Ritz Vectors-Based Deflation Preconditioner for Linear System with Multiple Right-Hand Sides

Computational mathematics have many tools to solve the large systems of equations which may be linear or nonlinear. Iterative methods are used to solve the nonsymmetric definite system of linear equations like Krylov methods. Linear systems with multiple right-hand sides find application in many areas of engineering and science. Considering generality and indefiniteness, Krylov subspace methods are frequently used for such problems. However, problem with system with multiple right-hand side vectors requires constructing subspace for every right-hand side. GMRES produces Ritz vectors, approximation to eigenvectors during iterations. These Ritz values, recycled vectors are used in Krylov solve while solving system with second and subsequent right-hand side vectors. This is applied as a deflation preconditioner to GMRES. The numerical results show that the computational time, residuals and number of iterations is reduced as compare to simple GMRES. Deflation technique with Ritz vectors is not expensive as compare to the GMRES and as well as exact eigenvectors deflations.