We introduce a novel eigenvalue algorithm for near-diagonal matrices inspired by Rayleigh-Schrödinger perturbation theory and termed Iterative Perturbative Theory (IPT). Contrary to standard eigenvalue algorithms, which are either 'direct' (to compute all eigenpairs) or 'iterative' (to compute just a few), IPT computes any number of eigenpairs with the same basic iterative procedure. Thanks to this perfect parallelism, IPT proves more efficient than classical methods (LAPACK or CUSOLVER for the full-spectrum problem, preconditioned Davidson solvers for extremal eigenvalues). We give sufficient conditions for linear convergence and demonstrate performance on dense and sparse test matrices, including one from quantum chemistry.
In matrix theory and numerical analysis there are two very famous and important results. One is Gersgorin circle theorem, the other is strictly diagonally dominant theorem. They have important application and research value, and have been widely used and studied. In this paper, we investigate generalized diagonally dominant matrices and matrix eigenvalue inclusion regions. A class of G-function pairs is proposed, which extends the concept of G-functions. Thirteen kind of G-function pairs are established. Their properties and characteristics are studied. By using these special G-function pairs,…
We propose a new way of visualising the dynamics of iterative eigenvalue algorithms such as the QR algorithm, over the important special case of PSD (positive semi-definite) matrices. Many subtle and important properties of such algorithms are easily found this way. We believe that this may have pedagogical value to both students and researchers of numerical linear algebra. The fixed points of iterative algorithms are obtained visually, and their stability is analysed intuitively. It becomes clear that what it means for an iterative eigenvalue algorithm to "converge quickly" is an ambiguous qu…
Solving linear systems and computing eigenvalues are two fundamental problems in linear algebra. For solving linear systems, many efficient quantum algorithms have been discovered. For computing eigenvalues, currently, we have efficient quantum algorithms for Hermitian and unitary matrices. However, the general case is far from fully understood. Combining quantum phase estimation, quantum algorithm to solve linear differential equations and quantum singular value estimation, we propose two quantum algorithms to compute the eigenvalues of diagonalizable matrices that only have real eigenvalues …
Forward stable eigenvalue decomposition of rank-one modifications of diagonal matrices
Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split（斯普利特大学电气、机械与航海工程学院）；Department of Computer Science and Engineering, The Pennsylvania State University（宾夕法尼亚州立大学计算机科学与工程系）
We present two interactive visualisations of 2x2 real matrices, which we call v1 and v2. v1 is only valid for PSD matrices, and uses the spectral theorem in a trivial way -- we use it as a warm-up. By contrast, v2 is valid for *all* 2x2 real matrices, and is based on the lesser known theory of Lie Sphere Geometry. We show that the dynamics of iterative eigenvalue algorithms can be illustrated using both. v2 has the advantage that it simultaneously depicts many properties of a matrix, all of which are relevant to the study of eigenvalue algorithms. Examples of the properties of a matrix that v2…
We study Chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices. In this technique the subspace projection onto the target space of wanted eigenvectors is approximated with filter polynomials obtained from Chebyshev expansions of window functions. After the discussion of the conceptual foundations of Chebyshev filter diagonalization we analyze the impact of the choice of the damping kernel, search space size, and filter polynomial degree on the computational accuracy and effort, before we describe the necessary steps …
This paper presents a deterministic linear time complexity IDE-SimRank method to approximately compute SimRank with proved error bound. SimRank is a well-known similarity measure between graph vertices which relies on graph topology only and is built on intuition that "two objects are similar if they are related to similar objects". The fixed point equation for direct SimRank computation is the discrete Lyapunov equation with specific diagonal matrix in the right hand side. The proposed method is based on estimation of this diagonal matrix with GMRES and use this estimation to compute singe-so…
Contour integration techniques have become a popular choice for solving the linear and non-linear eigenvalue problems. They principally include the Sakurai-Sugiura methods, the Beyn's algorithm, the FEAST/NLFEAST algorithms and other rational filtering techniques. While these methods can result in effective 'black-box' approach for solving linear eigenvalue problems, they still present several shortcomings for addressing nonlinear eigenvalue problems which are both mathematically and practically far more challenging. In this paper, we introduce a new hybrid algorithm that advantageously combin…
Efficient solution of the lowest eigenmodes is studied for a family of related eigenvalue problems with common $2\times 2$ block structure. It is assumed that the upper diagonal block varies between different versions while the lower diagonal block and the range of the coupling blocks remains unchanged. Such block structure naturally arises when studying the effect of a subsystem to the eigenmodes of the full system. The proposed method is based on interpolation of the resolvent function after some of its singularities have been removed by a spectral projection. Singular value decomposition ca…
Convergence and mixed-precision preconditioning for the naive Jacobi eigenvalue algorithm
University of Zagreb Faculty of Chemical Engineering and Technology（萨格勒布大学化学工程与技术学院）；University of Osijek School of Applied Mathematics and Informatics（奥西耶克大学应用数学与信息学院）
We derive bounds on the eigenvalues of a generic form of double saddle-point matrices. The bounds are expressed in terms of extremal eigenvalues and singular values of the associated block matrices. Inertia and algebraic multiplicity of eigenvalues are considered as well. The analysis includes bounds for preconditioned matrices based on block diagonal preconditioners using Schur complements, and it is shown that in this case the eigenvalues are clustered within a few intervals bounded away from zero. Analysis for approximations of Schur complements is included. Some numerical experiments valid…
We propose two different strategies to find eigenvalues and eigenvectors of a given, not necessarily Hermitian, matrix $A$. Our methods apply also to the case of complex eigenvalues, making the strategies interesting for applications to physics, and to pseudo-hermitian quantum mechanics in particular. We first consider a {\em dynamical} approach, based on a pair of ordinary differential equations defined in terms of the matrix $A$ and of its adjoint $A^\dagger$. Then we consider an extension of the so-called power method, for which we prove a fixed point theorem for $A\neq A^\dagger$ useful in…
Nesterov's well-known scheme for accelerating gradient descent in convex optimization problems is adapted to accelerating stationary iterative solvers for linear systems. Compared with classical Krylov subspace acceleration methods, the proposed scheme requires more iterations, but it is trivial to implement and retains essentially the same computational cost as the unaccelerated method. An explicit formula for a fixed optimal parameter is derived in the case where the stationary iteration matrix has only real eigenvalues, based only on the smallest and largest eigenvalues. The fixed parameter…
Continuous meets discrete: geometry dependence of the performance of iterative solutions for nonsymmetric PDEs in terms of Laplace eigenvalues
Chalmers University and the University of Gothenburg（查尔姆斯理工大学和哥德堡大学）；ELTE Eötvös Loránd University（罗兰·厄特沃什大学）；Budapest University of Technology and Economics（布达佩斯技术与经济大学）
We show that the global minimum (resp. maximum) of a continuous function on a compact set can be approximated from above (resp. from below) by computing the smallest (rest. largest) eigenvalue of a hierarchy of (r x r) tri-diagonal univariate moment matrices of increasing size. Equivalently it reduces to computing the smallest (resp. largest) root of a certain univariate degree-r orthonormal polynomial. This provides a strong connection between the fields of optimization, orthogonal polynomials, numerical analysis and linear algebra, via asymptotic spectral analysis of tri-diagonal symmetric m…
