|Teoria||9||1st Semester, 2nd Semester||Leonard Peter Bos|
|Laboratorio||3||1st Semester, 2nd Semester||Manolo Venturin|
|Teoria||Thursday||4:30 PM - 5:30 PM||lesson||Lecture Hall E|
|Laboratorio||Tuesday||4:30 PM - 6:30 PM||lesson||Lecture Hall E, Delta|
|Teoria||Monday||11:30 AM - 1:30 PM||lesson||Lecture Hall E|
|Teoria||Friday||8:30 AM - 10:30 AM||lesson||Lecture Hall E|
|Laboratorio||Wednesday||8:30 AM - 10:30 AM||laboratorio||Laboratory Delta|
In this course we will study the most important methods for the numerical solution of the classical problems of Mathematical Analysis. Besides the required theoretical background material needed for understanding the subject, particular emphasis will be placed on algorithms, including their implementation, their complexity and efficiency, as well as the purely numerical problems of convergence and stability. The objective is thus not just to provide the student with a knowledge of the methods, but also to cultivate a "numerical intution" which is important for solving real world problems.
The implementation in Matlab and/or GNU Octave of the basic algorithms of Numerical Analysis.
* Error Analysis
Representation of numbers. Absolute error and relative error. Machine numbers and associated errors. Algoriths for evaluating an expression. Conditioning of problems and the stability of methods.
* Non-linear Equations
The bisection method. Fixed point methods: generality, convergence and stopping criteria. The secant method, Newton's method and Aitken acceleration. Algebraic polynomials: Horner's rule.
* Linear Systems
Direct methods: LU factorization and pivoting, forward and back substitution, fast algorithms for tridiagonal systems.
Iterative methods: Jacobi iteration, Gauss-Seidel and SOR. Iterative improvement. Richardson's method and the gradient method. Sparse and banded systems. Solution of over and under determined systems. Solution of poorly conditioned systems.
* Eigenvalues and Eigenvectors
Localization of eigenvalues: Gerschgorin circles. The power method and the inverse power method. The QR method and its variants. Eigenvalues of tridiagonal matrices: Schur's method.
* Interpolation and Approximation of Functions and Data.
Polynomial interpolation: the Newton and Lagrange form. Approximation error estimates. Trigonometric interpolation and the the Fast Fourier Transform (FFT). Piecewise polynomial interpolation and splines. Least squares and the SVD.
* Numerical Differentiation and Integration.
Simple formulas for approximating derivatives and their relative error.
Numerical integration or quadrature: interpolation based formulas, both simple and composite. Error in quadrature. Adaptive methods. Gaussian quadrature.
* Numerical Solution of Ordinary Differential equations (ODE).
What the student has learned will be tested by means of an Oral Exam. In the first part you will be asked to discuss a number of the exercises given during the Laboratory. Then you will be asked some questions regarding the theory discussed during the lectures. The student is to bring with them their exercise notes and problem solutions to the exam.
Hence attending the Laboratory and solving the assigned problems are necessary in order to be able to pass the exam.