- Basic command of modern numerical methods of applied maths
- Basic use of common packages and libraries for numeric and scientific computing
- Knowledge of mathematical foundations of numerical methods used in contemporary fundamental and applied research and development.
- Ability to choose an appropriate numerical method for a given problem.
- Experience working with modern software packages and libraries for numerical and scientific computing.
- Numerical mathematics.Overview of computational problems: set-ups and approaches. Correctness, accuracy, stability, condition numbers. Roundoff errors. Machine arithmetics. Uncertainties.
- Matrix factorizationsQR factorization of a matrix. Householder reflections and Givens rotations. SVD factorization. Properties of singular vectors and singular values. Eckhart-Young theorem. Moore-Penrose pseudoinverse. Over- and under-determined systems of linear equations. Solving linear systems in a least-squares sense. Ordinary least squares.
- Eigenvalue problem.The eigenvalue problem. Power iteration, inverse iteration. Shift-invert mode. QR algorithm.
- Systems of linear equations: direct methods.Vector and matrix norms. Condition number of a linear system. Gaussian elimination and LU decomposition of a matrix. Pivoting. Computational complexity. Cholesky decomposition for PSD matrices. Tridiagonal systems, Thomas algorithm.
- Systems of linear equations: iterative methods.Jacobi iteration. Seidel iteration. Convergence properties and convergence theorems. Canonic form of iterative methods. Variational approaches.
- Nonlinear equations and systems of equations.Localization of roots. Bisection, simple iteration and its variations. Modified schemes, Newton's method. Convergence theorems. A priori and a posteriori error estimates. Multiple roots. Inverse quadratic interpolation. Hybrid schemes.
- Interpolation of functions.Interpolation problem. Univariate interpolation. Interpolation polynomial in the Lagrange form. Runge phenomenon. Chebyshev nodes. Piecewise polynomial interpolation, splines. Hermite cubic polynomial. Construction of a spline interpolant. Boundary conditions. Monotone interpolants.
- Numerical differentiationFinite-difference formulas. Optimal differentiation step. Construction of higher-order FD schemes. Richardson extrapolation and Neville algorithm. Algorithmic differentiation.
- Numerical integrationElementary quadrature rules: rectangle formulas, trapezoids, Simpson's rule. Gaussian quadratures.
- Initial value problem for ODEsOrdinary differential equations. Initial value problem. Implicit and explicit finite difference schemes. Euler rule and its modifications. Runge-Kutta methods. Linear multistep methods: convergence, approximation and stability. Zero-stability of LMMs. A-stability, stiff systems. Dahlquist barriers.
- Fredholm integral equations.Fredholm equations of the first and second kinds. Fredholm alternative. Nystrom method of solving FIE of the 2nd kind.
- Monte Carlo methods.Buffon's needle. MC integration in d>1. Pseudo random and quasi-random (low-discrepancy) sequences. Sampling from non-uniform distributions. Markov chain MC.
- In-class activity
- ExamThe final grade can be set based on the accumulated grade, without taking the exam.
- Online course
- Homework problems
- Interim assessment (4 module)0.1 in-class activity + 0.2 online course + 0.5 HW + 0.2 exam
- Вычислительные методы для инженеров : учеб. пособие для вузов, Амосов, А. А., 2003
- William H. Press, Saul A. Teukolsky, William T. Vetterling, & Brian P. Flannery. (1992). Numerical Recipes in C: The Art of Scientific Computing. Second Edition. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.9CFCD6AE
- Wright, S. J., & Nocedal, J. (2015). Numerical optimization. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.92C9B6B0
- Численные методы : учеб. пособие для вузов, Калиткин, Н. Н., 2011
- Численные методы, Самарский, А. А., 1989