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AuthorHรคmmerlin, Gรผnther. author
TitleNumerical Mathematics [electronic resource] / by Gรผnther Hรคmmerlin, Karl-Heinz Hoffman
ImprintNew York, NY : Springer New York, 1991
Connect tohttp://dx.doi.org/10.1007/978-1-4612-4442-4
Descript XII, 425 p. online resource

SUMMARY

"In truth, it is not knowledge, but learning, not possessing, but production, not being there, but travelling there, which provides the greatest pleasure. When I have completely understood something, then I turn away and move on into the dark; indeed, so curious is the insatiable man, that when he has completed one house, rather than living in it peacefully, he starts to build another. " Letter from C. F. Gauss to W. Bolyai on Sept. 2, 1808 This textbook adds a book devoted to applied mathematics to the series "Grundwissen Mathematik. " Our goals, like those of the other books in the series, are to explain connections and common viewpoints between various mathematical areas, to emphasize the motivation for studying certain probยญ lem areas, and to present the historical development of our subject. Our aim in this book is to discuss some of the central problems which arise in applications of mathematics, to develop constructive methods for the numerical solution of these problems, and to study the associated questions of accuracy. In doing so, we also present some theoretical results needed for our development, especially when they involve material which is beyond the scope of the usual beginning courses in calculus and linear algebra. This book is based on lectures given over many years at the Universities of Freiburg, Munich, Berlin and Augsburg


CONTENT

1 Computing -- ยง1. Numbers and Their Representation -- ยง2. Floating Point Arithmetic -- ยง3. Error Analysis -- ยง4. Algorithms -- 2. Linear Systems of Equations -- ยง1. Gauss Elimination -- ยง2. The Cholesky Decomposition -- ยง3. The QR Decomposition of Householder -- ยง4. Vector Norms and Norms of Matrices -- ยง5. Error Bounds -- ยง6. III-Conditioned Problems -- 3. Eigenvalues -- ยง1. Reduction to Tridiagonal or Hessenberg Form -- ยง2. The Jacobi Rotation and Eigenvalue Estimates -- ยง3. The Power Method -- ยง4. The QR Algorithm -- 4. Approximation -- ยง1. Preliminaries -- ยง2. The Approximation Theorems of Weierstrass -- ยง3. The General Approximation Problem -- ยง4. Uniform Approximation -- ยง5. Approximation in Pre-Hilbert Spaces -- ยง6. The Method of Least Squares -- 5. Interpolation -- ยง1. The Interpolation Problem -- ยง2. Interpolation Methods and Remainders -- ยง3. Equidistant Interpolation Points -- ยง4. Convergence of Interpolating Polynomials -- ยง5. More on Interpolation -- ยง6. Multidimensional Interpolation -- 6. Splines -- ยง1. Polynomial Splines -- ยง2. Interpolating Splines -- ยง3. B-splines -- ยง4. Computing Interpolating Splines -- ยง5. Error Bounds and Spline Approximation -- ยง6. Multidimensional Splines -- 7. Integration -- ยง1. Interpolatory Quadrature -- ยง2. Extrapolation -- ยง3. Gauss Quadrature -- ยง4. Special Quadrature Methods -- ยง5. Optimality and Convergence -- ยง6. Multidimensional Integration -- 8. Iteration -- ยง1. The General Iteration Method -- ยง2. Newtonโ{128}{153}s Method -- ยง3. Iterative Solution of Linear Systems of Equations -- ยง4. More on Convergence -- 9. Linear. Optimization -- ยง1. Introductory Examples and the General Problem -- ยง2. Polyhedra -- ยง3. The Simplex Method -- ยง4. Complexity Analysis -- References -- Symbols


Mathematics Numerical analysis Mathematics Numerical Analysis



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