Introduction to Numerical Analysis
Numerical analysis is an increasingly important link between pure mathematics and its application in science and technology. This textbook provides an introduction to the justification and development of constructive methods that provide sufficiently accurate approximations to the solution of numerical problems, and the analysis of the influence that errors in data, finite-precision calculations, and approximation formulas have on results, problem formulation and the choice of method. It also serves as an introduction to scientific programming in MATLAB, including many simple and difficult, theoretical and computational exercises. A unique feature of this book is the consequent development of interval analysis as a tool for rigorous computation and computer assisted proofs, along with the traditional material.
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absolute error accuracy Algorithm analysis approximation arbitrary arithmetic expression automatic differentiation bisection method calculation Cholesky factorization coefficients column computed condition number continuously differentiable functions convergence order decimal defined derivative determined diagonal differential equations differential numbers digits divided differences eigenvalues error bounds Euler's method evaluation example extrapolation function values Gaussian elimination gives grid hence Hermitian holds Horner scheme integral interpolation points interpolation polynomial interval arithmetic iterative refinement linear systems M-matrix MATLAB matrix mean value form monotone multiple Neumaier Newton method nodes nonlinear nonsingular norm numerically stable obtain operations permutation permutation matrix piecewise linear pivoting polynomial of degree positive definite Proof Proposition quadrature formula quadrature rules recursively relative error root rounding errors satisfies secant method Section sequence Show sign change simple zero singular solving spline stable Suppose system of equations triangular factorization vector