Numerical Analysis: A Mathematical Introduction
This book provides professionals and students with a thorough understanding of the interface between mathematics and scientific computation. Ranging from classical questions to modern techniques, it explains why numerical computations succeed or fail. The book is divided into four sections, with an emphasis on the use of mathematics as a tool in determining the success rate of numerical methods. The text requires only a modest level of mathematical training, and is ideally suited for scientists and students in mathematics, physics and engineering.
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The entrance fee
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