Numerical Analysis


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





Responsible teacher

António Manuel Morais Fernandes de Oliveira, Maria Cecília Marques Rodrigues


Weekly - 3

Total - 39

Teaching language



Basic knowledge in analysis and linear algebra


  • Atkinson K., An Introduction to Numerical Analysis, Wiley, Second Edition, 1989.
  • Burden R. e Faires J. , Numerical Analysis, Brooks-Cole Publishing Company, 9th Edition, 2011.
  • Conte S. e Boor C., Elementary Numerical Analysisan algorithmic approach, Mc Graw Hill, 1981
  • Isaacson E. e Keller H., Analysis of Numerical Methods, Dover, 1994
  • Apontamentos de apoio às aulas teórico-práticas de Cálculo Numérico, Cecília Rodrigues, 2018, disponível no Clip.
  • Pina H., Métodos Numéricos, Mc Graw Hill, 1995
  • Valença M. R., Métodos Numéricos,Livraria Minho, Terceira Edição, 1993

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. In order to implement some of the methods covered, some classes will be taught in the Lab where there are computers with wxMaxima software installed.

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

1. Frequency

All students are  dismissed from obtaining attendance, however, in the laboratory classes, a record of the students present will be made for contact control, in case of infection by Covid.

Any student must be registered in one of the laboratory classes.


2. 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.

The student who obtains a final grade at U.C. equal to or higher than 17.5 values can choose to get the final grade of 17 values or take a complementary evaluation for grade defence. If the student does not take this evaluation will have the final grade of 17 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 one hour and a half and a computational work to be done in groups of 3 or 4 students, in wxmaxima language. Let NT1 and NT2 be the grades of tests 1 and 2, respectively, and NTC the classification of computational work. The student may choose to do the computational work or not. 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

Note: the weights of the test scores can be changed depending on the test dates.

If Nav

If 9.5 ≤ Nav

If Nav ≥ 17.5 values the student can choose to get the final grade of 17 values or take a complementary test for grade defence.


2.2 Examination Appeal

An appeal exam may be submitted to any student not yet approved in the course. 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.

If 9.5 ≤ NR

If NR ≥ 17.5 values the student can choose to get the final grade of 17 values or take a complementary test for grade defence.


3. Grade improvement

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


1.1 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 Direct methods: Gauss Method; pivoting strategies.

5.3 Iterative methods: general procedure.

5.4 Jacobi and Gauss-Seidel methods.


6. Numerical solution of ordinary differential equations

6.1 Euler methods.

6.2 Taylor methods.

6.3 Runge-Kutta methods.