An improved Wilkinson algorithm for solving linear equations

An improved Wilkinson iterative algorithm for solving linear equations is proposed. An amendment factor is introduced to reduce the condition number of the coefficient matrix of linear equations. An automatic step size is adopted to estimate the local error and change the step size correspondingly. The convergence was proved for 5500-order Hilbert linear equations solved using the improved algorithm combined with the amended conjugate gradient method. The relative error for the solution was less than 1.6%. The numerical results demonstrate that this new iterative algorithm is superior to other methods such as the amended conjugate gradient, the Wilkinson iterative algorithm and the improved algorithm described by Wu and Fang [Appl. Math. Comput. 193 (2007) 506-513]. The new algorithm is more applicable for solving ill-conditioned linear equations.