# Numerical Analysis B

## Objectives

We will illustrate several numerical methods for the computer solution of certain classes of mathematical problems. We will show how to use these methods in order to solve nonlinear equations, linear systems, integrate and construct accurate approximations for the solution of differential equations.

## General characterization

12686

6.0

##### Responsible teacher

Maria Paula da Costa Couto

Weekly - 4

Total - 56

Português

### Prerequisites

Basic knowledge in analysis (calculus) and linear algebra

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

• Santos, F. Correia dos; Duarte, Jorge; Lopes, Nuno D.,
Fundamentos de Análise Numérica (Com Python 3 e R), Edições Sílabo, 2019 (2ª edição).

### Teaching method

The course works with theoretical-practical classes (TP), in which the successive topics of the UC program will be explained and discussed. In order to consolidate a given subject, exercises related to each of the topics covered will be solved.

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

Evaluation

Assessment are classified from 0 to 20 values. A student obtains approval if the final grade at U.C. is greater than or equal to 9.5 values.

For any evaluation (test or exam) the student must have his Citizen Card or official identification document.

For tests or exams, the student may use a calculating machine, which may be graphical. Other electronic material is forbidden.

In the assessment the student can choose between continuous evaluation or by appeal exam.

2.1 Continuous evaluation

The evaluation during the semester consists of two tests lasting two hours and a computational work to be done in groups of 3 or 4 students. The student may choose to do the computational work or not. Let NT1 and NT2 be the grades of tests 1 and 2, respectively, and NTC the classification of computational work. If a student does not attend one of the assessments, this assessment element will be rated 0.

If the student performs the computational work, the grade of continuous assessment (Nav) is given by:

Nav = 0.45 × NT1 + 0.45 × NT2 + 0.10 × NTC.

If the student does not perform the computational work, the grade of continuous assessment (Nav) is

Nav = 0.5 × NT1 + 0.5 × NT2

2.2 Examination Appeal

Any student not yet approved in the course may perform the appeal exam. The appeal exam lasts for 3 hours.

If the CR exam grade is lower than 9.5, the student fails.

If CR ≥ 9.5 and the student has performed computational work, the appeal grade; NR will be given by:

NR = max {CR, 0.90 × CR + 0.10 × NTC};

If CR ≥ 9.5 and the student has not performed computational work, the appeal grade; NR will only be CR.

Students wishing to take the exam to improve their grade must, in advance, request such an improvement from the academic services.

The grade improvement exam is graded in a similar manner to the Season of Appeal.

If the result is higher than that already obtained in U.C, it will be taken as a final grade. Otherwise, there is no grade improvement, maintaining the previous grade.

4. Special exam

The special season exam classification is carried out analogously to the Examination appeal.

## 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: