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.
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