Numerical Mathematics and Probability

Elements of probability. Events, complementary events, discrete and continuous random variables, limit theorems. Metric spaces. The Banach Theorem. Polynomial approximation to functions: approximate numbers and operation with them, Taylor's polynomials, Lagrange's interpolation polynomial, finite and divided differences, Newton's interpolation polynomial, numerical differentiation, splines, the method of least squares. Numerical integration: quadrature formulas, Newton-Cotes quadrature formulas, general quadrature formulas. Numerical methods of linear algebra: the Gauss elimination method, norms and condition of matrices, the method of simple iterations, Seidel's method, partial eigenvalue problems. Methods of solving nonlinear equations and systems: the iteration method, Newton's method, the method of dividing a line segment into two equal parts, the method of steepest (gradient) descent. Numerical methods for ordinary differential equations: Euler's methods, Runge-Kutta method, Adam's method.

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