Code: | EM0016 | Acronym: | AN |
Keywords | |
---|---|
Classification | Keyword |
OFICIAL | Mathematics |
Active? | Yes |
Web Page: | http//moodle.up.pt |
Responsible unit: | Mathematics Section |
Course/CS Responsible: | Master in Mechanical Engineering |
Acronym | No. of Students | Study Plan | Curricular Years | Credits UCN | Credits ECTS | Contact hours | Total Time |
---|---|---|---|---|---|---|---|
MIEM | 203 | Syllabus since 2006/2007 | 2 | - | 6 | 39 | 162 |
General:
The students will be able to choose the most efficient methods for the solution of each basic Numerical Analysis problem. The students are expected to understand the theorems and convergence conditions of each of the methods described, to be able to program them, to test them effectively on a computer and discuss the results obtained.
Specific:
For each chapter in the program the successful students
will be able to list the applicability conditions of the numerical methods and state the corresponding theorems of convergence;
they will be able to apply the methods, formula and algorithms taught to simple problems;
they will be able to describe the behavior of the methods, translate them into algorithms and
‘Matlab Functions’ as well as test them on examples comparing and analyzing the results;
they will explain the proofs of the theorems given and apply the proof techniques involved to other related situations;
they will be able to solve new problems with the numerical tools here taught and compare the performance of the various numerical methods in terms of speed and accuracy.
For each chapter in the program the successful students
will be able to list the applicability conditions of the numerical methods and state the corresponding theorems of convergence;
they will be able to apply the methods, formula and algorithms taught to simple problems;
they will be able to describe the behavior of the methods, translate them into algorithms and
‘Matlab Functions’ as well as test them on examples comparing and analyzing the results;
they will explain the proofs of the theorems given and apply the proof techniques involved to other related situations;
they will be able to solve new problems with the numerical tools here taught and compare the performance of the various numerical methods in terms of speed and accuracy.
The students are supposed to know the subjects taught in Linear Algebra and Mathematical Analysis as well as in Computer programming
Chapter 1. Number systems and errors ; number systems on computers; representation of integers and floating point arithmetic; round-off error; absolute error and relative error, significant digits, Taylor's formula and error estimation; error analysis.
Chapter 2. Linear systems of equations: Gaussian elimination. Round off errors and possible instability of the numerical methods, pivoting strategies. Solution of triangular systems. Tridiagonal systems. LU factorization; application to the computation of determinants and to the inversion of matrices.
Iterative methods: Jacobi and Gauss-Seidel; convergence theorems and algorithms. solution of triangular systems.
Chapter 3. Least squares approximation. Orthogonal polynomials. Curve fitting. Over-determined systems of equations.
Chapter 4. Non linear equations: general conditions for the solution, stopping criteria for iterative methods; some iterative methods: successive bisection, fixed point iteration, Newton's method, secant method. Convergence theorems and algorithms; polynomial equations.
Chapter 5. Numerical integration: Newton-Cotes formulae (ex: Trapezoidal and Simpson rules); composite rules; numerical quadrature errors. Gaussian quadrature.
Chapter 6. Polynomial interpolation: finite differences; methods of Newton and Lagrange; error of the interpolating polynomial.
Chapter 7. Ordinary Differential equations: Euler s method for ODE of order 1; Taylor methods. Order of a method for ODE of order1. Runge-Kutta methods of order 2 and 4
Small computer projects using WINDOWS or UNIX and MATLAB.
This course has both technological and scientific component with weight of 50% and 50%, respectively
Lectures with "Powerpoint". Small illustrating computer projects supervised by teachers in the computer room, with Matlab.
Designation | Weight (%) |
---|---|
Exame | 75,00 |
Participação presencial | 0,00 |
Teste | 25,00 |
Total: | 100,00 |
Designation | Time (hours) |
---|---|
Estudo autónomo | 120,00 |
Frequência das aulas | 42,00 |
Total: | 162,00 |
Minimum requirements to be admitted to the exam:
registration and not to exceed the maximum number of absences allowed.
Type of assessment: Avaliação distribuída com exame final
The students will answer 7 to 10 quizzes in the Moodle elearning platform, and the best 5 will be taken counting 25% of the final grading and a writen exam counting 75% to the final grade.