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EM 207 - NUMERICAL ANALYSIS (Análise Numérica) Series expansions computation of transcendental functions using series developments. 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. Non linear equations: general conditions for the solution, stopping criteria for iterative methods; some iterative methods: successive bissection, fixed point iteration, Newton's method, secant method; polynomial equations. Linear systems of equations: solution of triangular systems; Gaussian elimination, pivoting strategies; application to the computation of determinants and to the inversion of matrices. Iterative methods: Jacobi and Gauss-Seidel; convergence theorems. Polynomial interpolation: divided differences; methods of Newton and Lagrange; error of the interpolating polynomial. Approximation. Least squares approximation. Orthogonal polynomials. Numerical integration: Newton-Cotes formulae (ex: Trapezoidal and Simpson rules) Gaussian quadrature; composite rules; numerical quadrature errors. 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. Objectives To learn the most efficient and common methods for the solution of each basic Numerical Anaysis problem. The students are expected to learn the theorems and conditions of convergence of each method, to be able to program them, and to test them effectively on a computer and discuss the results obtained. |