## Exploring Numerical Methods: An Introduction to Scientific Computing Using MATLABExploring Numerical Methods is designed to provide beginning engineering and science students, as well as upper-level mathematics students, with an introduction to numerical analysis that emphasizes insight and hands-on experience. To serve the needs of both the younger and the more experienced audience, each chapter begins with an intuitive presentation of motivation and simple algorithms. Topics are developed progressively within each chapter and the advanced material, which reveals underlying theory and discusses complicated methods, is clearly marked. The text takes a focused approach to introducing the more important numerical algorithms and exposes students to partial differential equations by using simple prototypes. This text provides a strong experiential basis for future study. |

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### Contents

Numerical Analysis and Scientific Computing | 7 |

The Solution of Systems of Linear Equations | 31 |

Polynomials and Polynomial Approximations | 57 |

Chapter 5 | 85 |

3 | 125 |

The Solution of Nonlinear Equations | 171 |

Advanced Matrix Algorithms | 209 |

Eigenvalue Problems | 237 |

Boundary and Eigenvalue Problems for Ordinary | 313 |

Initial Value Problems for Partial | 343 |

Boundary Value Problems for Partial | 365 |

IllConditioned and IllPosed Problems | 383 |

Numerical Computing in MATLAB | 413 |

NASOFT User Manual and Example Scripts | 427 |

Explorations | 449 |

465 | |

### Common terms and phrases

accuracy algorithms approximate solution bound boundary conditions boundary value problem characteristic polynomial Chebyshev polynomials coefficients collocation complicated condition number cubic spline degree polynomial denote derivative diagonal discretization error eigenvalue problem eigenvectors elements estimate Euler's method evaluated Example Exercise f(xi finite difference method floating-point formulas function grid heat equation Hilbert matrix IEEE ill-conditioned implementation initial value problems integral integrand interpolation interval inverse iteration knots least squares solution linear system MATLAB matrix mesh points minimization multistep method Newton's method nonlinear nonzero norm numerical methods order of convergence orthogonal panel partial differential equations piecewise polynomials pivot pn(x polynomial approximation polynomial of degree power method quadrature points right side root root-finding rounding errors satisfies secant method sequence Show significant digits simple singular values solve stability step Suppose symmetric symmetric matrix Table Theorem tion vector weights Xi+i y(xi zero