Numerical Analysis

General characterization

Prerequisites

Basic knowledge in analysis and linear algebra

Bibliography

• Quarteroni A., Saleri F.,  Scientific Computing with MATLAB and Octave, Series: Texts in Computational Science and Engineering , Vol. 2 Springer, 2006

• Quarteroni A., Sacco R., Saleri F., Numerical Mathematics, Springer, 2000

• Burden R. e Faires J. , Numerical Analysis, Brooks-Cole Publishing Company, 9th Edition, 2011.

• Pina H., Métodos Numéricos, Mc Graw Hill, 1995

Teaching method

The course works with theoretical-practical classes (TP), in which the successive topics of the chair program will be explained and discussed. In order to consolidate a given subject, exercises related to each of the topics covered will be solved. In order to implement some of the methods covered, TP classes will be taught in the Lab where there are computers with wxMaxima software installed.

Students must attend, at least 2/3 of TPs classes or attend the previous year.

The evaluation of the course consists of two tests that address the knowledge acquired in TPs classes and / or a computational work to be prepared in group.

Evaluation method

Important:

In order to be evaluated, the student must attend at least 2/3 of the TP classes.

Evaluation Methods:

1-Continuous evaluation

The continuous evaluation consists of two tests during the semester and or a computational work.

•   If the student chooses the evaluation which consists of two test and a computational work the final grade, FG, is given by 

FG ​​= 0.35 × NT1 + 0.55 × NT2 + 0.1 × NTC,

             where NT1, NT2 are the grades of the Test 1 and Test 2, respectively and NTC is the garde of the computational work.

•   If the student chooses the evaluation which consists of two test, the final grade, FG, is given by 

FG ​​= 0.4 × NT1 + 0.6 × NT2,

             where NT1, NT2 are the grades of the Test 1 and Test 2, respectively.
  

 The student is approved if the final grade is greater than or equal to 9.5.

The first test lasts for 1h30m and the second test lasts for 2h ..

2-Final exam evaluation.

The student is approved if the grade of the final exam, FE, is greater than or equal to 9.5 and the final grade is given by
FG = max {FE, 0.9xFE + 0.1xNTC}

The final exam lasts for 3h.

Subject matter

1.Introduction

1.1 Errors, significant digits.

1.2 Conditioning of a problem and stability of a method. 

1.3 Introduction to a computational program for Numerical Analysis.

2. Polynomial approximation and interpolation

2.1 Interpolation and Lagrange polynomial

2.2 Divided differences, interpolating polynomial of Newton.

2.3 Cubic Spline interpolation.

2.4 Least squares approximation. 

3. Numerical integration

3. 1 Newton-Cotes integration formulas (Single and composite rules)

3.2 Gaussian integration. Other integration methods.

4. Root finding for nonlinear equations

4.1 Bisection method.

4.2 Fixed-point iteration method. Newton method. Secant method. 

5. Iterative methods for solving linear systems of equations

5.1 Norms of vectors and matrices. Conditioning of a system.

5.2 Eigenvalues and eigenvectors. Gershgorin theorem.

5.3 Iterative methods: general procedure.

5.4 Jacobi, Gauss-Seidel and relaxation methods.

6. Numerical solution of ordinary differential equations

6.1 Euler methods.

6.2 Taylor methods.

6.3 Runge-Kutta methods.

Programs

Programs where the course is taught: