Kuniyoshi Abe

A variant algorithm of the ORTHOMiN(m) method for solving linear systems

Abstract We propose a variant of O rthomin (m) for solving linear systems A x = b . It is mathematically equivalent to the original O rthomin (m) method, but uses recurrence formulas that are different from those of O rthomin (m); they contain alternative expressions for the auxiliary vectors and the recurrence coefficients. Our implementation has the same computational costs as O rthomin (m). As a result of numerical experiments on nonsingular linear systems, we have confirmed the equivalence of our proposed variant of O rthomin (m) with the original O rthomin (m) using finite precision arithmetic; numerical experiments on singular linear systems show that our proposed algorithm is more accurate and less affected by rounding errors than the original O rthomin (m).

A variant of IDRstab with reliable update strategies for solving sparse linear systems

The IDRStab method is often more effective than the IDR(s) method and the BiCGstab(@?) method for solving large nonsymmetric linear systems. IDRStab can have a large so-called residual gap: the convergence of recursively computed residual norms does not coincide with that of explicitly computed residual norms because of the influence of rounding errors. We therefore propose an alternative recursion formula for updating the residuals to narrow the residual gap. The formula requires extra matrix-vector multiplications, but we reduce total computational costs by giving an alternative implementation which reduces the number of vector updates. Numerical experiments show that the alternative recursion formula reliably reduces the residual gap, and that our proposed variant of IDRStab is effective for sparse linear systems.

Hybrid Bi-CG methods with a Bi-CG formulation closer to the IDR approach

The Induced Dimension Reduction(s) (IDR(s)) method has recently been developed. Sleijpen et al. have reformulated the Bi-Conjugate Gradient STABilized (BiCGSTAB) method to clarify the relationship between BiCGSTAB and IDR(s). The formulation of Bi-Conjugate Gradient (Bi-CG) part used in the reformulated BiCGSTAB is different from that of the original Bi-CG method; the Bi-CG coefficients are computed by a formulation that is closer to the IDR approach. In this paper, we will redesign variants of the Conjugate Gradient Squared method (CGS) method, BiCGSTAB and the Generalized Product-type method derived from Bi-CG (GPBiCG)/BiCG MR2 by using the Bi-CG formulation that is closer to the IDR approach. Although our proposed variants are mathematically equivalent to their counterparts, the computation of one of the Bi-CG coefficients differs, and the recurrences of the variants are also partly different from those of the original hybrid Bi-CG methods. Numerical experiments show that the variants of BiCGSTAB and GPBiCG/BiCG MR2 are more stable and lead to faster convergence typically for linear systems for which the methods converge slowly (long stagnation phase), and that the variants of CGS attain more accurate approximate solutions.

Application of MRTR Method for Solving Complex Symmetric Linear Systems

The MRTR method has been recognized as an effective iterative method for singular systems of linear equations. The MRTR method is based on the three-term recurrence formula of the CG method and the algorithm is proven to be mathematically equivalent to the CR method. This paper will describe the algorithm of the cs_MRTR method for solving complex symmetric linear systems, and prove that this method is mathematically equivalent to the COCR method. Numerical experiments indicate that the cs_MRTR method convergences more stably compared with the COCR method.

An extended GS method for dense linear systems

Davey and Rosindale [K. Davey, I. Rosindale, An iterative solution scheme for systems of boundary element equations, Internat. J. Numer. Methods Engrg. 37 (1994) 1399-1411] derived the GSOR method, which uses an upper triangular matrix @W in order to solve dense linear systems. By applying functional analysis, the authors presented an expression for the optimum @W. Moreover, Davey and Bounds [K. Davey, S. Bounds, A generalized SOR method for dense linear systems of boundary element equations, SIAM J. Comput. 19 (1998) 953-967] also introduced further interesting results. In this note, we employ a matrix analysis approach to investigate these schemes, and derive theorems that compare these schemes with existing preconditioners for dense linear systems. We show that the convergence rate of the Gauss-Seidel method with preconditioner PG is superior to that of the GSOR method. Moreover, we define some splittings associated with the iterative schemes. Some numerical examples are reported to confirm the theoretical analysis. We show that the EGS method with preconditioner PG(@copt) produces an extremely small spectral radius in comparison with the other schemes considered.

Variant Implementations of SCBiCG Method for Linear Equations with Complex Symmetric Matrices

SCBiCG (Bi-Conjugate Gradient method for Symmetric Complex matrices) has been proposed for solving linear equations with complex symmetric matrices, where coefficients ci need to be set by users in SCBiCG. We have had the numerical results that the residual norms of SCBiCG do not converge when the coefficients ci are real. We therefore design an efficient implementation such that the coefficients ci which are complex are given by a computation. Numerical experiments show that the residual norms of our proposed variant with the complex coefficients ci converge slightly faster than those of COCG (Conjugate Orthogonal Conjugate Gradient method) and some implementations of SCBiCG.

A variable preconditioned GCR(m) method using the GSOR method for singular and rectangular linear systems

The Generalized Conjugate Residual (GCR) method with a variable preconditioning is an efficient method for solving a large sparse linear system Ax=b. It has been clarified by some numerical experiments that the Successive Over Relaxation (SOR) method is more effective than Krylov subspace methods such as GCR and ILU(0) preconditioned GCR for performing the variable preconditioning. However, SOR cannot be applied for performing the variable preconditioning when solving such linear systems that the coefficient matrix has diagonal entries of zero or is not square. Therefore, we propose a type of the generalized SOR (GSOR) method. By numerical experiments on the singular linear systems, we demonstrate that the variable preconditioned GCR using GSOR is effective.

Converting BiCR method for linear equations with complex symmetric matrices

Abstract 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.

Performance of Krylov Subspace Methods for Symmetric Matrices in Hybrid Parallelization

We deal with Krylov subspace methods such as the Conjugate Gradient (CG) method for solving linear equations with symmetric matrices on a parallel computer. The algorithm which has the less number of synchronization (abbreviated as synchro) points is crucial for reducing the communication time on the parallel computer. CG has two synchro points per iteration, and the AZMJ variant of Orthomin(2) (abbreviated as AZMJ) which has just one synchro point has been proposed. A number of strategies to generate preconditioners have been known for obtaining successful and rapid convergence. We apply the Symmetric Successive Over Relaxation (SSOR) preconditioner to their methods. Then extra computational costs are required and we need to compute the forward and back substitution in the preconditioned algorithm. We therefore propose an alternative SSOR splitting for the parallel computing, and a computation procedure to parallelize the forward and back substitution and to reduce the computational costs. The numerical results show that the convergence behavior of AZMJ is superior to that of CG, and the parallel performance of AZMJ, which has the less number of synchro points than CG, is higher using the hybrid parallelization on the parallel computer. AZMJ and CG with the preconditioner using our proposed procedure are efficient on the parallel computer, and are useful for obtaining rapid convergence.

On MrR (Mister R) Method for Solving Linear Equations with Symmetric Matrices

Krylov subspace methods, such as the Conjugate Gradient (CG) and Conjugate Residual (CR) methods, are treated for efficiently solving a linear system of equations with symmetric matrices. AZMJ variant of Orthomin(2) (abbreviated as AZMJ) [1] has recently been proposed for solving the linear equations. In this paper, an alternative AZMJ variant is redesigned, i.e., an alternative minimum residual method for symmetric matrices is proposed by using the coupled two-term recurrences formulated by Rutishauser. The recurrence coefficients are determined by imposing the A-orthogonality on the residuals as well as CR. Our proposed variant is referred to as MrR. It is mathematically equivalent to CR and AZMJ, but the implementations are different; the recurrence formulae contain alternative expressions for the auxiliary vectors and the recurrence coefficients. Through numerical experiments on the linear equations with real symmetric matrices, it is demonstrated that the residual norms of MrR converge faster than those of CG and AZMJ.