# Numerical Analysis

## Objectives

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

10541

6.0

Weekly - 5

Total - 56

Português

### Prerequisites

Basic knowledge in analysis (calculus) and linear algebra

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### Evaluation method

Evaluations consists in two tests and one computational project. The student final grade is given by NF=0.45×NT1+0.45×NT2+0.10×NTC, where NT1, NT2 represent the grades obtained from test 1 and 2, respectively, and NTC the grade of the computational project. NF is the so called "avaliação continua". For students who fail in the "avaliação contínua", there is a general exam in the end of the semester ("exame de recurso").

## Subject matter

1.Introduction

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 Eigenvalues and eigenvectors. Gershgorin theorem.

5.3 Iterative methods: general procedure.

5.4 Jacobi, Gauss-Seidel and relaxation methods.

6. Numerical solution of ordinary differential equations

6.1 Euler methods.

6.2 Taylor methods.

6.3 Runge-Kutta methods.

## Programs

Programs where the course is taught: