# Numerical Computation

## You are here

Credits
6
Types
Specialization complementary (Computing)
Requirements
• Prerequisite: M1
• Prerequisite: M2
Department
MAT
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.

## Teachers

### Person in charge

• Irene María De Parada Muñoz ( )

## Weekly hours

Theory
2
Problems
0
Laboratory
2
Guided learning
0
Autonomous learning
6

## Competences

### Transversal Competences

#### Reasoning

• G9 [Avaluable] - Capacity of critical, logical and mathematical reasoning. Capacity to solve problems in her study area. Abstraction capacity: capacity to create and use models that reflect real situations. Capacity to design and perform simple experiments and analyse and interpret its results. Analysis, synthesis and evaluation capacity.
• G9.3 - Critical capacity, evaluation capacity.

### Technical Competences of each Specialization

#### Computer science specialization

• CCO1 - To have an in-depth knowledge about the fundamental principles and computations models and be able to apply them to interpret, select, value, model and create new concepts, theories, uses and technological developments, related to informatics.
• CCO1.1 - To evaluate the computational complexity of a problem, know the algorithmic strategies which can solve it and recommend, develop and implement the solution which guarantees the best performance according to the established requirements.
• CCO2 - To develop effectively and efficiently the adequate algorithms and software to solve complex computation problems.
• CCO2.3 - To develop and evaluate interactive systems and systems that show complex information, and its application to solve person-computer interaction problems.
• CCO2.6 - To design and implement graphic, virtual reality, augmented reality and video-games applications.

## Objectives

1. Analysis, programming, interpretation and verification of results, documentation and prediction of the mathematical model to study. Knowledge of the capacity of the machine where epsilon is working. Calculus of functions and numerical error propagation and representation of data. Ability to study the problem and its numerical stability: ill conditioned problems. Calculation of effective capacity and series acceleration of convergence.
Related competences: G9.3, CCO1.1,
2. Distinguish between methods of interpolation and approximation of functions. Master the interpolation methods: linear system, Lagrange, Newton and Txebixev. Learn the advantages and disadvantages of each. Differentiate between Lagrange polynomial interpolation and hermitiana, and know to use them as appropriate. Choose the method of approximation: error in the choice of nodes, minimum squared error and the standard error of sub-infinite interval.
Related competences: G9.3, CCO1.1,
3. Evaluation of the technical resolution to use depending on the size of the system: direct or iterative. Estimate condition number of the matrix system. Calculation of cash values ​​and their application in various models.
Related competences: G9.3, CCO2.3,
4. Get dominate the methods of numerical integration of differential equations and simpler problems involving the integration step reduction or improvement of computation time with a step too large.
Related competences: G9.3, CCO2.3, CCO2.6, CCO1.1,
5. Analyze and decide the most efficient method to compute solutions of a nonlinear equation. Studying the concept of order and the computational cost for iterative methods. Learn some tolerance requiring the calculation, counting the number of iterations necessary to introduce a set of initial approximations, the problem applied to several examples with varying difficulty.
Related competences: G9.3, CCO2.3, CCO1.1,
6. Discretize the equations, analyze the failure of local and global problem solving associated systems of equations.
Related competences: G9.3, CCO2.3, CCO1.1,
7. Consider the possibilities that may present a problem, achieving a versatility that makes possible wider application in terms of the diversity question.
Related competences: G9.3, CCO2.3, CCO2.6, CCO1.1,

## Contents

1. PRELIMINARIES
Introduction to the course; Methodology; Programme; Bibliography; Evaluation.
What is CN? Mathematical modelling. Sources of error, and the stability of algorithms.
Floating point arithmetical representation. Error analysis.
Calculating series. Accelerating convergence.
2. POLYNOMIAL INTERPOLATION
Polynomial interpolation: Lagrange Method. Newton divided difference method.
Interpolation errors. Choice of nodes. Tchebichev polynomials.
Runge"s phenomenon. Hermite interpolation.
3. NUMERICAL LINEAR ALGEBRA
System of Linear Equations. Directe methods: Gaussian elimination. LU decomposition. Iterative methods.
Eigenvalues and Eigenvectors. The power method. The QR method. Singular values.
4. ZEROS OF NONLINEAR FUNCTIONS
Nested interval methods and iterative methods.
Convergence order and method efficiency.
Accelerating convergence.
5. NUMERICAL INTEGRATION
Numerical derivation. Truncation error. Richardson extrapolation
Numerical integration: Newton-Côtes formulae. Romberg"s Method.
Gaussian integration.
6. INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS
Initial value problems: Introductory examples. Pass methods. Multi-pass methods.
Differential equations. Consistency, stability, and convergence. Stiff equations.
Boundary value problems. The Finite Difference Method applied to linear problems.
7. INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS
Introductory examples: heat and wave equations. Finite Difference Method and the Finite Elements Method.
Consistency, stability and convergence. Numerical resolution.

## Activities

Activity Evaluation act

### Introduction to Matlab

Assistir a la classe, fer els exercicis proposats i redactar un document amb els enunciats, estratègia, programació, resolució i discussió dels resultats que s'haurà d'entregar.
Objectives: 1
Contents:
Theory
0h
Problems
0h
Laboratory
2h
Guided learning
0h
Autonomous learning
2h

### Preliminaries.

Assistir a classe, participar activament i resoldre els exercicis proposats en el termini prefixat.
Objectives: 1
Contents:
Theory
4h
Problems
0h
Laboratory
2h
Guided learning
0h
Autonomous learning
4h

### Polynomial interpolation.

Assistir a classe, participar activament i resoldre els exercicis proposats en el termini prefixat.
Objectives: 2
Contents:
Theory
2h
Problems
0h
Laboratory
2h
Guided learning
0h
Autonomous learning
6h

### Numerical linear algebra.

Assistir a classe, participar activament i resoldre els exercicis proposats en el termini prefixat.
Objectives: 3
Contents:
Theory
8h
Problems
0h
Laboratory
8h
Guided learning
0h
Autonomous learning
11h

### Zeros of nonlinear functions.

Assistir a classe, participar activament i resoldre els exercicis proposats en el termini prefixat.
Objectives: 5
Contents:
Theory
4h
Problems
0h
Laboratory
4h
Guided learning
0h
Autonomous learning
4h

### Practical delivery 1

Objectives: 1 2 3 5
Week: 7
Type: assigment
Theory
0h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
14h

### First test using MATLAB®

The set of problems to be solved deal with the following contents: - PRELIMINARIES - POLYNOMIAL INTERPOLATION - NUMERICAL LINEAR ALGEBRA. - ZEROS OF NONLINEAR FUNCTIONS
Objectives: 1 3 5 7
Week: 10
Type: lab exam
Theory
0h
Problems
0h
Laboratory
2h
Guided learning
0h
Autonomous learning
4h

### Numerical integration.

Assistir a classe, participar activament i resoldre els exercicis proposats en el termini prefixat.
Objectives: 4
Contents:
Theory
4h
Problems
0h
Laboratory
4h
Guided learning
0h
Autonomous learning
6h

### Practical delivery 2

Objectives: 1 5 4 7
Week: 13
Type: assigment
Theory
0h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
14h

### Differential Equations.

Assistir a classe, participar activament i resoldre els exercicis proposats en el termini prefixat
Objectives: 6 7
Contents:
Theory
4h
Problems
0h
Laboratory
4h
Guided learning
0h
Autonomous learning
5h

### Second test.

The set of problems to be solved deal with the following contents: - ZEROS OF NONLINEAR FUNCTIONS - NUMERICAL INTEGRATION - INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS
Objectives: 1 2 5 4 6 7
Week: 14
Type: lab exam
Theory
0h
Problems
0h
Laboratory
2h
Guided learning
0h
Autonomous learning
4h

### Third partial test. Basic theoretical concepts and exercises

Content associated with this activity: - PRELIMINARIES - POLYNOMIAL INTERPOLATION - NUMERICAL LINEAR ALGEBRA. - ZEROS OF NONLINEAR FUNCTIONS - NUMERICAL INTEGRATION - INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS
Objectives: 1 2 3 5 7
Week: 15 (Outside class hours)
Type: theory exam
Theory
1h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
4h

### Final assessment exam: Basic theoretical concepts and exercises, problems and practices with Matlab.

Content associated with this activity: - PRELIMINARIES - ZEROS OF NONLINEAR FUNCTIONS - NUMERICAL LINEAR ALGEBRA - NUMERICAL INTERPOLATION - NUMERICAL INTEGRATION - INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS - INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS.
Objectives: 1 2 3 5 4 6 7
Week: 15 (Outside class hours)
Type: theory exam
Theory
3h
Problems
0h
Laboratory
0h
Guided learning
0h
Autonomous learning
12h

## Teaching methodology

Classes of Theory: The theory classes will consist of presenting a real problem and the definition and construction of concepts, methods and techniques necessary to resolve the situation and to do, in addition, a prediction for problems or situations presented to the next. To solving problems that complement and / or extend the theoretical and presented examples of the theory classes.

Practical Classes: Classes will consist of laboratory studies and visualization algorithms worked on the theory class, using a numerical software -Matlab, Octave- more input from symbolic manipulator -Maple- . These exercises will be introduced initially by the teacher in a classroom PCs and the students continue to interactively according to a previously prepared script of the session.

Practices: Each student will perform more than five short practices in Matlab corresponding to the first five chapters. These practices consist of one or more application routines proposed by the teacher to solve a particular practical problem numerically.

## Evaluation methodology

Continuous assessment.

It is the recommended option for students who attend class regularly. In the evaluation of the course will participate together several concepts that will lead to the final grade:

NOTA_CURS = 0,3*PRAC+0,3*TEO+0,4*PROBS

1.- Grade PRAC. Reports of MATLAB® practices (3 points).
2.- Grade TEO. Test for the most basic concepts of theory and practice (3 points). It consists of a short answer test questions.
3.- Grade PROBS. Two or more tests of problems with Matlab (4 points).

Single assessment.

It is the option recommended for the students that does NOT attend class regularly.
The single assessment. consists of a single exam with theory, problems and practice, which evaluates the knowledge of the whole subject. In the practice part and problems part, the student is asked to use the MATLAB software. The date is set by the Faculty in the calendar of final exams.

The technical skills are worth 60% of the course. The cross-competition is worth 40%. The note will be calculated cross competition from activities in the classroom and laboratory practices delivered.