We theoretically prove that this method converges to the unique solution of the system of linear equations. Is the product of symmetric positive semidefinite matrices positive definite. On a class of nonhermitian matrices with positive definite schur complements authors. Let h be any complex inner product space with inner product. Part of themathematics commons this dissertation is brought to you for free and open access by the iowa state university capstones, theses and dissertations at iowa state university.
New hermitian and skewhermitian splitting methods for non. Convergence properties of preconditioned hermitian and. These methods include a hermitianskewhermitian splitting hss iteration and its inexact variant, the inexact hermitianskew. Extensions to certain nonhermitian matrices are discussed in 12,14,24,25,26.
You have to be careful about what you mean by positive semidefinite in the case of nonhermitian matrices. On the convergence of a new splitting iterative method for. Hermitian matrices are fundamental to the quantum theory of matrix mechanics created by werner heisenberg, max born, and pascual jordan in 1925 examples. Linear systems, splitting method, nonhermitian matrix, positive definite. By further generalizing the concept of hermitian or normal and skewhermitian splitting for a nonhermitian and positivedefinite matrix, we introduce a new splitting, called positivedefinite and skewhermitian splitting pss, and then establish a class of pss methods similar to the hermitian or normal and skewhermitian splitting hss or nss method for iteratively solving the positive. This line of code is the only part im not sure about, i have a feeling that mathematica doesnt determine the eigensystem of nonhermitians properly. The solutions of many problems in scientific computing are eventually turned into the solutions of the large linear systems, that is, axb 1 where is a large sparse nonhermitian positive definite matrix, and. For the positive semidefinite system of linear equations of a block twobytwo structure, by making use of the hermitianskewhermitian splitting iteration technique we establish a class of preconditioned hermitianskewhermitian splitting iteration methods. Inexact version of the method which employs conjugate gradient as its inner process is presented. Convergence properties of the preconditioned hss methods 5 theorem 3. Note that as its a symmetric matrix all the eigenvalues are real, so it makes sense to talk about them being positive or negative.
A matrix is positive definite fxtax ofor all vectors x 0. Buy your school textbooks, materials and every day products here. A considerable part of the research in numerical linear algebra since the midseventies has been devoted to generalizations of cg to indefinite and nonhermitian matrices see e. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Furthermore, x denotes the vector whose components is the absolute value of the corresponding components of x and. But avoid asking for help, clarification, or responding to other answers. Berman and plemmons l and wang and bai 3 gave some sufficient. In order to solve system by iterative methods, usually, efficient splittings of the coefficient matrix a are required.
Also if anyone knows if mathematica actually does properly determine these quantities of nonhermitian matrix automatically when running eigensystem please let me know. In this section, the conjugate transpose of matrix is denoted as, the transpose of matrix is denoted as. Citeseerx document details isaac councill, lee giles, pradeep teregowda. In general, if h has an unbroken pt symmetry, then it has another symmetry represented by a linear operator c. Does nonsymmetric positive definite matrix have positive. If aand bare both hermitian, we write a b a b if and only if a. Symmetric matrices are good their eigenvalues are real and each has a com plete set of. Nonstationary multisplittings with general weighting. By assumption, a has full pivots, so it is non singular. Crossref qinqin shen, yang cao and li wang, two improvements of the deteriorated pss preconditioner for generalized saddle point problems, numerical algorithms, 10.
A note on the modified hermitian and skewhermitian. For example, the classic jacobi and gaussseidel iterations. Accelerated circulant and skew circulant splitting methods. Hermitian positive definite system of linear equations when its coefficient matrix. Cholesky decomposition in linear algebra, the cholesky decomposition or cholesky factorization is a decomposition of a hermitian, positivedefinite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e. Frequently in physics the energy of a system in state x is represented as. Pdf on a class of nonhermitian matrices with positive. The following properties are equivalent to m being positive definite. We present sufficient conditions for the convergent splitting of a nonhermitian positive definite matrix. Positivedefinite matrix wikipedia republished wiki 2. It should be noted that the quasioptimal parameters of the pss iteration method preconditioner have been studied in 9, 10 when the coefficient matrix is nonhermitian positive definite. Theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system.
On the positive definite solutions of a nonlinear matrix equation liu, panpan, zhang, shugong, and li, qingchun, journal of applied mathematics, 20 the reflexive and hermitian reflexive solutions of the generalized sylvesterconjugate matrix equation hajarian, masoud and dehghan, mehdi, bulletin of the belgian mathematical society simon. Hermitian, positive definite matrices page 2 physics. Modified parallel multisplitting iterative methods the aim of this paper is to investigate the convergent parallel multisplitting iterative algorithms for the nonhermitian positive definite systems of linear equations. For example, the classic jacobi and gaussseidel iteration split the matrix into its diagonal and offdiagonal parts. A possible method to investigate nonhermitian hamiltonians is suggested through finding a hermitian operator. Hermitian positive definite system of linear equations when its coefficient matrix possesses either a dominant hermitian part or a dominant skew. Recall that b is assumed to be hermitian and positive definite. In this paper, a generalization of the positivedefinite and skewhermitian splitting gpss iteration is considered for solving nonhermitian and positive definite systems of linear equations. Xueyuan tan, shifted ssorlike preconditioner for nonhermitian positive definite matrices, numerical algorithms, 75, 1, 245, 2017. Nonhermitian positive definite matrices, pregular splitting, convergence, sor methods, precon ditioned gmres. Preconditioned hermitian and skewhermitian splitting. These results are applicable to identify the convergence of.
As an example, a nonhermitian and nonptsymmetric hamiltonian with imaginary linear coordinate and. The thing about positive definite matrices is xtax is always positive, for any nonzerovector x, not just for an eigenvector. An efficient hermitian and skewhermitian splitting method is presented for solving nonhermitian and normal positive definite linear systems with strong hermitian parts. Recall that a hermitian matrix is called positive or positive semidefinite if. By making use of the standard quadratic optimization technique we choose the optimal weighting matrices at each step. Hermitian matrices we conclude this section with an observation that has important implications for algorithms that approximate eigenvalues of very large hermitian matrix a with those of the small matrix h q.
Citeseerx hermitian and skewhermitian splitting methods. In this case i think what you mean is that all eigenvalues are positive or nonnegative. Let be a hermitian positive definite toeplitz matrix, and let be its hermitian positive circulant and skew circulant parts, be a nonnegative constant, and be a positive constant. The tts iterative method is proposed to solve nonhermitian positive definite linear systems and some convergence conditions are presented. It should be noted that the quasioptimal parameters of the pss iteration method preconditioner have been studied in 9, 10 when the coefficient matrix is nonhermitian positivedefinite. To the extent that it applies to matrices, it should only apply to hermitian ones. Thanks for contributing an answer to physics stack exchange. On a class of nonhermitian matrices with positive definite schur complements thomas berger, juan giribet, francisco mart inez per ia, and carsten trunk abstract. Then the tts method converges to the unique solution of for any choice of the initial guess if and only if, where,, and.
Amazingly, the energy levels of these hamiltonians are all real and positive. Nonhermitian random matrices cn is an n n real random matrix with i. Subsequently, these convergence conditions are applied to the alus method proposed by xiang et al. Hermitian and skewhermitian splitting methods for nonhermitian positive definite linear systems abstract. Also, for solving the large sparse system of linear equations, the new splitting iteration method is much more practical and efficient than the hss iteration method. To solve 1 iteratively, the efficient splitting of the coefficient matrix are usually required. Then the iteration matrix of the acscs method is and its spectral radius is bounded by where, are eigenvalues of, respectively. Can a positive definite matrix have complex eigenvalues. Siam journal on scientific computing siam society for.
Then the iteration matrix of the iterative scheme is proof. Convergence of tts iterative method for nonhermitian. Research article convergence of tts iterative method for. On a class of nonhermitian matrices with positive definite schur complements article pdf available in proceedings of the american mathematical society 1476. Convergence on successive overrelaxed iterative methods. In engineering applications n 106 is common, and n 109 22 2. However, its not hard to compute the pivots, and the signs. Optimization of extrapolated cayley transform with non. A possible method for nonhermitian and nonptsymmetric.
Let be nonhermitian positive definite with the tts as in. Theoretical analysis shows that the gpss method converges unconditionally to the exact solution of the linear system, with the upper bound of its convergence factor dependent only on the spectrum of the. Modified parallel multisplitting iterative methods for non. Splitting iteration methods for nonhermitian positive definite systems of linear equations bai, zhongzhi, hokkaido mathematical journal, 2007. An algorithm for eigenvectors of nonhermitian matrices albert maurice erisman iowa state university follow this and additional works at. An algorithm for eigenvectors of nonhermitian matrices. A square matrix is hermitian if and only if it is unitarily diagonalizable with real eigenvalues applications. We study efficient iterative methods for the large sparse nonhermitian positive definite system of linear equations based on the hermitian and skewhermitian splitting of. These methods converge unconditionally to the exact solution of the linear system, with the upper bounds of their convergence factors being only dependent on the spectrum of the hermitian or normal, or positivedefinite splitting matrix, but independent of the spectrum of the skewhermitian splitting matrix as well as the eigenvectors of all matrices involved. Sufficient conditions for the convergent splittings of nonhermitian. On the maximal size of largeaverage and anovafit submatrices in a gaussian random matrix sun, xing and nobel, andrew b.
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