Numerical Analisys

Knowledge corresponding to the syllabus from 1st year Mathematics I and Mathematics II curricular units: elementary mathematics, matrix calculus, differential calculus and integral calculus.

Expository method will be used to explain theoretical subjects and to analyze and solved problems.

Theoretical subjects and application examples will be presented. Each chapter will begins with motivation examples and will ends with the analysis of application examples. Exercises will be analyzed, discussed and solved, using mathematical software (Geogebra and Matlab) and spreadsheet program (Excel). Emphasis will be placed on the critical analysis of the selection criteria for mathematical techniques and consequent results.

Each chapter will ends with application examples to Civil Engineering.

In Moodle, will be available all the necessary documents for this course unit: textbook, exercise book, forms, math tables, educational videos and exams from past years.

– Understand the limitation of analytical techniques to solve mathematical problems.
– Understand why numerical errors exist and how they can be controlled.
– Master the fundamentals of the different numerical methods, choose and use of the most effective methods to solve mathematical problems and interpret the correspondent results. 
– Use numerical methods to solve problems on Civil Engineering.
– Analyse and solve mathematical problems using Geogebra, Matlab and Excel.

1. Theory of errors (brief remarks).

2. Roots of nonlinear equations.
– Introduction.
– Location of roots: graphical method and Bolzano’s theorem.
– Bisection and Newton’s methods: bases, iterative formulas, error and stopping criteria.
– Analysis and calculus using Geogebra, Excel and Matlab.
– Roots of polynomials: counting, location, separation and approximation.
– Applications to Civil Engineering.

3. Polynomial interpolation.
– Introduction.
– Uniqueness of the interpolating polynomial.
– Interpolating polynomial: Lagrange’s and Newton’s form and interpolation error.
– Analysis and calculus using Geogebra.
– Inverse polynomial interpolation.
– Applications to Civil Engineering.

4. Numerical integration.
– Introduction.
– Trapezoidal and Simpson’s rules: bases, formulas and errors.
– Analysis and calculus using Geogebra.
– Numerical calculus using Matlab.
– Applications to Civil Engineering.

5. Numerical integration of ordinary differential problems of first order initial value.
– Introduction.
– Euler and second order Runge-Kutta methods: bases, iterative formulas and errors.
– Analysis and calculus using Geogebra.
– Numerical calculus using Matlab and Excel.

6. Iterative methods for solving linear systems.
– Introduction.
– Jacobi and Gauss-Seidel’s methods: bases and iterative formulas.
– Analysis and calculus using Geogebra.
– Numerical calculus using Matlab.

Evaluation will be carried out through written examination and will include problem solving, using the calculator, mathematical software (MatLab and Geogebra) and spreadsheet program (Excel), theoretical and explanatory questions.

The assessment may be done in two ways (chosen by the student):

Continuous assessment:

Two tests (test 1 and test 2), each one quoted for 10 values, on dates to be defined.

- test 1 will focus on chapters 1, 2 and 3;

- test 2 will focus on chapters 4, 5 and 6, as well as the contents of previous chapters essential to understanding those ones.

Final result will be given by the summation of the grades of both tests. Student will be approved if final result, rounded, is greater or equal than 10 values and the result of each test is greater or equal than 4 values (out of 10).

Final assessment:

Final exam, quoted for 20 values, to be carried out in the periods defined in the school calendar (regular season and appeal season). Student will be approved if the exam result, rounded, is greater or equal than 10 values.

NAO