x_{i}, y_{i}), 1\leq i \leq n$. For this, $P n$ the usual procedure consists in searching for a polynomial of degree $n, p(z) = \sum^{n}_{i=0} ai\cdot z_{i}$ , such that $p(x_{i}) = y_{i}$. Imposing these conditions we find a linear system such that its solutions yield the values of a i , so that this polynomial will be the desired curve. Another field in which the problem is used is in digital signal processing, an area in engineering dedicated to the analysis and processing of signals (audio, voice, image or video). The problem also appears in structural analysis, the resolution of equations of material endurance to find internal strain, deformation and internal tensions that happen in a given structure. Finally, where it is more frequently found is in the field of linear programming and non-linear problem approximation.

For all the above, we want to study different methods for solving equation systems. In the first section we show some preliminary questions related to the topic that will be needed in the sequel. In the second section we present some the more frequent methods, such as the direct methods (to which the gaussian methods belong), iterative methods and Krylov methods. Next we study the efficiency of these methods by analysing some of the most important aspects such as the error, running time and number of iterations needed for iterative and Krylov methods, all by the use of some examples. In the case of iterative methods we see how the spectral radius is a useful tool for these methods, specially for convergence issues. This is why in the third section some methods for finding eigenvalues are presented, the most important of which is the maximum modulus method. In the fourth section we present some of the most frequent methods for non-linear systems, such as Newton’s methods for various variables or the continuation method. Finally, by the use of the results we have obtained, we reach various conclusions about the variety of methods studied in this piece of work.