# Numerical Analysis II

## Objectives

At the end of this course the student will have acquired knowledge and skills that will enable him (i) to use the conjugate gradient method to approximate the solution of a linear system of equations  (ii) to approximate the eigenvalues of a matrice using the power method (iii)to approximate the first and higher order derivatives of a function (iv) to approximate the solution of initial value problems.

Furthermore, the student should be able to implement, using a computational language, the algorithms related with the covered numerical methods and analyze the numerical results obtained.

## General characterization

10982

6.0

##### Responsible teacher

Magda Stela de Jesus Rebelo

##### Hours

Weekly - 5

Total - Available soon

Português

Available soon

### Bibliography

BURDEN, R.L.; FAIRES, J.D. (1993) -- Numerical Analysis (fifth edition), Prindle, Weber & Schmidt, Boston.

PINA, H. (1995) -- Métodos Numéricos, McGrawHill.

CIARLET, P.G. (1985) -- Introduction à l''''''''Analyse Numérique
Matricielle et à l''''''''Optimisation, Masson, Paris.

CROUZEIX, M. and A. MIGNOT (1984) -- Analyse
Numérique des Equations Differentielles, Masson, Paris.

RAVIART, P.A. and J.M. THOMAS (1983) -- Introduction
a l''''''''Analyse Numérique des Equations aux Derivées Partielles, Masson,
Paris.

BRAUN, M. (1993) -- Differential Equations and their applications (4th edition). Springer-Verlag.

ISAACSON, E. and H.B. KELLER (1994) -- Analysis of Numerical Methods, Dover.

### Teaching method

The theory is explained and illustrated with examples. Main results are proved. The students are given the opportunity of working some problems, with the instructor´s support if needed, and the instructor´s comments on relevant results highlighted in the problems.

### Evaluation method

1. The assessment will be done through two tests (T1, T2) and rwo homeworks (

TC1, TC2) .

2. Each test will be will be graded 0-20 values​​ and the test T1 will be worth 30% of the final grade and the test T2  will be worth 40% of the final grade .

3. Each homework will be will be graded 0-20 values​​ and will be worth 15%  of the final grade.

4. To pass, students must obtain a final classification equal to,  or higher than 10.

## Subject matter

Numerical Analysis II

1-Numerical Matricial Analysis

• Matrix condition number
• Iterative methods for the solution of a system of equations: Jacobi, Gauss-Seidel, Relaxation, Gradient Methods.
• Iterative methods for the calculation of eigenvalues and eigenvectors; Power method.

2- Numerical Differentiation

• derivatives of first order (forward, backward and central difference formulas),
higher order derivatives.
• Richardson''s extrapolation.

3-Numerical solution of Ordinary Differential Equations

• Euler Method;
• Taylor method;
• Runge-Kutta methods;
• Multistep methods (implicit and explicit);
• Predictor–corrector methods;
• Higher-order equations and systems of differential equations.
• Finite Difference Methods

## Programs

Programs where the course is taught: