Extrapolated Iterative Methods for Linear Systems Approximation

In this note, we present results on extrapolated iterative methods, especially the extrapolated successive overrelaxation (ESOR). We show that they converge even if the original iteration diverges which increases considerably the scope of application of iterative schemes. The ESOR method is discussed under this aspect for consistently ordered systems and complex eigenvalues of the Jacobi iteration matrix. Comparison theorems are given to show that the ESOR is particularly useful if the SOR diverges or its optimum parameter $\omega _b $ cannot be determined; but even if $\omega _b $ is known the ESOR may be faster than the SOR method. Further insight into the structure of the method is obtained by relating it to Euler’s integration method.