The optimal ω is not best for the SOR iteration method

Abstract

The SOR iteration method is popular for solving many of the large sparse systems of linear algebraic equations which are used to approximate many of the partial differential equations which arise in engineering. We consider the matrix equation Au= w when David Young's SOR theory is applicable (for example, when the matrix A is a symmetric, block tridiagonal matrix). It is well known that using the “optimal relaxation factor” ω b produces the smallest possible spectral radius of the associated SOR iteration matrix ( L ω). It is also known that no polynomial acceleration can reduce the size of the spectral radius when the relaxation factor ω= ω b is used. Therefore it has been assumed that it is best to use the combination of the unaccelerated SOR iteration together with the “optimal relaxation factor” ω b . (The spectral radius is ω b −1.) But we show that a smaller average spectral radius can be achieved by using a polynomial acceleration together with a suboptimal relaxation factor ( ω< ω b ).