Cholesky decomposition

About: Cholesky decomposition is a research topic. Over the lifetime, 3823 publications have been published within this topic receiving 99297 citations.

Numerical Recipes in C: The Art of Scientific Computing

Abstract: From the Publisher: This is the revised and greatly expanded Second Edition of the hugely popular Numerical Recipes: The Art of Scientific Computing. The product of a unique collaboration among four leading scientists in academic research and industry, Numerical Recipes is a complete text and reference book on scientific computing. In a self-contained manner it proceeds from mathematical and theoretical considerations to actual practical computer routines. With over 100 new routines (now well over 300 in all), plus upgraded versions of many of the original routines, this book is more than ever the most practical, comprehensive handbook of scientific computing available today. The book retains the informal, easy-to-read style that made the first edition so popular, with many new topics presented at the same accessible level. In addition, some sections of more advanced material have been introduced, set off in small type from the main body of the text. Numerical Recipes is an ideal textbook for scientists and engineers and an indispensable reference for anyone who works in scientific computing. Highlights of the new material include a new chapter on integral equations and inverse methods; multigrid methods for solving partial differential equations; improved random number routines; wavelet transforms; the statistical bootstrap method; a new chapter on "less-numerical" algorithms including compression coding and arbitrary precision arithmetic; band diagonal linear systems; linear algebra on sparse matrices; Cholesky and QR decomposition; calculation of numerical derivatives; Pade approximants, and rational Chebyshev approximation; new special functions; Monte Carlo integration in high-dimensional spaces; globally convergent methods for sets of nonlinear equations; an expanded chapter on fast Fourier methods; spectral analysis on unevenly sampled data; Savitzky-Golay smoothing filters; and two-dimensional Kolmogorov-Smirnoff tests. All this is in addition to material on such basic top

Orthogonal least squares methods and their application to non-linear system identification

Abstract: Identification algorithms based on the well-known linear least squares methods of gaussian elimination, Cholesky decomposition, classical Gram-Schmidt, modified Gram-Schmidt, Householder transformation, Givens method, and singular value decomposition are reviewed. The classical Gram-Schmidt, modified Gram-Schmidt, and Householder transformation algorithms are then extended to combine structure determination, or which terms to include in the model, and parameter estimation in a very simple and efficient manner for a class of multivariate discrete-time non-linear stochastic systems which are linear in the parameters.

Iterative Methods for Solving Linear Systems

Abstract: List of Algorithms Preface 1. Introduction. Brief Overview of the State of the Art Notation Review of Relevant Linear Algebra Part I. Krylov Subspace Approximations. 2. Some Iteration Methods. Simple Iteration Orthomin(1) and Steepest Descent Orthomin(2) and CG Orthodir, MINRES, and GMRES Derivation of MINRES and CG from the Lanczos Algorithm 3. Error Bounds for CG, MINRES, and GMRES. Hermitian Problems-CG and MINRES Non-Hermitian Problems-GMRES 4. Effects of Finite Precision Arithmetic. Some Numerical Examples The Lanczos Algorithm A Hypothetical MINRES/CG Implementation A Matrix Completion Problem Orthogonal Polynomials 5. BiCG and Related Methods. The Two-Sided Lanczos Algorithm The Biconjugate Gradient Algorithm The Quasi-Minimal Residual Algorithm Relation Between BiCG and QMR The Conjugate Gradient Squared Algorithm The BiCGSTAB Algorithm Which Method Should I Use? 6. Is There A Short Recurrence for a Near-Optimal Approximation? The Faber and Manteuffel Result Implications 7. Miscellaneous Issues. Symmetrizing the Problem Error Estimation and Stopping Criteria Attainable Accuracy Multiple Right-Hand Sides and Block Methods Computer Implementation Part II. Preconditioners. 8. Overview and Preconditioned Algorithms. 9. Two Example Problems. The Diffusion Equation The Transport Equation 10. Comparison of Preconditioners. Jacobi, Gauss--Seidel, SOR The Perron--Frobenius Theorem Comparison of Regular Splittings Regular Splittings Used with the CG Algorithm Optimal Diagonal and Block Diagonal Preconditioners 11. Incomplete Decompositions. Incomplete Cholesky Decomposition Modified Incomplete Cholesky Decomposition 12. Multigrid and Domain Decomposition Methods. Multigrid Methods Basic Ideas of Domain Decomposition Methods.

A Sequential Ensemble Kalman Filter for Atmospheric Data Assimilation

Abstract: An ensemble Kalman filter may be considered for the 4D assimilation of atmospheric data. In this paper, an efficient implementation of the analysis step of the filter is proposed. It employs a Schur (elementwise) product of the covariances of the background error calculated from the ensemble and a correlation function having local support to filter the small (and noisy) background-error covariances associated with remote observations. To solve the Kalman filter equations, the observations are organized into batches that are assimilated sequentially. For each batch, a Cholesky decomposition method is used to solve the system of linear equations. The ensemble of background fields is updated at each step of the sequential algorithm and, as more and more batches of observations are assimilated, evolves to eventually become the ensemble of analysis fields. A prototype sequential filter has been developed. Experiments are performed with a simulated observational network consisting of 542 radiosonde and 615 satellite-thickness profiles. Experimental results indicate that the quality of the analysis is almost independent of the number of batches (except when the ensemble is very small). This supports the use of a sequential algorithm. A parallel version of the algorithm is described and used to assimilate over 100 000 observations into a pair of 50-member ensembles. Its operation count is proportional to the number of observations, the number of analysis grid points, and the number of ensemble members. In view of the flexibility of the sequential filter and its encouraging performance on a NEC SX-4 computer, an application with a primitive equations model can now be envisioned.

Direct Methods for Sparse Linear Systems

Abstract: Preface 1. Introduction 2. Basic algorithms 3. Solving triangular systems 4. Cholesky factorization 5. Orthogonal methods 6. LU factorization 7. Fill-reducing orderings 8. Solving sparse linear systems 9. CSparse 10. Sparse matrices in MATLAB Appendix: Basics of the C programming language Bibliography Index.