Operator Coefficient Methods for Linear Equations

Abstract: New iterative methods for solving linear equations are presented that are easy to use, generalize good existing methods, and appear to be faster. The new algorithms mix two kinds of linear recurrence formulas. Older methods have either high order recurrence formulas with scalars for coefficients, as in truncated orthomin, or have 1st order recurrence formulas with matrix polynomials for coefficients, as in restarted gcr/gmres. The new methods include both: high order recurrence formulas and matrix polynomials for coefficients. These methods provide a trade-off between recurrence order and polynomial degree that can be exploited to achieve greater efficiency. Convergence results are obtained for both constant coefficient and varying coefficient methods.