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About this book :-
Numerical Analysis written by Ridgway Scott.
This book is an invitation to study more deeply advanced topics in mathematics. It may require a short detour to understand completely what is being said regarding operator theory in infinite-dimensional vector spaces or regarding algebraic concepts like tensors and flags. Numerical analysis provides, in a way that is accessible to advanced undergraduates, an introduction to many of the advanced concepts of modern analysis.
We have assumed that the general style of a course using this book will be to prove theorems. Indeed, we have attempted to facilitate a “Moore2 method” style of learning by providing a sequence of steps to be verified as exercises. This has also guided the set of topics to some degree. We have tried to hit the interesting points, and we have kept the list of topics covered as short as possible. Completeness is left to graduate level courses using the texts we mention at the end of many chapters.
Book Detail :-
Title: Numerical Analysis
Author(s): L. Ridgway Scott
Publisher: Princeton University Press
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About Author :-
The author L. Ridgway Scott has been Professor of Computer Science and of Mathematics at the University of Chicago from 1998 to 2017 and the Louis Block Professor since 2001.
He obtained the BSc from Tulane University in 1969 and the PhD degree in Mathematics from the Massachusetts Institute of Technology in 1973.
Professor Scott was a founding member of the Advanced Computer Architecture Laboratory at University of Michigan, an early center for the study of parallel computing and a “beta-site” for one of the first-generation of hypercube computers, the nCUBE-1. He also helped to establish a program in parallel scientific computing at the Pennsylvania State University, which became a “beta-site” for the second-generation Intel hypercube, the iPSC-2. He co-founded what later became the W. G. Pritchard Fluid Mechanics Laboratory at Penn State.
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Book Contents :-
Numerical Analysis written by Ridgway Scott cover the following topics. '
Finding roots, Analyzing Heron’s algorithm, Where to start, An unstable algorithm, General roots: effects of floating-point, Exercises, Solutions
Fixed-point iteration, Particular methods, Complex roots, Error propagation, More reading, Exercises, Solutions
Gaussian elimination, Factorization, Triangular matrices, Pivoting, More reading, Exercises, Solutions
Direct factorization, Caution about factorization, Banded matrices, More reading, Exercises, Solution
Normed vector spaces, Proving the triangle inequality, Relations between norms, Inner-product spaces, More reading, Exercises, Solutions
Operators, Schur decomposition, Convergent matrices, Powers of matrices, Exercises, Solutions
Functional iteration for systems, Newton’s method, Limiting behavior of Newton’s method, Mixing solvers, More reading, Exercises, Solutions
Stationary iterative methods, General splittings, Necessary conditions for convergence, More reading, Exercises, Solutions
Minimization methods, Conjugate Gradient iteration, Optimal approximation of CG, Comparing iterative solvers, More reading, Exercises, Solutions
Local approximation: Taylor’s theorem, Distributed approximation: interpolation, Norms in infinite-dimensional spaces, More reading, Exercises, Solutions
Chebyshev and Hermite Interpolation
Error term ω, Chebyshev basis functions, Lebesgue function, Generalized interpolation, More reading, Exercises, Solutions
Best approximation by polynomials, Weierstrass and Bernstein, Least squares, Piecewise polynomial approximation, Adaptive approximation, More reading, Exercises, Solutions
Interpolatory quadrature, Peano kernel theorem, Gregorie-Euler-Maclaurin formulas, Other quadrature rules, More reading, Exercises, Solutions Eigenvalue Problems
Eigenvalue examples, Gershgorin’s theorem, Solving separately, How not to eigen, Reduction to Hessenberg form, More reading, Exercises, Solution
Power method, Inverse iteration, Singular value decomposition, Comparing factorizations, More reading, Exercises, Solutions
Ordinary Differential Equations
Basic theory of ODEs, Existence and uniqueness of solutions, Basic discretization methods, Convergence of discretization methods, More reading, Exercises, Solutions
Higher-order ODE Discretization Methods
Higher-order discretization, Convergence conditions, Backward differentiation formulas, More reading, Exercises, Solutions
Floating-point arithmetic, Errors in solving systems, More reading, Exercises, Solutions
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