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Very suitable for the scientific computing part of the everywhere growing CSE courses
Eminently suitable for physicists, chemists, engineers and other scientists who have to upgrade their knowledge on numerical methods
This textbook is an introduction to Scientific Computing, in which several numerical methods for the computer solution of certain classes of mathematical problems are illustrated. The authors show how to compute the zeros or the integrals of continuous functions, solve linear systems, approximate functions by polynomials and construct accurate approximations for the solution of differential equations. To make the presentation concrete and appealing, the programming environment Matlab is adopted as a faithful companion. All the algorithms introduced throughout the book are shown, thus furnishing an immediate quantitative assessment of their theoretical properties such as stability, accuracy and complexity. The book also contains the solution to several problems raised through exercises and examples, often originating from specific applications. A specific section is devoted to subjects which were not addressed in the book and indicate the bibliographical references for a more comprehensive treatment of the material.
1. What can’t be ignored.- 1.1 Real numbers.- 1.1.1 How do we represent them.- 1.1.2 How do we operate with floating-point numbers.- 1.2 Complex numbers.- 1.3 Matrices.- 1.3.1 Vectors.- 1.4 Real functions.- 1.4.1 The zeros.- 1.4.2 Polynomials.- 1.4.3 Integration and differentiation.- 1.5 To err is not only human.- 1.5.1 Talking about costs.- 1.6 A few more words about MATLAB.- 1.6.1 MATLAB statements.- 1.6.2 Programming in MATLAB.- 1.7 What we haven’t told you.- 1.8 Exercises.- 2. Nonlinear equations.- 2.1 The bisection method.- 2.2 The Newton method.- 2.3 Fixed point iterations.- 2.3.1 How to terminate fixed point iterations.- 2.4 What we haven’t told you.- 2.5 Exercises.- 3. Approximation of functions and data.- 3.1 Interpolation.- 3.1.1 Lagrangian polynomial interpolation.- 3.1.2 Chebyshev interpolation.- 3.1.3 Trigonometric interpolation and FFT.- 3.2 Piecewise linear interpolation.- 3.3 Approximation by spline functions.- 3.4 The least squares method.- 3.5 What we haven’t told you.- 3.6 Exercises.- 4. Numerical differentiation and integration.- 4.1 Approximation of function derivatives.- 4.2 Numerical integration.- 4.2.1 Midpoint formula.- 4.2.2 Trapezoidal formula.- 4.2.3 Simpson formula.- 4.3 Simpson adaptive formula.- 4.4 What we haven’t told you.- 4.5 Exercises.- 5. Linear systems.- 5.1 The LU factorization method.- 5.2 The technique of pivoting.- 5.3 How accurate is the LU factorization?.- 5.4 How to solve a tridiagonal system.- 5.5 Iterative methods.- 5.5.1 How to construct an iterative method.- 5.6 When should an iterative method be stopped?.- 5.7 Richardson method.- 5.8 What we haven’t told you.- 5.9 Exercises.- 6. Eigenvalues and eigenvectors.- 6.1 The power method.- 6.1.1 Convergence analysis.- 6.2 Generalization of the power method.- 6.3 How to compute the shift.- 6.4 Computation of all the eigenvalues.- 6.5 What we haven’t told you.- 6.6 Exercises.- 7. Ordinary differential equations.- 7.1 The Cauchy problem.- 7.2 Euler methods.- 7.2.1 Convergence analysis.- 7.3 The Crank-Nicolson method.- 7.4 Zero-stability.- 7.5 Stability on unbounded intervals.- 7.5.1 Absolute stability controls perturbations.- 7.6 High order methods.- 7.7 The predictor-corrector methods.- 7.8 Systems of differential equations.- 7.9 What we haven’t told you.- 7.10 Exercises.- 8. Numerical methods for boundary-value problems.- 8.1 Approximation of boundary-value problems.- 8.1.1 Approximation by finite differences.- 8.1.2 Approximation by finite elements.- 8.2 Finite differences in 2 dimensions.- 8.2.1 Consistency and convergence.- 8.3 What we haven’t told you.- 8.4 Exercises.- 9. Solutions of the exercises.- 9.1 Chapter 1.- 9.2 Chapter 2.- 9.3 Chapter 3.- 9.4 Chapter 4.- 9.5 Chapter 5.- 9.6 Chapter 6.- 9.7 Chapter 7.- 9.8 Chapter 8.- Index of MATLAB Programs.