Cholesky Factorization And Matrix Inversion
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Cholesky Factorization And Matrix Inversion
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Author : Erwin Schmid
language : en
Publisher:
Release Date : 1973
Cholesky Factorization And Matrix Inversion written by Erwin Schmid and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1973 with Geodesy categories.
The Cholesky square root algorithm used in the solution of linear equations with a positive definite matrix of coefficients is developed by elementary matrix algebra, independent of the Gaussian elimination from which it was originally derived. The Cholesky factorization leads to a simple inversion procedure for the given matrix. A simple transformation makes the inversion applicable to nonsymmetric matrices. The least squares hypothesis is shown to be the simplest and most general unique solution of a system of linear equations with a nonsquare matrix of coefficients. The method of proof is extended to develop the Gaussian elimination algorithm in a readily comprehensible procedure.
Numerical Methods Of Statistics
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Author : John F. Monahan
language : en
Publisher: Cambridge University Press
Release Date : 2001-02-05
Numerical Methods Of Statistics written by John F. Monahan and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-02-05 with Computers categories.
This 2001 book provides a basic background in numerical analysis and its applications in statistics.
Generalized Inverses
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Author : Adi Ben-Israel
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-04-18
Generalized Inverses written by Adi Ben-Israel and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-04-18 with Mathematics categories.
This second edition accounts for many major developments in generalized inverses while maintaining the informal and leisurely style of the 1974 first edition. Added material includes a chapter on applications, new exercises, and an appendix on the work of E.H. Moore.
Eigenvalue Algorithms For Symmetric Hierarchical Matrices
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Author : Thomas Mach
language : en
Publisher: Thomas Mach
Release Date : 2012
Eigenvalue Algorithms For Symmetric Hierarchical Matrices written by Thomas Mach and has been published by Thomas Mach this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
This thesis is on the numerical computation of eigenvalues of symmetric hierarchical matrices. The numerical algorithms used for this computation are derivations of the LR Cholesky algorithm, the preconditioned inverse iteration, and a bisection method based on LDL factorizations. The investigation of QR decompositions for H-matrices leads to a new QR decomposition. It has some properties that are superior to the existing ones, which is shown by experiments using the HQR decompositions to build a QR (eigenvalue) algorithm for H-matrices does not progress to a more efficient algorithm than the LR Cholesky algorithm. The implementation of the LR Cholesky algorithm for hierarchical matrices together with deflation and shift strategies yields an algorithm that require O(n) iterations to find all eigenvalues. Unfortunately, the local ranks of the iterates show a strong growth in the first steps. These H-fill-ins makes the computation expensive, so that O(n³) flops and O(n²) storage are required. Theorem 4.3.1 explains this behavior and shows that the LR Cholesky algorithm is efficient for the simple structured Hl-matrices. There is an exact LDLT factorization for Hl-matrices and an approximate LDLT factorization for H-matrices in linear-polylogarithmic complexity. This factorizations can be used to compute the inertia of an H-matrix. With the knowledge of the inertia for arbitrary shifts, one can compute an eigenvalue by bisectioning. The slicing the spectrum algorithm can compute all eigenvalues of an Hl-matrix in linear-polylogarithmic complexity. A single eigenvalue can be computed in O(k²n log^4 n). Since the LDLT factorization for general H-matrices is only approximative, the accuracy of the LDLT slicing algorithm is limited. The local ranks of the LDLT factorization for indefinite matrices are generally unknown, so that there is no statement on the complexity of the algorithm besides the numerical results in Table 5.7. The preconditioned inverse iteration computes the smallest eigenvalue and the corresponding eigenvector. This method is efficient, since the number of iterations is independent of the matrix dimension. If other eigenvalues than the smallest are searched, then preconditioned inverse iteration can not be simply applied to the shifted matrix, since positive definiteness is necessary. The squared and shifted matrix (M-mu I)² is positive definite. Inner eigenvalues can be computed by the combination of folded spectrum method and PINVIT. Numerical experiments show that the approximate inversion of (M-mu I)² is more expensive than the approximate inversion of M, so that the computation of the inner eigenvalues is more expensive. We compare the different eigenvalue algorithms. The preconditioned inverse iteration for hierarchical matrices is better than the LDLT slicing algorithm for the computation of the smallest eigenvalues, especially if the inverse is already available. The computation of inner eigenvalues with the folded spectrum method and preconditioned inverse iteration is more expensive. The LDLT slicing algorithm is competitive to H-PINVIT for the computation of inner eigenvalues. In the case of large, sparse matrices, specially tailored algorithms for sparse matrices, like the MATLAB function eigs, are more efficient. If one wants to compute all eigenvalues, then the LDLT slicing algorithm seems to be better than the LR Cholesky algorithm. If the matrix is small enough to be handled in dense arithmetic (and is not an Hl(1)-matrix), then dense eigensolvers, like the LAPACK function dsyev, are superior. The H-PINVIT and the LDLT slicing algorithm require only an almost linear amount of storage. They can handle larger matrices than eigenvalue algorithms for dense matrices. For Hl-matrices of local rank 1, the LDLT slicing algorithm and the LR Cholesky algorithm need almost the same time for the computation of all eigenvalues. For large matrices, both algorithms are faster than the dense LAPACK function dsyev.
Adjustment Computations
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Author : Charles D. Ghilani
language : en
Publisher: John Wiley & Sons
Release Date : 2011-08-26
Adjustment Computations written by Charles D. Ghilani and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-08-26 with Technology & Engineering categories.
the complete guide to adjusting for measurement error expanded and updated no measurement is ever exact. Adjustment Computations updates a classic, definitive text on surveying with the latest methodologies and tools for analyzing and adjusting errors with a focus on least squares adjustments, the most rigorous methodology available and the one on which accuracy standards for surveys are based. This extensively updated Fifth Edition shares new information on advances in modern software and GNSS-acquired data. Expanded sections offer a greater amount of computable problems and their worked solutions, while new screenshots guide readers through the exercises. Continuing its legacy as a reliable primer, Adjustment Computations covers the basic terms and fundamentals of errors and methods of analyzing them and progresses to specific adjustment computations and spatial information analysis. Current and comprehensive, the book features: Easy-to-understand language and an emphasis on real-world applications Analyzing data in three dimensions, confidence intervals, statistical testing, and more An updated support web page containing a 150-page solutions manual, software (STATS, ADJUST, and MATRIX for Windows computers), MathCAD worksheets, and more at http://www.wiley.com/college/ghilani The latest information on advanced topics such as the tau criterion used in post-adjustment statistical blunder detection Adjustment Computations, Fifth Edition is an invaluable reference and self-study resource for working surveyors, photogrammetrists, and professionals who use GNSS and GIS for data collection and analysis, including oceanographers, urban planners, foresters, geographers, and transportation planners. It's also an indispensable resource for students preparing for licensing exams and the ideal textbook for courses in surveying, civil engineering, forestry, cartography, and geology.
Numerical Matrix Analysis
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Author : Ilse C. F. Ipsen
language : en
Publisher: SIAM
Release Date : 2009-01-01
Numerical Matrix Analysis written by Ilse C. F. Ipsen and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-01-01 with Mathematics categories.
The purpose of this book is to promote understanding of two phenomena: sensitivity of linear systems and least squares problems, and numerical stability of algorithms. Sensitivity and stability are analyzed as mathematical properties, without reference to finite precision arithmetic. The material is presented at a basic level, emphasizing ideas and intuition, but in a mathematically rigorous fashion. The derivations are simple and elegant, and the results are easy to understand and interpret. The book is self-contained. It was written for students in all areas of mathematics, engineering, and the computational sciences, but can easily be used for self-study. This text differs from other numerical linear algebra texts by offering the following: a systematic development of numerical conditioning; a simplified concept of numerical stability in exact arithmetic; simple derivations; a high-level view of algorithms; and results for complex matrices.
Beginning Matlab And Simulink
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Author : Sulaymon Eshkabilov
language : en
Publisher: Apress
Release Date : 2019-11-28
Beginning Matlab And Simulink written by Sulaymon Eshkabilov and has been published by Apress this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-11-28 with Computers categories.
Employ essential and hands-on tools and functions of the MATLAB and Simulink packages, which are explained and demonstrated via interactive examples and case studies. This book contains dozens of simulation models and solved problems via m-files/scripts and Simulink models which help you to learn programming and modeling essentials. You’ll become efficient with many of the built-in tools and functions of MATLAB/Simulink while solving engineering and scientific computing problems. Beginning MATLAB and Simulink explains various practical issues of programming and modelling in parallel by comparing MATLAB and Simulink. After reading and using this book, you'll be proficient at using MATLAB and applying the source code from the book's examples as templates for your own projects in data science or engineering. What You Will Learn Get started using MATLAB and Simulink Carry out data visualization with MATLAB Gain the programming and modeling essentials of MATLAB Build a GUI with MATLAB Work with integration and numerical root finding methods Apply MATLAB to differential equations-based models and simulations Use MATLAB for data science projects Who This Book Is For Engineers, programmers, data scientists, and students majoring in engineering and scientific computing.
Linear Models And Time Series Analysis
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Author : Marc S. Paolella
language : en
Publisher: John Wiley & Sons
Release Date : 2018-10-10
Linear Models And Time Series Analysis written by Marc S. Paolella and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-10-10 with Mathematics categories.
A comprehensive and timely edition on an emerging new trend in time series Linear Models and Time-Series Analysis: Regression, ANOVA, ARMA and GARCH sets a strong foundation, in terms of distribution theory, for the linear model (regression and ANOVA), univariate time series analysis (ARMAX and GARCH), and some multivariate models associated primarily with modeling financial asset returns (copula-based structures and the discrete mixed normal and Laplace). It builds on the author's previous book, Fundamental Statistical Inference: A Computational Approach, which introduced the major concepts of statistical inference. Attention is explicitly paid to application and numeric computation, with examples of Matlab code throughout. The code offers a framework for discussion and illustration of numerics, and shows the mapping from theory to computation. The topic of time series analysis is on firm footing, with numerous textbooks and research journals dedicated to it. With respect to the subject/technology, many chapters in Linear Models and Time-Series Analysis cover firmly entrenched topics (regression and ARMA). Several others are dedicated to very modern methods, as used in empirical finance, asset pricing, risk management, and portfolio optimization, in order to address the severe change in performance of many pension funds, and changes in how fund managers work. Covers traditional time series analysis with new guidelines Provides access to cutting edge topics that are at the forefront of financial econometrics and industry Includes latest developments and topics such as financial returns data, notably also in a multivariate context Written by a leading expert in time series analysis Extensively classroom tested Includes a tutorial on SAS Supplemented with a companion website containing numerous Matlab programs Solutions to most exercises are provided in the book Linear Models and Time-Series Analysis: Regression, ANOVA, ARMA and GARCH is suitable for advanced masters students in statistics and quantitative finance, as well as doctoral students in economics and finance. It is also useful for quantitative financial practitioners in large financial institutions and smaller finance outlets.
Engineering Design Via Surrogate Modelling
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Author : Alexander Forrester
language : en
Publisher: John Wiley & Sons
Release Date : 2008-09-15
Engineering Design Via Surrogate Modelling written by Alexander Forrester and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-09-15 with Technology & Engineering categories.
Surrogate models expedite the search for promising designs by standing in for expensive design evaluations or simulations. They provide a global model of some metric of a design (such as weight, aerodynamic drag, cost, etc.), which can then be optimized efficiently. Engineering Design via Surrogate Modelling is a self-contained guide to surrogate models and their use in engineering design. The fundamentals of building, selecting, validating, searching and refining a surrogate are presented in a manner accessible to novices in the field. Figures are used liberally to explain the key concepts and clearly show the differences between the various techniques, as well as to emphasize the intuitive nature of the conceptual and mathematical reasoning behind them. More advanced and recent concepts are each presented in stand-alone chapters, allowing the reader to concentrate on material pertinent to their current design problem, and concepts are clearly demonstrated using simple design problems. This collection of advanced concepts (visualization, constraint handling, coping with noisy data, gradient-enhanced modelling, multi-fidelity analysis and multiple objectives) represents an invaluable reference manual for engineers and researchers active in the area. Engineering Design via Surrogate Modelling is complemented by a suite of Matlab codes, allowing the reader to apply all the techniques presented to their own design problems. By applying statistical modelling to engineering design, this book bridges the wide gap between the engineering and statistics communities. It will appeal to postgraduates and researchers across the academic engineering design community as well as practising design engineers. Provides an inclusive and practical guide to using surrogates in engineering design. Presents the fundamentals of building, selecting, validating, searching and refining a surrogate model. Guides the reader through the practical implementation of a surrogate-based design process using a set of case studies from real engineering design challenges. Accompanied by a companion website featuring Matlab software at http://www.wiley.com/go/forrester
Matlab Guide
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Author : Desmond J. Higham
language : en
Publisher: SIAM
Release Date : 2016-12-27
Matlab Guide written by Desmond J. Higham and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-12-27 with Science categories.
MATLAB is an interactive system for numerical computation that is widely used for teaching and research in industry and academia. It provides a modern programming language and problem solving environment, with powerful data structures, customizable graphics, and easy-to-use editing and debugging tools. This third edition of MATLAB Guide completely revises and updates the best-selling second edition and is more than 30 percent longer. The book remains a lively, concise introduction to the most popular and important features of MATLAB and the Symbolic Math Toolbox. Key features are a tutorial in Chapter 1 that gives a hands-on overview of MATLAB; a thorough treatment of MATLAB mathematics, including the linear algebra and numerical analysis functions and the differential equation solvers; and a web page at http://www.siam.org/books/ot150 that provides example program files, updates, and links to MATLAB resources. The new edition contains color figures throughout; includes pithy discussions of related topics in new ?Asides" boxes that augment the text; has new chapters on the Parallel Computing Toolbox, object-oriented programming, graphs, and large data sets; covers important new MATLAB data types such as categorical arrays, string arrays, tall arrays, tables, and timetables; contains more on MATLAB workflow, including the Live Editor and unit tests; and fully reflects major updates to the MATLAB graphics system. This book is suitable for both beginners and more experienced users, including students, researchers, and practitioners.