We demonstrate that the combination of simply contracted basis functions and the lanczos algorithm yields an extremely efficient method for computing vibrational energy levels. We discuss ideas and present some result...
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We demonstrate that the combination of simply contracted basis functions and the lanczos algorithm yields an extremely efficient method for computing vibrational energy levels. We discuss ideas and present some results for HOOH and CH4. The basis functions we use are products of eigenfunctions of reduced-dimension Hamiltonians obtained by freezing coordinates at equilibrium. The basis functions represent the desired wave functions well yet are simple enough that matrix-vector products may be evaluated efficiently. (C) 2003 Wiley Periodicals, Inc.
An ''industrial strength'' algorithm for solving sparse symmetric generalized eigenproblems is described. The algorithm has its foundations in known techniques in solving sparse symmetric eigenproblems...
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An ''industrial strength'' algorithm for solving sparse symmetric generalized eigenproblems is described. The algorithm has its foundations in known techniques in solving sparse symmetric eigenproblems, notably the spectral transformation of Ericsson and Ruhe and the block lanczos algorithm. However, the combination of these two techniques is not trivial;there are many pitfalls awaiting the unwary implementor. The focus of this paper is on identifying those pitfalls and avoiding them, leading to a ''bomb-proof'' algorithm that can live as a black box eigensolver inside a large applications code. The code that results comprises a robust shift selection strategy and a block lanczos algorithm that is a novel combination of new techniques and extensions of old techniques.
In this paper, we report our parallel implementations of the lanczos sparse linear system solving algorithm over large prime fields, on a multi-core platform. We employ several load-balancing methods suited to these p...
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ISBN:
(纸本)9783642113215
In this paper, we report our parallel implementations of the lanczos sparse linear system solving algorithm over large prime fields, on a multi-core platform. We employ several load-balancing methods suited to these platforms. We have carried out process-level and thread-level parallel implementations under two different arithmetic libraries, and the best speedup obtained is 6.57 on eight cores. To the best of our knowledge;no implementation of the lanczos algorithm on a multi-core platform is ever reported in the literature. Moreover, we seem to have achieved significantly larger speedup compared to all previously reported implementations of this algorithm.
We demonstrate that the combination of simply contracted basis functions and the lanczos algorithm yields an extremely efficient method for computing vibrational energy levels. We discuss ideas and present some result...
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We demonstrate that the combination of simply contracted basis functions and the lanczos algorithm yields an extremely efficient method for computing vibrational energy levels. We discuss ideas and present some results for HOOH and CH4. The basis functions we use are products of eigenfunctions of reduced-dimension Hamiltonians obtained by freezing coordinates at equilibrium. The basis functions represent the desired wave functions well yet are simple enough that matrix-vector products may be evaluated efficiently. (C) 2003 Wiley Periodicals, Inc.
In theory, the lanczos algorithm generates an orthogonal basis of the corresponding Krylov subspace. However, in finite precision arithmetic the orthogonality and linear independence of the computed lanczos vectors is...
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In theory, the lanczos algorithm generates an orthogonal basis of the corresponding Krylov subspace. However, in finite precision arithmetic the orthogonality and linear independence of the computed lanczos vectors is usually lost quickly. In this paper we study a class of matrices and starting vectors having a special nonzero structure that guarantees exact computations of the lanczos algorithm whenever floating point arithmetic satisfying the IEEE 754 standard is used. Analogous results are formulated also for an implementation of the conjugate gradient method called cglanczos. This implementation then computes approximations that agree with their exact counterparts to a relative accuracy given by the machine precision and the condition number of the system matrix. The results are extended to the Arnoldi algorithm, the nonsymmetric lanczos algorithm, the Golub-Kahan bidiagonalization, the block-lanczos algorithm, and their counterparts for solving linear systems.
An adaptation of the conventional lanczos algorithm is proposed to solve the general symmetric eigenvalue problem K phi = lambda K-G phi in the case when the geometric stiffness matrix K-G is not necessarily positive-...
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An adaptation of the conventional lanczos algorithm is proposed to solve the general symmetric eigenvalue problem K phi = lambda K-G phi in the case when the geometric stiffness matrix K-G is not necessarily positive-definite. The only requirement for the new algorithm to work is that matrix K must be positive-definite. Firstly, the algorithm is presented for the standard situation where no shifting is assumed. Secondly, the algorithm is extended to include shifting since this procedure may be important for enhanced precision or acceleration of convergence rates. Neither version of the algorithm requires matrix inversion, but more resources in terms of memory allocation are needed by the version with shifting.
This paper explores the interconnections between two methods which can be used to obtain rational interpolants. The first method, the behavioral approach, constructs a generating system in the frequency domain which e...
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This paper explores the interconnections between two methods which can be used to obtain rational interpolants. The first method, the behavioral approach, constructs a generating system in the frequency domain which explains a given data set composed of trajectories. The second method, the rational lanczos algorithm, can be used to construct a rational interpolant for the transfer function of a linear system defined by (potentially very high-order) state-space equations. This paper works to merge the theoretical attributes of the behavioral approach with the theoretical and computational properties of rational lanczos. As a result, it lays the foundation for the computation of reduced-order, stabilizing controllers through rational interpolation.
The time-ordered exponential of a time-dependent matrix A(t) is defined as the function of A(t) that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in A(t...
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The time-ordered exponential of a time-dependent matrix A(t) is defined as the function of A(t) that solves the first-order system of coupled linear differential equations with non-constant coefficients encoded in A(t). The authors have recently proposed the first lanczos-like algorithm capable of evaluating this function. This algorithm relies on inverses of time-dependent functions with respect to a non-commutative convolution-like product, denoted by *. Yet, the existence of such inverses, crucial to avoid algorithmic breakdowns, still needed to be proved. Here we constructively prove that *-inverses exist for all non-identically null, smooth, separable functions of two variables. As a corollary, we partially solve the Green's function inverse problem which, given a distribution G, asks for the differential operator whose fundamental solution is G. Our results are abundantly illustrated by examples.
We present methods for computing the controllable space of a Linear Dynamic System. The methods are based on orthogonalizing Krylov sequences and can use much less storage than other methods with little or no loss of ...
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We present methods for computing the controllable space of a Linear Dynamic System. The methods are based on orthogonalizing Krylov sequences and can use much less storage than other methods with little or no loss of numerical stability.
In this paper we propose a block lanczos algorithm suitable for MIMD distributed memory message passing architectures. It is based on an efficient parallelization of basic linear algebra operations, such as matrix-mat...
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