Advances in Civil Engineering

Volume 2019, Article ID 7527590, 5 pages

https://doi.org/10.1155/2019/7527590

## Nonrecursive Equivalent of the Conjugate Gradient Method without the Need to Restart

^{1}University of Zagreb, Faculty of Civil Engineering, Kaciceva 26, Zagreb 10 000, Croatia^{2}University of Zagreb, Faculty of Mining, Geology and Petroleum Engineering, Pierottijeva 6, Zagreb 10 000, Croatia

Correspondence should be addressed to Damir Lazarevic; rh.darg@rimad

Received 18 January 2019; Accepted 24 March 2019; Published 11 April 2019

Academic Editor: Dimitris Rizos

Copyright © 2019 Josip Dvornik et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A simple alternative to the conjugate gradient (CG) method is presented; this method is developed as a special case of the more general iterated Ritz method (IRM) for solving a system of linear equations. This novel algorithm is not based on conjugacy; i.e., it is not necessary to maintain overall orthogonalities between various vectors from distant steps. This method is more stable than CG, and restarting techniques are not required. As in CG, only one matrix-vector multiplication is required per step with appropriate transformations. The algorithm is easily explained by energy considerations without appealing to the -orthogonality in *n*-dimensional space. Finally, relaxation factor and preconditioning-like techniques can be adopted easily.

#### 1. Introduction

Letbe a real linear system with a symmetric positive definite (SPD) matrix of order *n*. By IRM, the solution is sought through successive minimisation of the corresponding energy functional, or the quadratic forminside a small subspace formed at each iteration step [1]. After the convergence criterion is reached, a solution is found that is close to the unique minimiser of . Geometrically, this is the point close to the centre of the hyperellipsoids , where c are arbitrary real constants.

#### 2. Briefly about IRM

The main idea here is to present the solution increment by the discretised Ritz method:where is a matrix of linearly independent coordinate vectors and is the vector of corresponding coefficients. The energy decrement associated with (3) can also be expressed as the quadratic function:where and are the SPD generalised (Ritz) matrix and the generalised residual vector, respectively, and both terms are of order *m*. After minimising (4), we obtain the system of equations that should be solved at each step:

The solution is used to find the increment in (3), and is updated afterwards. The residual is defined in a standard manner as .

Obviously, IRM represents an iterative procedure, where a discrete Ritz method is applied at each step and a suitable set of coordinate vectors which span a subspace are generated. A local energy minimum is sought within that subspace (therefore, equation (5) should be solved at each step), thereby decreasing the total energy of the system, which eventually converges to the required minimum. The subspace dimension, or the size of (5), is not limited. Rather, it aims to be small, much smaller than the number of unknowns (), because every iteration must be as fast as possible. Such a small system (though is usually full) can be solved by any direct method. Simple pseudocode, with input data and sequence of instructions common for the iterative solution methods, is given by Algorithm 1.