2024
Numerical Analysis Complements
Name: Numerical Analysis Complements
Code: MAT14356M
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Presentation
The students will acquire skills in theory and implementation of numerical methods to solve large-scale systems of linear equations, to calculate eigenvalues and eigenvectors, to solve ordinary differential equations and partial differential equations.
Sustainable Development Goals
Learning Goals
It is intended that students acquire skills in the theory and computational implementation of advanced numerical methods.
Contents
- Resolution of large Systems of Linear Equations. Direct methods: using factorizations, systems with special matrices (bands, tridiagonal, sparse), stability of solutions. Iterative methods: relaxation method, Krylov subspace methods, conjugate gradients method and preconditioning.
- Calculation of eigenvalues and eigenvectors. Gershgorin Circles. Power methods. Orthogonal transformations, similar matrices, Jacobi method, QR algorithm, Lanczos algorithm.
- Systems of Ordinary Differential Equation. Rigid problems.
- Partial Differential Equations. Examples of boundary value problems. Approximation by finite differences. Finite element approximation. Approximation of two-dimensional problems. Consistency and convergence. Poisson equation, heat equation, wave equation.
- Numerical resolution of Integral Equations.
- Implementation of some algorithms in an interactive numerical and symbolic calculation system.
- Calculation of eigenvalues and eigenvectors. Gershgorin Circles. Power methods. Orthogonal transformations, similar matrices, Jacobi method, QR algorithm, Lanczos algorithm.
- Systems of Ordinary Differential Equation. Rigid problems.
- Partial Differential Equations. Examples of boundary value problems. Approximation by finite differences. Finite element approximation. Approximation of two-dimensional problems. Consistency and convergence. Poisson equation, heat equation, wave equation.
- Numerical resolution of Integral Equations.
- Implementation of some algorithms in an interactive numerical and symbolic calculation system.
Teaching Methods
Structured exposition, examples with emphasis on applications and on solving exercises, a computational project to be carried out by students.
The student can choose one of the following forms of assessment:
- Continuous evaluation - a theory test and a mandatory computational project (individual or in small grups). A minimum score of 8 points is required for each test/project to obtain approval. Final classification = 40% of project classification + 60% of test classification.
- Evaluation by exam - one exam with a weight of 100% in the final result.
The student can choose one of the following forms of assessment:
- Continuous evaluation - a theory test and a mandatory computational project (individual or in small grups). A minimum score of 8 points is required for each test/project to obtain approval. Final classification = 40% of project classification + 60% of test classification.
- Evaluation by exam - one exam with a weight of 100% in the final result.
