18,107 research outputs found
Conjugate gradient type methods for linear systems with complex symmetric coefficient matrices
We consider conjugate gradient type methods for the solution of large sparse linear system Ax equals b with complex symmetric coefficient matrices A equals A(T). Such linear systems arise in important applications, such as the numerical solution of the complex Helmholtz equation. Furthermore, most complex non-Hermitian linear systems which occur in practice are actually complex symmetric. We investigate conjugate gradient type iterations which are based on a variant of the nonsymmetric Lanczos algorithm for complex symmetric matrices. We propose a new approach with iterates defined by a quasi-minimal residual property. The resulting algorithm presents several advantages over the standard biconjugate gradient method. We also include some remarks on the obvious approach to general complex linear systems by solving equivalent real linear systems for the real and imaginary parts of x. Finally, numerical experiments for linear systems arising from the complex Helmholtz equation are reported
Preconditioning complex symmetric linear systems
A new polynomial preconditioner for symmetric complex linear systems based on
Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear
systems is herein presented. It applies to Conjugate Orthogonal Conjugate
Gradient (COCG) or Conjugate Orthogonal Conjugate Residual (COCR) iterative
solvers and does not require any estimation of the spectrum of the coefficient
matrix. An upper bound of the condition number of the preconditioned linear
system is provided. Moreover, to reduce the computational cost, an inexact
variant based on incomplete Cholesky decomposition or orthogonal polynomials is
proposed. Numerical results show that the present preconditioner and its
inexact variant are efficient and robust solvers for this class of linear
systems. A stability analysis of the method completes the description of the
preconditioner.Comment: 26 pages, 4 figures, 4 table
A biconjugate gradient type algorithm on massively parallel architectures
The biconjugate gradient (BCG) method is the natural generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. Recently, Freund and Nachtigal have proposed a novel BCG type approach, the quasi-minimal residual method (QMR), which overcomes the problems of BCG. Here, an implementation is presented of QMR based on an s-step version of the nonsymmetric look-ahead Lanczos algorithm. The main feature of the s-step Lanczos algorithm is that, in general, all inner products, except for one, can be computed in parallel at the end of each block; this is unlike the other standard Lanczos process where inner products are generated sequentially. The resulting implementation of QMR is particularly attractive on massively parallel SIMD architectures, such as the Connection Machine
On choice of preconditioner for minimum residual methods for nonsymmetric matrices
Existing convergence bounds for Krylov subspace methods such as GMRES for nonsymmetric linear systems give little mathematical guidance for the choice of preconditioner. Here, we establish a desirable mathematical property of a preconditioner which guarantees that convergence of a minimum residual method will essentially depend only on the eigenvalues of the preconditioned system, as is true in the symmetric case. Our theory covers only a subset of nonsymmetric coefficient matrices but computations indicate that it might be more generally applicable
Characterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations
In this paper we study multivariate polynomial functions in complex variables
and the corresponding associated symmetric tensor representations. The focus is
on finding conditions under which such complex polynomials/tensors always take
real values. We introduce the notion of symmetric conjugate forms and general
conjugate forms, and present characteristic conditions for such complex
polynomials to be real-valued. As applications of our results, we discuss the
relation between nonnegative polynomials and sums of squares in the context of
complex polynomials. Moreover, new notions of eigenvalues/eigenvectors for
complex tensors are introduced, extending properties from the Hermitian
matrices. Finally, we discuss an important property for symmetric tensors,
which states that the largest absolute value of eigenvalue of a symmetric real
tensor is equal to its largest singular value; the result is known as Banach's
theorem. We show that a similar result holds in the complex case as well
Conditioning analysis of block incomplete factorization and its application to elliptic equations
The paper deals with eigenvalue estimates for block incomplete fac- torization methods for symmetric matrices. First, some previous results on upper bounds for the maximum eigenvalue of preconditioned matrices are generalized to each eigenvalue. Second, upper bounds for the maximum eigenvalue of the preconditioned matrix are further estimated, which presents a substantial im- provement of earlier results. Finally, the results are used to estimate bounds for every eigenvalue of the preconditioned matrices, in particular, for the maximum eigenvalue, when a modified block incomplete factorization is used to solve an elliptic equation with variable coefficients in two dimensions. The analysis yields a new upper bound of type Ī³hā1 for the condition number of the preconditioned matrix and shows clearly how the coefficients of the differential equation influ- ence the positive constant Ī³
QMR: A Quasi-Minimal Residual method for non-Hermitian linear systems
The biconjugate gradient (BCG) method is the natural generalization of the classical conjugate gradient algorithm for Hermitian positive definite matrices to general non-Hermitian linear systems. Unfortunately, the original BCG algorithm is susceptible to possible breakdowns and numerical instabilities. A novel BCG like approach is presented called the quasi-minimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a look-ahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported
- ā¦