## Concise Numerical MathematicsThis book succinctly covers the key topics of numerical methods. While it is basically a survey of the subject, it has enough depth for the student to walk away with the ability to implement the methods by writing computer programs or by applying them to problems in physics or engineering. The author manages to cover the essentials while avoiding redundancies and using well-chosen examples and exercises.The exposition is supplemented by numerous figures. Work estimates and pseudo codes are provided for many algorithms, which can be easily converted to computer programs. Topics covered include interpolation, the fast Fourier transform, iterative methods for solving systems of linear and nonlinear equations, numerical methods for solving ODEs, numerical methods for matrix eigenvalue problems, approximation theory, and computer arithmetic. In general, the author assumes only a knowledge of calculus and linear algebra. The book is suitable as a text for a first course in numerical methods for mathematics students or students in neighboring fields, such as engineering, physics, and computer science. |

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### Contents

1 | |

Spline Functions | 23 |

The Discrete Fourier Transform and Its Applications | 41 |

Solution of Linear Systems of Equations | 59 |

Nonlinear Systems of Equations | 105 |

The Numerical Integration of Functions | 123 |

Explicit OneStep Methods for Initial Value Problems in Ordinary Differential Equations | 161 |

Multistep Methods for Initial Value Problems of Ordinary Differential Equations | 189 |

The Conjugate Gradient and GMRES Methods | 311 |

Eigenvalue Problems | 339 |

Numerical Methods for Eigenvalue Problems | 355 |

Peanos Error Representation | 393 |

Approximation Theory | 401 |

Computer Arithmetic | 423 |

443 | |

449 | |

Boundary Value Problems for Ordinary Differential Equations | 247 |

Jacobi GaussSeidel and Relaxation Methods for the Solution of Linear Systems of Equations | 281 |

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Additional topics arbitrary arithmetic operations Arnoldi process boundary value problem calculation CNxN coefficients completes the proof consistency order convergence Corollary corresponding cubic spline denotes determined discrete Fourier transform eigenvalues error estimate error representation example Exercise factorization Figure floating point number following holds following theorem follows directly function f G RN Galerkin method Gauss-Seidel method Hessenberg Hessenberg matrix identity inequality initial value problem interpolating polynomial interpolation interval introduced iteration error iteration function Jacobi method Lemma linear subspace linear system Lipschitz condition m-step method matrix A G RNxN matrix norm multistep methods Newton's method notation obtains one-step method orthogonal pairwise distinct point number system positive definite proof of Theorem QR algorithm regular matrix Remark respect satisfied scalar product Section solution space statement support abscissas symmetric system of equations topics and literature upper triangular vector norm yields