# Numerical Analysis I

## Objectives

The main goal of this course is to supply the students with well-known numerical approximation techniques, particularly, in the following topics:

1. Solutions of equations in one variable;

2. Interpolation and polynomial approximation;

3. Numerical differentiation and integration;

4. Systems of linear equations.

Furthermore, a CAS (Computer Algebra System) is introduced as tool for the implementation of algorithms developed for those lectured techniques.

## General characterization

##### Code

10979

##### Credits

6.0

##### Responsible teacher

João Filipe Lita da Silva

##### Hours

Weekly - 6

Total - 126

##### Teaching language

Português

### Prerequisites

Basics in Real Analysis and Linear Algebra.

### Bibliography

1. R.L. BURDEN, J.D. FAIRES, Numerical Analysis (Ninth Edition), Brooks/Cole, Cengage Learning, 2011.

2. J.L. BUCHANAN, P.R. TURNER, Numerical Methods and Analysis, McGraw-Hill International Editions, 1992.

3. M.P.J. CARPENTIER, Análise Numérica, Departamento de Matemática do Instituto Superior Técnico, 1993.

4. W. GAUTSCHI, Numerical Analysis (Second Edition), Birkhäuser, 2012.

5. F.B. HILDEBRAND, Introduction to Numerical Analysis (Second Edition), Dover Publications, 1987.

6. D. KINCAID, W. CHENEY, Numerical Analysis: Mathematics of Scientific Computing (Third Edition), American Mathematical Society, 2002.

7. H. PINA, Métodos Numéricos (Segunda Edição), Escolar Editora, 2010.

### Teaching method

The theoretical statements are presented to the students in theoretical lectures with the corresponding proofs (or a sketch of the proof). Enlightening examples are also given as supplement. In the laboratory lectures, proposed problems are explored and solved, involving the students on that process.

### Evaluation method

1. The student''''s evaluation is determined by two components: theoretical-practical and project.

2. The theoretical-practical component encloses either two tests or a final exam.

(a) Each test is graded in the interval [0,20] and it establishes 40% of the final grade.

(b) Alternatively, the final exam sets up 80% of the final grade and it is graded in the interval [0,20].

3. The project is graded in the interval [0,20] and it establishes 20% of the final grade.

(a) The project is constituted by individual/team homeworks and its grade is obtained by computing the arithmetic mean of the individual/team homework grades (each individual/team homework is graded in the interval [0,20]).

4. To pass, the student must get, rounded to units, a final grade greater or equal to 10.

## Subject matter

**0. Preliminaries**

0.1 Errors

0.2 CAS: an overview

**1. Solutions of nonlinear equations**

1.1 The bisection method

1.2 Fixed-point iteration

1.3 Newton-Raphson''s method

1.4 Secant method, False Position method, and Müller''s method

1.5 Error analysis and convergence. Aitken''s method. Basics on polynomials (Horner''s method)

**2. Interpolation and polynomial approximation**

2.1 Interpolation and the Lagrange polynomial

2.2 Divided differences

2.3 Hermite interpolation

2.4 Cubic spline interpolation

**3. Numerical differentiation and integration**

3.1 Numerical differentiation

3.2 Elements of numerical integration

3.3 Composite numerical integration

3.4 Gaussian Quadrature

**4. Direct methods for solving linear systems**

4.1 Linear systems of equations (the method of Gaussian elimination)

4.2 Pivoting strategies

4.3 Matrix factorization

4.4 Special types of matrices

**5. Iterative methods for solving linear systems**

5.1 Norms of vectors and matrices

5.2 Eigenvalues and eigenvectors

5.3 Gershgorin theorem

5.4 Iterative techniques for solving linear systems (the Jacobi, Gauss-Seidel, and relaxation techniques)

**6. Approximation theory**

6.1 Least squares approximation

6.2 Orthogonal polynomials (Chebyshev polynomials)

6.3 Rational function approximation (Padé approximation technique)