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Credits  Dept. 

7.5 (6.0 ECTS)  MAT 
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This subject offers an extensive overview of numerical analysis in order for students to gain a good understanding of both fundamental topics and for them to familiarise themselves with the concepts, basic methods, current techniques, applications for PCs and current libraries in the working world. The first and second parts of the subject introduce more basic, fundamental material, while the third part places a greater emphasis on solving the sorts of equations that all engineers must understand and be able to apply: equations with derivatives in which a sufficiently close first approach to the subject matter means that students will come away with the concepts and tools they need to be able to interpret the results. The subject focuses on opening students' minds to as wide a range of methods and applications as possible, so that they end up with a solid background as programmers and users of numerical methods.
Estimated time (hours):
T  P  L  Alt  Ext. L  Stu  A. time 
Theory  Problems  Laboratory  Other activities  External Laboratory  Study  Additional time 

T  P  L  Alt  Ext. L  Stu  A. time  Total  

3,0  3,0  1,0  0  1,0  6,0  0  14,0  
Polynomial interpolation: Lagrange Method. Newton divided difference method.
Interpolation errors. Choice of nodes. Tchebichev polynomials. Runge"s phenomenon. Hermite interpolation. 

T  P  L  Alt  Ext. L  Stu  A. time  Total  

7,0  4,0  1,0  0  2,0  9,0  0  23,0  
Direct methods: Gaussian (GaussJordan) elimination and LU factorisation.
Compact methods. Inverse calculations. Introduction to matricial norms. Error constraints Concept of eigen values and eigen vector. Associated real problems. Iterative Methods: Jacobi and GaussSeidel methods. Convergence. Powers and derivatives method. Householder QR factorisation. Eigen values for symmetric tridiagonal matrices. QR method. 

T  P  L  Alt  Ext. L  Stu  A. time  Total  

5,0  3,0  1,0  0  2,0  5,0  0  16,0  
Richardson"s Extrapolation. Numerical integration: NewtonCôtes formulae. Romberg"s Method. Adaptive integration. Improper integrals. Gaussian integration. 

T  P  L  Alt  Ext. L  Stu  A. time  Total  

4,0  3,0  1,0  0  2,0  6,0  0  16,0  
Nested interval methods and iterative methods. Convergence order and method efficiency. Accelerating convergence. 

T  P  L  Alt  Ext. L  Stu  A. time  Total  

7,0  3,0  2,0  0  3,0  8,0  0  23,0  
Initial value problems: Introductory examples. Pass methods. Multipass methods.
Differential equations. Consistency, stability, and convergence. Stiff equations. Boundary value problems. The Finite Difference Method applied to linear problems. 

T  P  L  Alt  Ext. L  Stu  A. time  Total  

5,0  3,0  2,0  0  3,0  6,0  0  19,0  
Introductory examples: heat and wave equations. Finite Difference Method and the Finite Elements Method.
Consistency, stability, and convergence. Numerical resolution. 
Total per kind  T  P  L  Alt  Ext. L  Stu  A. time  Total 
34,0  22,0  10,0  0  16,0  46,0  0  128,0  
Avaluation additional hours  10,0  
Total work hours for student  138,0 
Theory classes: The theory classes will: present a real problem; define and construct concepts, methods and techniques needed to solve the problem; predict problems and indicate kindred situations.
Classes of problems: These classes will mainly be spent on solving problems that complement and/or extend the theory presented and examples thereof.
Lab classes: The lab classes will consist of study and visualisation of the algorithms covered in the theory class, using numerical software (Matlab, Octave, etc.) and symbolic manipulation tools (Maple, Maxima, etc). These exercises will be presented by the teacher in the PC classroom, and the students will continue work on them in interactive fashion in accordance with the prepared guidelines.
Practical assignments: each student has to carry out five short practical assignments in C++, corresponding to the first five chapters. These assignments will consist of an application for one or more routines set by the teacher. The aim is to solve a specific practical problem. Microsoft Developer Studio, which allows the use of Visual C, will be employed.
Overall grading of the assignments will be based on:
Lab classes: practical exercises with Matlab or Octave. The teacher will present the practical work for the lab classes. Students will then set to work on a series of prepared calculations. The work will not be submitted at the end of the session but rather by a set deadline (2 points);
Practical work in a classical programming language: Fortran or C in which the functions to be used have already been constructed, students having to write the main programme (2 points);
Two exams of problems, in which students may use calculators and books (2 + 2 points). The exams will be held in class hours (duration: two hours).
The final test is on fundamental theory (2 points). This consists of a series of questions requiring short answers.
Students must have passed M1 and M2 courses during the selection stage.