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A divide-and-conquer method for the tridiagonal generalized eigenvalue problem

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 1997年第1期86卷 141-148页

作者： Elsner, L Fasse, A Langmann, E UNIV BIELEFELD FAK MATHD-33501 BIELEFELDGERMANY

We introduce a divide-and-conquer method for the generalized eigenvalue problem Ax = lambda Bx, where A and B are real symmetric tridiagonal matrices and B is positive-definite. It is a generalization of Cuppen's method for the standard eigenvalue problem, B = I, which is based on rank-one modifications. Our method is an alternative to a method developed by Borges and Gragg using restrictions and extensions.

关键词： divide-and-conquer method tridiagonal eigenvalue problem generalized eigenvalue problem

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Implementation of divide-and-conquer method including Hartree-Fock exchange interaction

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JOURNAL OF COMPUTATIONAL CHEMISTRY 2007年第12期28卷 2003-2012页

作者： Akama, Tomoko Kobayashi, Masato Nakai, Hiromi Waseda Univ Sch Sci & Engn Dept Chem Tokyo 1698555 Japan

The divide-and-conquer (DC) method, which is one of the linear-scaling methods avoiding explicit diagonalization of the Fock matrix. has been applied mainly to pure density functional theory (DFT) or semiempirical molecular orbital calculations so far. The present study applies the DC method to such calculations including the Hartree-Fock (HF) exchange terms as the HF and hybrid HF/DFT. Reliability of the DC-HF and DC-hybrid HF/DFT is found to be strongly dependent oil the cut-off radius, which defines the localization region in the DC formalism. This dependence on the cut-off radius is assessed from various points of view: that is, total energy, energy components, local energies, and density of states. Additionally, to accelerate the self-consistent field convergence in DC calculations, a new convergence technique is proposed. (c) 2007 Wiley, Periodicals, Inc.

关键词： divide-and-conquer method Hartree-Fock exchange linear-scaling method local energy density of states

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A divide-and-conquer method for the Takagi factorization

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2008年第1期30卷 142-153页

作者： Xu, Wei Qiao, Sanzheng McMaster Univ Dept Comp & Software Hamilton ON L8S 4K1 Canada

This paper presents a divide-and-conquer method for computing the symmetric singular value decomposition, or Takagi factorization, of a complex symmetric and tridiagonal matrix. An analysis of accuracy shows that our method produces accurate Takagi values and orthogonal Takagi vectors. Our preliminary numerical experiments have confirmed our analysis and demonstrated that our divide-and-conquer method is much more efficient than the implicit QR method even for moderately large matrices.

关键词： divide-and-conquer method symmetric SVD Takagi factorization

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Two-Level Hierarchical Parallelization of Second-Order Moller-Plesset Perturbation Calculations in divide-and-conquer method

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JOURNAL OF COMPUTATIONAL CHEMISTRY 2011年第13期32卷 2756-2764页

作者： Katouda, Michio Kobayashi, Masato Nakai, Hiromi Nagase, Shigeru Inst Mol Sci Dept Theoret & Computat Mol Sci Okazaki Aichi 4448585 Japan Waseda Univ Sch Adv Sci & Engn Dept Chem & Biochem Tokyo 1698555 Japan Waseda Univ Res Inst Sci & Engn Tokyo 1698555 Japan Japan Sci & Technol Agcy CREST Tokyo 1020075 Japan

A two-level hierarchical parallelization scheme including the second-order Moller-Plesset perturbation (MP2) theory in the divide-and-conquer method is presented. The scheme is a combination of coarse-grain parallelization assigning each subsystem to a group of processors, with fine-grain parallelization, where the computational tasks for evaluating MP2 correlation energy of the assigned subsystem are distributed among processors in the group. Test calculations demonstrate that the present scheme shows high parallel efficiency and makes MP2 calculations practical for very large molecules. (C) 2011 Wiley Periodicals, Inc. J Comput Chem 32: 2756-2764, 2011

关键词： divide-and-conquer method second-order Moller-Plesset perturbation theory parallel implementation parallel algorithm parallel performance granularity linear-scaling method electron correlation large molecule extended system

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An extension of the divide-and-conquer method for a class of symmetric block-tridiagonal eigenproblems

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ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE 2002年第1期28卷 45-58页

作者： Gansterer, WN Ward, RC Muller, RP Univ Tennessee Dept Comp Sci Knoxville TN 37996 USA CALTECH Beckman Inst Mat & Proc Simulat Ctr Pasadena CA 91125 USA

A divide-and-conquer method for computing eigenvalues and eigenvectors of a block-tridiagonal matrix with rank-one off-diagonal blocks is presented. The implications of unbalanced merging operations due to unequal block sizes are analyzed and illustrated with numerical examples. It is shown that an unfavorable order for merging blocks in the synthesis phase of the algorithm may lead to a significant increase of the arithmetic complexity. A strategy to determine a good merging order that is at least close to optimal in all cases is given. The method has been implemented and applied to test problems from a quantum chemistry application.

关键词： algorithms performance block-tridiagonal matrix divide-and-conquer method

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ANALYSIS OF THE divide-and-conquer method FOR ELECTRONIC STRUCTURE CALCULATIONS

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MATHEMATICS OF COMPUTATION 2016年第302期85卷 2919-2938页

作者： Chen, Jingrun Lu, Jianfeng Univ Calif Santa Barbara Dept Math South Hall 6705 Santa Barbara CA 93106 USA Soochow Univ Math Ctr Interdisciplinary Res Zuzhou 215006 Peoples R China Soochow Univ Sch Math Sci Zuzhou 215006 Peoples R China Duke Univ Dept Math Box 90320 Durham NC 27708 USA Duke Univ Dept Phys Box 90320 Durham NC 27708 USA Duke Univ Dept Chem Box 90320 Durham NC 27708 USA

We study the accuracy of the divide-and-conquer method for electronic structure calculations. The analysis is conducted for a prototypical subdomain problem in the method. We prove that the pointwise difference between electron densities of the global system and the subsystem decays exponentially as a function of the distance away from the boundary of the subsystem, under the gap assumption of both the global system and the subsystem. We show that the gap assumption is crucial for the accuracy of the divide-and-conquer method by numerical examples. In particular, we show examples with the loss of accuracy when the gap assumption of the subsystem is invalid.

关键词： Density functional theory divide-and-conquer method gap assumption exponential decay

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A low-complexity divide-and-conquer method for computing eigenvalues and eigenvectors of symmetric band matrices

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BIT 2001年第5期41卷 967-976页

作者： Gansterer, WN Schneid, J Ueberhuber, CW Univ Tennessee Dept Comp Sci Knoxville TN 37996 USA Vienna Univ Technol Inst Appl Math & Numer Anal A-1040 Vienna Austria

A framework for an efficient low-complexity divide-and-conquer algorithm for computing eigenvalues and eigenvectors of an n x n symmetric band matrix with semibandwidth b << n is proposed and its arithmetic complexity analyzed. The distinctive feature of the algorithm-after subdivision of the original problem into p subproblems and their solution-is a separation of the eigenvalue and eigenvector computations in the central synthesis problem. The eigenvalues are computed recursively by representing the corresponding symmetric rank b (p-1) modi cation of a diagonal matrix as a series of rank-one modi cations. Each rank-one modi cation problem can be solved using techniques developed for the tridiagonal divide-and-conquer algorithm. Once the eigenvalues are known, the corresponding eigenvectors can be computed efficiently using modified QR factorizations with restricted column pivoting. It is shown that the complexity of the resulting divide-and-conquer algorithm is O(n(2)b(2)) (in exact arithmetic).

关键词： banded symmetric eigenproblem divide-and-conquer method complexity analysis

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Large-Scale Two-Component Relativistic Quantum-Chemical Theory: Combination of the Infinite-Order Douglas-Kroll-Hess method with the Local Unitary Transformation Scheme and the divide-and-conquer method

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INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 2015年第5期115卷 253-257页

作者： Seino, Junji Nakai, Hiromi Waseda Univ Dept Chem & Biochem Sch Adv Sci & Engn Tokyo 1698555 Japan Waseda Univ Res Inst Sci & Engn Tokyo 1698555 Japan Japan Sci & Technol Agcy CREST Tokyo 1020075 Japan Kyoto Univ ESICB Nishigyo Ku Kyotodaigaku Katsura Kyoto 6158520 Japan

Large-scale two-component (2c) relativistic quantum-chemical (RQC) theory is reviewed. We briefly discuss the theories, advantages, and extensibilities of an overall linear-scaling scheme in 2c relativistic theory. The theory is based on the infinite-order Douglas-Kroll-Hess method, with the local unitary transformation scheme to produce the 2c relativistic Hamiltonian, and the divide-and-conquer method to achieve linear-scaling of Hartree-Fock and electron correlation methods. Furthermore, perspectives for large-scale RQC are explained to bring the practical usage and treatment of light and heavy elements in 2c relativistic calculations close to those in non-relativistic methods. (c) 2014 Wiley Periodicals, Inc.

关键词： two-component relativistic method large-scale molecular theory local unitary transformation divide-and-conquer method

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A divide-and-conquer method with approximate Fermi levels for parallel computations

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THEORETICAL CHEMISTRY ACCOUNTS 2015年第5期134卷 1-11页

作者： Yoshikawa, Takeshi Nakai, Hiromi Waseda Univ Sch Adv Sci & Engn Dept Chem & Biochem Tokyo 1698555 Japan Waseda Univ Res Inst Sci & Engn Tokyo 1698555 Japan Japan Sci & Technol Agcy CREST Kawaguchi Saitama 3320012 Japan Kyoto Univ ESICB Kyoto 6158520 Japan

This study proposes two efficient algorithms of the linear-scaling divide-and-conquer self-consistent field (DC-SCF) method for parallel computations. These algorithms minimize the amount of network communication required for determining the common Fermi level by adopting an approximate Fermi level. One algorithm adopts the quasi-converged Fermi level during DC-SCF iterations, while the other uses the quasi-converged electron numbers of individual subsystems. A numerical assessment demonstrates the high parallel efficiency for both methods without loss in accuracy.

关键词： Linear-scaling computation Common Fermi level Self-consistent field calculation Parallel computation divide-and-conquer method

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A divide-and-conquer method Based Ensemble Regression Model for Water Quality Prediction 1

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8th International Conference on Rough Sets and Knowledge Technology (RSKT)

作者： Zou, Xuan Wang, Guoyin Gou, Guanglei Li, Hong Chongqing Univ Posts & Telecommun Chongqing Key Lab Computat Intelligence Chongqing Peoples R China Chinese Acad Sci Chongqing Inst Green & Intelligent Technol Inst Elect Informat & Technol Chongqing Peoples R China Southwest Jiao Tong Univ Sch Informat Sci & Technol Chengdu Peoples R China

ISBN: (数字)9783642412998

ISBN: (纸本)9783642412998;9783642412981

This paper proposes a novel ensemble regression model to predict time series data of water quality. The proposed model consists of multiple regressors and a classifier. The model transforms the original time series data into subsequences by sliding window and divides it into several parts according to the fitness of regressor so that each regressor has advantages in a specific part. The classifier decides which part the new data should belong to so that the model could divide the whole prediction problem into small parts and conquer it after computing on only one part. The ensemble regression model, with a combination of Support Vector Machine, RBF Neural Network and Grey Model, is tested using 450-week observations of CODMn data provided by Ministry of Environmental Protection of the People's Republic of China during 2004 and 2012. The results show that the model could approximately convert the problem of prediction into a problem of classification and provide better accuracy over each single model it has combined.

关键词： Water quality prediction ensemble regression divide-and-conquer method time series data mining

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