The Ehrlich-Aberth method for palindromic matrix polynomials represented in the Dickson basis

An algorithm based on the Ehrlich-Aberth root-finding method is presented for the computation of the eigenvalues of a T-palindromic matrix polynomial. A structured linearization of the polynomial represented in the Dickson basis is introduced in order to exploit the symmetry of the roots by halving the total number of the required approximations. The rank structure properties of the linearization allow the design of a fast and numerically robust implementation of the root-finding iteration. Numerical experiments that confirm the effectiveness and the robustness of the approach are provided.Comment: in press in Linear Algebra Appl. (2011

GENERALIZATIONS OF AN INVERSE FREE KRYLOV SUBSPACE METHOD FOR THE SYMMETRIC GENERALIZED EIGENVALUE PROBLEM

Symmetric generalized eigenvalue problems arise in many physical applications and frequently only a few of the eigenpairs are of interest. Typically, the problems are large and sparse, and therefore traditional methods such as the QZ algorithm may not be considered. Moreover, it may be impractical to apply shift-and-invert Lanczos, a favored method for problems of this type, due to difficulties in applying the inverse of the shifted matrix. With these difficulties in mind, Golub and Ye developed an inverse free Krylov subspace algorithm for the symmetric generalized eigenvalue problem. This method does not rely on shift-and-invert transformations for convergence acceleration, but rather a preconditioner is used. The algorithm suffers, however, in the presence of multiple or clustered eigenvalues. Also, it is only applicable to the location of extreme eigenvalues. In this work, we extend the method of Golub and Ye by developing a block generalization of their algorithm which enjoys considerably faster convergence than the usual method in the presence of multiplicities and clusters. Preconditioning techniques for the problems are discussed at length, and some insight is given into how these preconditioners accelerate the method. Finally we discuss a transformation which can be applied so that the algorithm extracts interior eigenvalues. A preconditioner based on a QR factorization with respect to the B-1 inner product is developed and applied in locating interior eigenvalues

스태거드 쿼크의 카이랄성과 테이스트 대칭성

학위논문 (박사) -- 서울대학교 대학원 : 자연과학대학 물리·천문학부(물리학전공), 2020. 8. 이원종.This thesis investigates the general properties of the eigenvalue spectrum for improved staggered quarks and underlying chiral symmetry and SU(4) taste symmetry on it. Staggered fermion formalism is a methodology of lattice gauge theory studying quantum chromodynamics. Unfortunately, both chiral symmetry and taste symmetry are not conserved exactly for staggered quarks on the lattice. However, results of this work indicate that those symmetries are considerably preserved and retain some continuum behaviors for the improved staggered quarks such as HYP staggered quarks, Asqtad staggered quarks, and HISQ staggered quarks. Numerical simulations in this thesis are performed in quenched QCD approximation using HYP staggered quarks. Here, a new chirality operator and a new shift operator are introduced. Unlike Goltermans irreducible representations, new definitions respect the same recursion relations as the continuum chirality operator γ_5 and related with each other by chiral Ward identity of conserved U(1) axial symmetry of staggered fermion actions. Chirality of staggered Dirac eigenmodes is measured by this new chirality operator, by which the would-be zero modes are identified, and their correspondence with topological charge via the index theorem is discussed. By extending this standard chirality measurement, the transition of chirality from an eigenmode to another is also studied by measuring the matrix elements of the chirality operator and the shift operator on the staggered Dirac eigenspace. This quantity is named leakage. The chiral Ward identity ensures eight leakage elements of the chirality operator and the shift operator are identical, which holds within numerical precision. Further investigation on the leakage reveals that leakages for would-be zero modes and non-zero modes exhibit opposite patterns so that one can discriminate them rigorously. Besides, the leakage pattern for non-zero modes reveals the existence of the SU(4) taste symmetry clearly, by which two barometers of the taste symmetry breaking are measured. A machine learning analysis confirms the universality of leakage patterns. As a byproduct of this research, the renormalization of chirality is also discussed.본 학위논문에서는 향상된 스태거드 쿼크의 고유값이 가지는 일반적인 성질과 고유값 스펙트럼에서 나타나는 카이랄 대칭성과 SU(4) 테이스트 대칭성의 형태를 살펴본다. 스태거드 페르미온은 양자색역학을 연구하는 데 사용되는 격자 게이지 이론의 한 방법론이다. 아쉽게도 격자 위에서 정의된 스태거드 쿼크에서 카이랄 대칭성과 테이스트 대칭성이 정확하게 보존되지 않는다. 하지만, 본 연구의 결과는 HYP 스태거드 쿼크, Asqtad 스태거드 쿼크, HISQ 스태거드 쿼크와 같은 향상된 스태거드 쿼크들에 대해서 이러한 대칭성들이 상당히 잘 보존되며 연속계의 성질을 어느 정도 유지하고 있다는 것을 보인다. 본 학위논문에서 수행 된 계산 시뮬레이션들은 Quenched 양자색역학 근사법과 HYP 스태거드 쿼크를 사용하여 수행되었다. 본 학위논문에서는 새로운 카이랄성 오퍼레이터와 시프트(이동) 오퍼레이터를 정의한다. 골터만의 기약 표현 정의와는 달리, 이 새로운 정의는 연속계의 카이랄성 오퍼레이터인 γ_5가 만족하는 것과 같은 회귀 공식을 만족하며, 스태거드 페르미온 액션에서 보존되는 U(1) 축대칭성의 Ward identity 에 의해 두 오퍼레이터가 연결된다. 이러한 새로운 카이랄성 오퍼레이터를 사용하여 스태거드 디랙 고유모드에 대해서 카이랄성을 측정하고, 이로부터 제로 모드를 구분하고 그 결과가 인덱스 정리에 따라 위상전하 값과 일치하는 것을 보인다. 이러한 통상적인 카이랄성 측정방법을 확장하여, 카이랄 오퍼레이터와 시프트 오퍼레이터의 스태거드 디랙 고유공간에서의 행렬원소를 측정하여 하나의 고유모드에서 다른 고유모드로 전이되는 카이랄성의 정도를 연구한다. 이 측정값을 리키지(누출)라고 명명한다. 카이랄 Ward identity에 의해 카이랄 오퍼레이터와 시프트 오퍼레이터의 리키지 원소 중 8개가 서로 같다는 것을 보일 수 있고, 이 결과는 계산 오차 내에서 잘 성립된다. 추가적인 연구를 통해 제로 모드와 비제로(제로가 아닌) 모드의 리키지가 서로 반대되는 패턴을 나타내는 것을 살펴본다. 이를 이용하여 제로모드와 비제로모드를 정밀하게 구분할 수 있을 것이다. 또한, 비제로모드의 리키지에서는 SU(4) 테이스트 대칭성의 존재가 명확하게 드러나는데, 이로부터 테이스트 대칭성의 깨짐의 정도를 측정할 수 있는 2가지 지표를 측정한다. 이러한 리키지 패턴이 보편적으로 만족됨을 기계 학습 분석법을 사용하여 보인다. 더불어, 본 연구의 부산물로서 카이랄성의 재규격화를 논의한다.1 Introduction 1 2 Staggered fermion 4 2.1 Lattice gauge theory 4 2.2 Staggered fermions 7 2.3 Quark bilinears 11 2.3.1 Staggered bilinear operator 11 2.3.2 Golterman's irreducible representation 12 3 Physics of eigenvalues 14 3.1 Eigenvalues of Dirac operator 14 3.1.1 Eigenvalues of continuum Dirac operator 14 3.1.2 Eigenvalues of staggered Dirac operator 15 3.2 Quark condensates 17 3.2.1 Quark condensate in the continuum 17 3.2.2 Calculation of quark condensates on the lattice 20 3.3 Topological charge 23 3.3.1 Index theorem 23 3.3.2 Measurement of topological charge 27 4 Eigenvalue spectrum of staggered fermions 33 4.1 Calculation of eigenvalues and eigenvectors 33 4.1.1 Eigenvalues of D_s^+ D_s 33 4.1.2 Lanczos algorithm 36 4.1.3 Even-odd preconditioning and phase ambiguity 40 4.2 Eigenvalue spectrum: numerical results 45 4.2.1 Eigenvalue spectrum for Q = 0 and Q = -1 45 4.2.2 Eigenvalue spectrum for Q = -2 and Q = -3 50 5 Chiral symmetry of staggered fermions 52 5.1 Chirality of staggered fermions 52 5.1.1 Chirality operator 52 5.1.2 Recursion relationships for chirality operators 58 5.1.3 Chirality measurement 61 5.2 Chiral Ward identity 64 5.2.1 Ward identities on eigenvalue spectrum 64 5.2.2 Ward identities: numerical results 67 6 Leakage of chirality 70 6.1 Taste symmetry of staggered fermions 70 6.1.1 Taste symmetry in the continuum 70 6.1.2 Taste symmetry breaking on eigenvalue spectrum 73 6.2 Leakage pattern and symmetry 75 6.2.1 Quartet index 76 6.2.2 Leakage pattern of chirality and shift operators 77 6.2.3 Examples of the leakage pattern for zero modes 87 6.2.4 Examples of the leakage pattern for non-zero modes 91 6.3 Machine learning of leakage pattern 99 7 Renormalization of chirality 108 7.1 Zero modes and renormalization 108 8 Conclusion 111 A Lanczos iteration method 124 A.1 Iterative eigenvalue-finding procedure 124 A.2 Krylov subspace and Arnoldi algorithm 128 A.3 Arnoldi iteration method 133 A.4 Lanczos algorithm and Lanczos iteration method 137 A.5 Improvements on Arnoldi and Lanczos iterations 139 A.5.1 implicit restart 140 A.5.2 polynomial acceleration 148 A.6 QR iteration 149 A.7 modified Gram-Schmidt algorithm 156Docto

Numerical analysis of the Lyapunov equation with application to interconnected power systems

Bibliography: p.109-111.Prepared under grant ERDA-E(49-18)-2087. Originally presented as the author's thesis, (M.S. and E.E.), M.I.T. Dept. of Electrical Engineering and Computer Science, 1976.by Thomas Mac Athay

A fast iterative algorithm for near-diagonal eigenvalue problems

We introduce a novel iterative eigenvalue algorithm for near-diagonal matrices termed iterative perturbative theory (IPT). Built upon a "perturbative" partitioning of the matrix into diagonal and off-diagonal parts, IPT computes one or all eigenpairs with a complexity per iteration of one matrix-vector or one matrix-matrix multiplication respectively. Thanks to the high parallelism of these basic linear algebra operations, we obtain excellent performance on multi-core processors and GPUs, with large speed-ups over standard methods (up to $\sim50$x with respect to LAPACK and ARPACK). For matrices which are not close to being diagonal but have well-separated eigenvalues, IPT can be be used to refine low-precision eigenpairs obtained by other methods. We give sufficient conditions for linear convergence and demonstrate performance on dense and sparse test matrices. In a real-world application from quantum chemistry, we find that IPT performs similarly to the Davidson algorithm.Comment: Based on arXiv:2002.1287