## Scientific Computing with MATLAB"It is important to prove, is more but it important to improve." This textbook is an introduction to Scientific Computing. We will illustrate several numerical methods for the computer solution of cer tain classes of mathematical problems that cannot be faced by paper and pencil. We will 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. With this aim, in Chapter 1 we will illustrate the rules of the game that computers adopt when storing and operating with real and complex numbers, vectors and matrices. In order to make our presentation concrete and appealing we will adopt the programming environment MATLAB ® 1 as a faithful companion. We will gradually discover its principal commands, statements and con structs. We will show how to execute all the algorithms that we intro duce throughout the book. This will enable us to furnish an immediate quantitative assessment of their theoretical properties such as stability, accuracy and complexity. We will solve several problems that will be raised through exercises and examples, often stemming from specific ap plications. |

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### Contents

1 | |

Nonlinear equations 37 | 36 |

Approximation of functions and data | 57 |

Numerical differentiation and integration | 83 |

Linear systems | 103 |

Eigenvalues and eigenvectors | 137 |

Ordinary differential equations | 153 |

Numerical methods for boundaryvalue problems | 187 |

Solutions of the exercises | 207 |

245 | |

251 | |

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### Common terms and phrases

absolute stability algorithm approximation backward Euler method bisection method called Cauchy problem Chebyshev nodes coefficients column vector complex number components compute the zeros condition number corresponding Crank-Nicolson method dashed line denote derivative differential equations eigenvalues eigenvector equal exact solution Example Exercise Figure finite difference fixed point iterations following instructions forward Euler method function f Gauss factorization Gauss–Seidel method graph grid Heun method i-th implemented initial vector instance interpolating polynomial iterative method Jacobi method linear system MATLAB MATLAB command MATLAB program method converges midpoint Newton method niter nmax null number of iterations numerical solution obtain perturbation error pivoting polynomial of degree positive definite power method quadrature formula real numbers requires residual respect to h result roundoff errors Runge-Kutta method satisfies sequence significant digits Simpson formula solve square matrix subintervals symmetric and positive trapezoidal formula tridiagonal trigonometric interpolant un+1 values variable verify