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Numerical Analysis and Scientific Computation


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Table of Contents

1. Nonlinear Equations.
Biscetion and Inverse Linear Interpolation.Newton's Method.The Fixed Point Theorem.Quadratic Convergence of Newton's Method.Variants of Newton's Method.Brent's Method.Effects of Finite Precision Arithmetic.Newton's Method for Systems.Broyden's Method.

2. Linear Systems.
Gaussian Elimination with Partial Pivoting.The LU Decomposition.The LU Decomposition with Pivoting.The Cholesky Decomposition.Condition Numbers.The QR Decomposition.Householder Triangularization and the QR Decomposition.Gram-Schmidt Orthogonalization and the QR Decomposition.The Singular Value Decomposition.

3. Iterative Methods.
Jacobi and Gauss-Seidel Iteration.Sparsity.Iterative Refinement.Preconditioning.Krylov Space Methods.Numerical Eigenproblems.

4. Polynomial Interpolation.
Lagrange Interpolating Polynomials.Piecewise Linear Interpolation.Cubic Splines.Computation of the Cubic Spline Coefficients.

5. Numerical Integration.
Closed Newton-Cotes Formulas.Open Newton-Cotes Formulas and Undetermined Coeffients.Gaussian Quadrature.Gauss-Chebyshev Quadrature.Radau and Lobatto Quadrature.Adaptivity and Automatic Integration.Romberg Integration.

6. Differential Equations.
Numerical Differentiation.Euler's Method.Improved Euler's Method.Analysis of Explicit One-Step Methods.Taylor and Runge-Kutta Methods.Adaptivity and Stiffness.Multi-Step Methods.

7. Nonlinear Optimization.
One-Dimensional Searches.The Method of Steepest Descent.Newton Methods for Nonlinear Optimization.Multiple Random Start Methods.Direct Search Methods.The Nelder-Mead Method.Conjugate Direction Methods.

8. Approximation Methods.
Linear and Nonlinear Least Squares.The Best Approximation Problem.Best Uniform Approximation.Applications of the Chebyshev Polynomials.





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