Chapter 4 - Direct algorithms of solution of linear equations

Abstract

Chapter 4 presents Direct Algorithms of Solution of Linear Equations. The chapter firstly introduces several theoretical tools for the solution of linear equations, pseudo inverse, minimum norm/residual solutions, solutions with linear constraints. Then it presents the algorithms of 5 special structured matrices: zero matrix, diagonal matrix, orthogonal matrix and lower/upper triangular matrix. It follows by 4 elimination algorithms: Gauss and Gauss-Jordan eliminations by non-orthogonal transformations, Householder and Givens eliminations by orthogonal transformations. It continues with decomposing algorithms that utilize all the matrix decomposition algorithms of Direct Algorithms of Decompositions of Matrices by Non-orthogonal/Orthogonal Transformations. Lastly it presents two special algorithms arising from the interpolation problems: Vandermonde matrix and Fast Fourier Transform.