Introduction to modeling Introduction to programming in MATLAB. Concept and definitions of error (absolute, relative, rounding, truncation, significant digits) and their propagation.

Specific Objectives

Be able to develop simple programs in MATLAB. To know the representation of integers and real numbers in a computer. To know the concept and definitions of error and understand how they affect the numerical calculation.

Dedication

Basic concepts of iterative methods: consistency, linear, superlinear or p-order convergence, asymptotic factor. Methods: Newton, secant, Whittaker. Solving engineering problems that deal with nonlinear scalar equations.

Specific Objectives

Understand the operation of iterative methods, differentiating them from methods with a finite number of operations. To know the properties, advantages and disadvantages of the usual iterative schemes. To know how to choose the most appropriate method in each case. To know how to analyze, implement and interpret the results of iterative methods.

Dedication

Classification and definitions. Elimination methods: Gauss Factorization methods: Crout and Cholesky Solving engineering problems that involve solving systems of linear equations

Specific Objectives

To know the classification of methods for solving systems of linear equations. To know the range of applicability of each method and its computational advantages and disadvantages. To know how to implement the resolution methods presented. To know how to identify the practical influence of the number of condition, preconditioners ...

Dedication

General approach: types and criteria of approximation Polynomial interpolation Least squares Sectional approximation Solving engineering problems involving the approximation of functions and data

Specific Objectives

To demonstrate knowledge and understanding of: - the criteria and types of functional approximation and their advantages and disadvantages, - Lagrange interpolation and its error and an ability to use it, - the least squares problem, namely to deduce the normal equations and understand the approximation orthogonality, - splines. To demonstrate an ability to use and code some intrinsic functions to approximate a data set. Be able to solve functional approximation problems

Dedication

Resolution of assessment #1

Dedication

General approach, eg with trapezoidal rule Definition of order of a quadrature Quadrature classification Newton-Cotes formulas Gauss quadrature Composite formulae Analyze and discuss convergence of the following quadratures: - Newton-Cotes and Gauss-Legendre as the number of integration points increases, - composite formulae as the number of intervals increases. Solving engineering problems involving the evaluation of integrals numerically

Specific Objectives

o demonstrate knowledge and understanding of: - The basis of numerical integration, - The classification of quadratures, - The basis of the Newton-Cotes and Gaussian quadratures, - The composite quadratures and their advantages and disadvantages. To demonstrate an ability to: - Define a quadrature if the integration points are given, - Use Newton-Cotes and Gaussian quadratures, choosing the correct one in terms of accuracy and computational cost, - Use composite quadratures. To demonstrate an ability to apply all the concepts of numerical integration to the FEM. To demonstrate an ability to implement an algorithm for numerical integration. To demonstrate an ability to implement an algorithm for composite formulae.

Dedication

General approach: reduction to first order, initial value (IVP). Methods based on the approximation of the derivative: Euler, backward Euler. Truncation error, consistency, local and global error, order. Single step methods (Runge-Kutta) methods: second and fourth order. Solving engineering problems described using ODEs

Specific Objectives

Understand the concept of well-posed initial value problems (IVP). Ability to identify and classify a problem of ODEs (in any order and dimension). Ability to rewrite high-order ODEs as a system of first order ODEs. Ability to identify Initial Value Problems (IVP) and Boundary Problem (BP). Understand the concepts of convergence and order of convergence. Knowledge of the basic properties of Runge- Kutta methods. To demonstrate an ability to model an engineering problem as a system of ODEs. To demonstrate an ability to use a library for the numerical solution of ODEs. Modelling and numerical resolution of engineering problems governed by ODEs.

Dedication

Test #2

Dedication