Description

Theory/Practice
2

Laboratory
2

Instructors

Luís Afonso

Contents

1. Introduction and Error Theory (CP1)
1.1 Finite representation, sources, and types of errors
1.2 Error propagation in numerical calculations

2. Numerical Solution of Nonlinear Equations (CP2)
2.1 Bisection, fixed-point, Newton, and secant methods
2.2 Convergence criteria

3. Systems of Linear Equations (CP3)
3.1 Jacobi method
3.2 Seidel method

4. Systems of Nonlinear Equations (CP4)
4.1 Newton-Raphson method
4.2 Application to optimization problems

5. Curve Fitting by Least Squares (CP5)
5.1 Linear and polynomial regression
5.2 Regression of linearizable functions
5.3 Measures of goodness of fit

6. Polynomial Interpolation (CP6)
6.1 Newton interpolation
6.2 Lagrange interpolation

7. Numerical Differentiation and Integration (CP7)
7.1 First and second-order finite differences
7.2 Boundary value problems
7.3 Newton-Cotes rules

8. Ordinary Differential Equations (CP8)
8.1 Euler, RK2, and RK4 methods
8.2 Application to Initial-value problems

Learning Outcomes

Provide students with both theoretical and practical knowledge of selected numerical methods. Students should become familiar with the concept of error, as well as the necessity of its analysis and control.
The goal is for students to learn how to apply previously acquired knowledge of Mathematical Analysis and Linear Algebra to study approximate models that represent physical phenomena and to solve the associated numerical problems in an approximate manner.

By the end of the semester, students should be able to:
OB1: Analyze errors and their propagation.
OB2: Use iterative methods to solve nonlinear equations and systems of linear and nonlinear equations.
OB3: Approximate functions using polynomial interpolation. Calculate estimates and Error Bounds.
OB4: Apply numerical methods for the approximate calculation of derivatives and integrals.
OB5: Fit function models to experimental data using the method of least squares.
OB6: Use numerical methods to solve problems expressed by differential equations and boundary conditions.