Topic

# Numerical analysis

About: Numerical analysis is a research topic. Over the lifetime, 52236 publications have been published within this topic receiving 1224230 citations. The topic is also known as: numerical computation & numerical mathematics.

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31 Jan 1986

TL;DR: Numerical Recipes: The Art of Scientific Computing as discussed by the authors is a complete text and reference book on scientific computing with over 100 new routines (now well over 300 in all), plus upgraded versions of many of the original routines, with many new topics presented at the same accessible level.

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

12,662 citations

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TL;DR: In this paper, a method for accurate and efficient local density functional calculations (LDF) on molecules is described and presented with results using fast convergent threedimensional numerical integrations to calculate the matrix elements occurring in the Ritz variation method.

Abstract: A method for accurate and efficient local density functional calculations (LDF) on molecules is described and presented with results The method, Dmol for short, uses fast convergent three‐dimensional numerical integrations to calculate the matrix elements occurring in the Ritz variation method The flexibility of the integration technique opens the way to use the most efficient variational basis sets A practical choice of numerical basis sets is shown with a built‐in capability to reach the LDF dissociation limit exactly Dmol includes also an efficient, exact approach for calculating the electrostatic potential Results on small molecules illustrate present accuracy and error properties of the method Computational effort for this method grows to leading order with the cube of the molecule size Except for the solution of an algebraic eigenvalue problem the method can be refined to quadratic growth for large molecules

8,673 citations

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01 Jan 1978

TL;DR: This report contains a description of the typical topics covered in a two-semester sequence in Numerical Analysis, and describes the accuracy, efficiency and robustness of these algorithms.

Abstract: Introduction. Mathematical approximations have been used since ancient times to estimate solutions, but with the rise of digital computing the field of numerical analysis has become a discipline in its own right. Numerical analysts develop and study algorithms that provide approximate solutions to various types of numerical problems, and they analyze the accuracy, efficiency and robustness of these algorithms. As technology becomes ever more essential for the study of mathematics, learning algorithms that provide approximate solutions to mathematical problems and understanding the accuracy of such approximations becomes increasingly important. This report contains a description of the typical topics covered in a two-semester sequence in Numerical Analysis.

7,315 citations

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01 Jan 1996

TL;DR: This text develops and applies the techniques used to solve problems in fluid mechanics on computers and describes in detail those most often used in practice, including advanced techniques in computational fluid dynamics.

Abstract: Preface. Basic Concepts of Fluid Flow.- Introduction to Numerical Methods.- Finite Difference Methods.- Finite Volume Methods.- Solution of Linear Equation Systems.- Methods for Unsteady Problems.- Solution of the Navier-Stokes Equations.- Complex Geometries.- Turbulent Flows.- Compressible Flow.- Efficiency and Accuracy Improvement. Special Topics.- Appendeces.

7,066 citations

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TL;DR: The numerical tests indicate that the L-BFGS method is faster than the method of Buckley and LeNir, and is better able to use additional storage to accelerate convergence, and the convergence properties are studied to prove global convergence on uniformly convex problems.

Abstract: We study the numerical performance of a limited memory quasi-Newton method for large scale optimization, which we call the L-BFGS method. We compare its performance with that of the method developed by Buckley and LeNir (1985), which combines cycles of BFGS steps and conjugate direction steps. Our numerical tests indicate that the L-BFGS method is faster than the method of Buckley and LeNir, and is better able to use additional storage to accelerate convergence. We show that the L-BFGS method can be greatly accelerated by means of a simple scaling. We then compare the L-BFGS method with the partitioned quasi-Newton method of Griewank and Toint (1982a). The results show that, for some problems, the partitioned quasi-Newton method is clearly superior to the L-BFGS method. However we find that for other problems the L-BFGS method is very competitive due to its low iteration cost. We also study the convergence properties of the L-BFGS method, and prove global convergence on uniformly convex problems.

7,004 citations