Methods of conjugate gradients for solving linear systems

TLDR

General algorithms for solving linear systems are essentially methods for finding an n‑dimensional ellipsoid, with connections to orthogonal polynomials and continued fractions. The study introduces an iterative algorithm for solving the linear system Ax = k. The algorithm iteratively solves the system in n steps. The algorithm is shown to be a special case of a very general method that includes Gaussian elimination.

Abstract

An iterative algorithm is given for solving a system Ax=k of n linear equations in n unknowns. The solution is given in n steps. It is shown that this method is a special case of a very general method which also includes Gaussian elimination. These general algorithms are essentially algorithms for finding an n dimensional ellipsoid. Connections are made with the theory of orthogonal polynomials and continued fractions.