Syllabus
1. What is numerical mathematics. Examples of applications.
2. Problem types and errors (forward, backward, residual). Stability of algorithms.
3. Schur theorem and its consequences.
4. Orthogonal transformations and QR factorization.
5. Least-squares problems and their solution by SVD and QR factorization.
6. Partial eigenvalue problem. Power method, Arnoldi and Lanczos method.
7. Systems of linear algebraic equations. LU factorization and its stability. Stationary iterative methods.
8. Nonlinear algebraic equations, Newton's method, fixed point iteration.
9. Numerical optimization, descent methods, Newton's method.
10. Orthogonal polynomials.
11. Interpolation of functions, Lagrange interpolation, spline functions.
12. Numerical quadrature, Newton-Cotes and Gauss formulas.
13. Numerical methods for ordinary differential equations, single step and Runge-Kutta methods, multistep methods, stability, orders.
Annotation
The first course of numerical analysis for students of General Mathematics.