Topic
Matrix multiplication
About: Matrix multiplication is a research topic. Over the lifetime, 7844 publications have been published within this topic receiving 214576 citations. The topic is also known as: matrix product & product of matrices.
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TL;DR: Although the derivation of the algorithm is presented in terms of matrices, no matrix matrix multiplications are needed and only the nonzero matrix elements have to be stored, making the method useful for very large molecules.
Abstract: In this article, we present a new LINear Constraint Solver (LINCS) for molecular simulations with bond constraints. The algorithm is inherently stable, as the constraints themselves are reset instead of derivatives of the constraints, thereby eliminating drift. Although the derivation of the algorithm is presented in terms of matrices, no matrix matrix multiplications are needed and only the nonzero matrix elements have to be stored, making the method useful for very large molecules. At the same accuracy, the LINCS algorithm is three to four times faster than the SHAKE algorithm. Parallelization of the algorithm is straightforward. (C) 1997 John Wiley & Sons, Inc.
12,699 citations
Book•
01 Jun 1984
TL;DR: In this article, the Routh-Hurwitz problem of singular pencils of matrices has been studied in the context of systems of linear differential equations with variable coefficients, and its applications to the analysis of complex matrices have been discussed.
Abstract: Volume 2: XI. Complex symmetric, skew-symmetric, and orthogonal matrices: 1. Some formulas for complex orthogonal and unitary matrices 2. Polar decomposition of a complex matrix 3. The normal form of a complex symmetric matrix 4. The normal form of a complex skew-symmetric matrix 5. The normal form of a complex orthogonal matrix XII. Singular pencils of matrices: 1. Introduction 2. Regular pencils of matrices 3. Singular pencils. The reduction theorem 4. The canonical form of a singular pencil of matrices 5. The minimal indices of a pencil. Criterion for strong equivalence of pencils 6. Singular pencils of quadratic forms 7. Application to differential equations XIII. Matrices with non-negative elements: 1. General properties 2. Spectral properties of irreducible non-negative matrices 3. Reducible matrices 4. The normal form of a reducible matrix 5. Primitive and imprimitive matrices 6. Stochastic matrices 7. Limiting probabilities for a homogeneous Markov chain with a finite number of states 8. Totally non-negative matrices 9. Oscillatory matrices XIV. Applications of the theory of matrices to the investigation of systems of linear differential equations: 1. Systems of linear differential equations with variable coefficients. General concepts 2. Lyapunov transformations 3. Reducible systems 4. The canonical form of a reducible system. Erugin's theorem 5. The matricant 6. The multiplicative integral. The infinitesimal calculus of Volterra 7. Differential systems in a complex domain. General properties 8. The multiplicative integral in a complex domain 9. Isolated singular points 10. Regular singularities 11. Reducible analytic systems 12. Analytic functions of several matrices and their application to the investigation of differential systems. The papers of Lappo-Danilevskii XV. The problem of Routh-Hurwitz and related questions: 1. Introduction 2. Cauchy indices 3. Routh's algorithm 4. The singular case. Examples 5. Lyapunov's theorem 6. The theorem of Routh-Hurwitz 7. Orlando's formula 8. Singular cases in the Routh-Hurwitz theorem 9. The method of quadratic forms. Determination of the number of distinct real roots of a polynomial 10. Infinite Hankel matrices of finite rank 11. Determination of the index of an arbitrary rational fraction by the coefficients of numerator and denominator 12. Another proof of the Routh-Hurwitz theorem 13. Some supplements to the Routh-Hurwitz theorem. Stability criterion of Lienard and Chipart 14. Some properties of Hurwitz polynomials. Stieltjes' theorem. Representation of Hurwitz polynomials by continued fractions 15. Domain of stability. Markov parameters 16. Connection with the problem of moments 17. Theorems of Markov and Chebyshev 18. The generalized Routh-Hurwitz problem Bibliography Index.
9,334 citations
Book•
12 Jul 2010TL;DR: The field of values as discussed by the authors is a generalization of the field of value of matrices and functions, and it includes singular value inequalities, matrix equations and Kronecker products, and Hadamard products.
Abstract: 1. The field of values 2. Stable matrices and inertia 3. Singular value inequalities 4. Matrix equations and Kronecker products 5. Hadamard products 6. Matrices and functions.
7,013 citations
Book•
01 Jan 1960
TL;DR: In this article, the Second Edition Preface is presented, where Maximization, Minimization, and Motivation are discussed, as well as a method of Hermite and Quadratic Form Index.
Abstract: Foreword Preface to the Second Edition Preface 1. Maximization, Minimization, and Motivation 2. Vectors and Matrices 3. Diagonalization and Canonical Forms for Symmetric Matrices 4. Reduction of General Symmetric Matrices to Diagonal Form 5. Constrained Maxima 6. Functions of Matrices 7. Variational Description of Characteristic Roots 8. Inequalities 9. Dynamic Programming 10. Matrices and Differential Equations 11. Explicit Solutions and Canonical Forms 12. Symmetric Function, Kronecker Products and Circulants 13. Stability Theory 14. Markoff Matrices and Probability Theory 15. Stochastic Matrices 16. Positive Matrices, Perron's Theorem, and Mathematical Economics 17. Control Processes 18. Invariant Imbedding 19. Numerical Inversion of the Laplace Transform and Tychonov Regularization Appendix A. Linear Equations and Rank Appendix B. The Quadratic Form of Selberg Appendix C. A Method of Hermite Appendix D. Moments and Quadratic Forms Index.
3,500 citations