Numerical Analysis and Simulation

Floating point arithmetic. Numerical stability. Iterative methods for high-dimensional linear systems. Seidel method, conjugated gradient. Krylov subspace method. Convergence analysis. Pre-conditioners. Numerical solution of non-linear equations. Fixed point methods. Newton's method. Interpolation and polynomial approximation: Lagrange, Newton, Hermite, Chebyshev. Interpolation error. Splines. Approximation theory. Numerical Integration: Composite Newton-Cotes formulas, Romberg method, Gauss methods. Adaptive integration. Numerical integration of ODE systems: convergence, A-stability, B-stability. Stiff systems. Taylor, Runge-Kutta, predictor-corrector, exponential methods; EDP discretization: Finite difference methods for Parabolic, Elliptical, Hyperbolic EDP. Stochastic Simulation. Monte Carlo methods. Numerical integration of stochastic differential equations (EDEs): Strong and weak approximation. Euler-Maruyama method, Milstein, Ito-Taylor. Convergence and numerical stability. Computer simulation of EDEs.

Basic Information

Workload
45 hours
Requirements
Functional Analysis: Fundamentals

Mandatory: 

  • Stoer & Bulirsch (2002). Introduction to Numerical Analysis. (Third Edition). TAM
  • Conte, S.D., de Boor, C. (2017). Elementary Numerical Analysis, an Algorithmic Approach. SIAM.
  • Timothy Sauer (2011). Numerical Analysis (2nd Edition).  Pearson
     

Complementary: 

  • Datta, N. Nuerical Linear Algebra and Applications (2010) (Second Edition) SIAM
  • Faire, D., & Burden, R. L. (2002) Numerical Methods (3 ed.). Brooks Cole
  • Griffiths D., & Higham, D. (2010) Numerical Methods for Ordinary Differential Equations. Springer.
  • Kloeden P., Platen E. (1999) Numerical solution of stochastic differential equations. Springer·    
  • Cuminato J, Menegmuette M (2013) Discretização de Equações Diferenciais Parciais: Técnicas de Diferenças Finitas. SBM.