Converting BiCR method for linear equations with complex symmetric matrices

The Bi-Conjugate Gradient (BiCG) method for Symmetric Complex matrices (SCBiCG), which can be derived from BiCG, has been proposed for solving linear equations with complex symmetric matrices. However, an alternative method derived from the Bi-Conjugate Residual (BiCR) method for complex symmetric matrices has not previously been proposed. We therefore design BiCR for Symmetric Complex matrices (SCBiCR) by using the same analogy as that discussed in SCBiCG. Coefficients ci with real number defined in SCBiCG need to be set by users before starting the iteration, and we have had the numerical results, with several combinations when the coefficients ci are real, that the residual norms of SCBiCG do not converge. We therefore design an alternative implementation such that the coefficients ci can be complex and are appropriately determined at each step of the algorithm. We give the preconditioned algorithms. Moreover, the factor in the loss of convergence speed is analyzed to clarify the difference of convergence between SCBiCG and our proposed SCBiCR. Numerical experiments demonstrate that the residual norms of our proposed variant with the complex coefficients ci converge fairly faster than those of the Conjugate Orthogonal Conjugate Gradient (COCG) method and several implementations of SCBiCG.