Numerical Analysis and Implementation Environments

1. Introduction to Numerical Analysis. 
2. Problem Solving Environments. Numerical software. Introduction to MATLAB and MATLAB programming.
3. Finite precision arithmetic, errors, floating point numbers, the IEEE floating point standard.
4. Error propagation in numerical computations. Condition number of mathematical problems. Forward and backward algorithm stability.
5. Numerical solution of non-linear equations (bisection, fixed-point iteration, Newton, secant). MATLAB functions.
6. Review of numerical linear algebra concepts: norms, special matrices, singular values, the SVD.
7. Solving linear systems: Direct methods (LU, Cholesky and their variants). MATLAB functions and the backslash operator.
8. Linear least squares problems: Normal equations. QR factorization. Householder reflections and Gram-Schmidt orthogonalization. MATLAB functions.
9. Iterative methods for solving systems: Jacobi, Gauss-Seidel, Richardson, SOR. Descent methods, conjugate directions. A brief introduction to Krylov subspaces. The Conjugate Gradient method and preconditioning for symmetric positive definite matrices. MATLAB functions.
10. Approximation of eigenvalues ​​and eigenvectors: Power method, inverse power method and variants, Schur decomposition. Brief description of subspace methods and the QR algorithm. MATLAB functions.
11. Nonlinear systems and optimization: Newton, quasi-Newton, some methods for unconstrained optimization. MATLAB functions.
12. Polynomial approximation and basic theorems. Polynomial interpolation and its representations (Newton, Lagrange, barycentric, Hermite). The Runge phenomenon. Chebyshev points and barycentric interpolation. The Chebfun package.
13. Piecewise polynomial interpolation and splines.  Adaptive methods. MATLAB functions.
14. Numerical differentiation: Derivation of simple formulas from the Taylor series, Richardson extrapolation, methods from Lagrange interpolation.
15. Numerical integration: simple and composite methods (midpoint, trapezoidal, Simpson). Adaptive methods. MATLAB functions.
16. Numerical solution of differential equations: Introduction to the numerical solution of DEs, classifications. Methods for initial value problems. Stability, consistency, convergence. Single-step methods: Forward Euler, backward Euler,  the matrix exponential. Stiffness. Runge-Kutta methods and the application of Richardson extrapolation. MATLAB functions.

Bibliography:

The textbook for the course is: U. Ascher and C. Greif, A First Course in Numerical Methods, SIAM, 2011. Other good references include: A. Quarteroni, F. Saleri and P. Gervasio , Scientific Computing with MATLAB and Octave, 4th ed., Springer, 2014. C. Moler, Numerical Computing with MATLAB, SIAM, available online.