Numerical Analysis: A Mathematical Introduction
Numerical analysis explains why numerical computations work, or fail. This book is divided into four parts. Part I starts Part I starts with a guided tour of floating number systems and machine arithmetic. The exponential and the logarithm are constructed from scratch to present a new point ofview on questions well-known to the reader, and the needed knowledge of linear algebra is summarized. Part II starts with polynomial approximation (polynomial interpolation, mean-square approximation, splines). It then deals with Fourier series, providing the trigonometric version of least squareapproximations, and one of the most important numerical algorithms, the fast Fourier transform. Any scientific computation program spends most of its time solving linear systems or approximating the solution of linear systems, even when trying to solve non-linear systems. Part III is therefore aboutnumerical linear algebra, while Part IV treats a selection of non-linear or complex problems: resolution of linear equations and systems, ordinary differential equations, single step and multi-step schemes, and an introduction to partial differential equations. The book has been written having inmind the advanced undergraduate students in mathematics who are interested in the spice and spirit of numerical analysis. The book does not assume previous knowledge of numerical methods. It will also be useful to scientists and engineers wishing to learn what mathematics has to say about thereason why their numerical methods work - or fail.
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A flavour of numerical analysis
Polynomial and trigonometric approximation
Leastsquares approximation for polynomials
Nonlinear equations and systems
Solving differential systems
Linear multistep schemes
Towards partial differential equations
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algebra algorithm approximation Assume B-splines basis block bounded calculate column Consequently construct continuous function convergence deduce defined denote diagonal differential equations dimension divided differences eigenvalues eigenvectors equal error estimate Euler scheme Exercise finite floating-point Fourier series give given Hermitian Hermitian matrix Hessenberg identity inequality initial condition integral interpolation polynomial interval invertible invertible matrix iterative method knots Lemma linear system LU decomposition modulus multiplication multistep Newton's method nonzero norm notation numerical analysis obtain operator norm orthogonal orthogonal polynomials orthonormal polynomial of degree positive definite problem Proof properties prove quadrature formula real number relation respect result satisfies scalar product sequence Show solution of eqn solve spectral radius splines stable strictly positive Subsection Suppose tends to infinity Theorem triangular matrix trigonometric polynomials unique upper triangular vanishes variable vector space zero