IErrors,
accuracy, stability of computer-based calculations.
Survey of Numerical software.
II. Linear systems
of algebraic equations: direct methods
Elements of matrix calculus
LR factorization of quadratic matrix
Eigenvectors and eigenvalues of matrix
Direct methods in linear algebra
Introductory notes
Gauss elimination method with choice of
pivot element
Matrix inversion using Gauss method
Factorization methods
Program realization
Assignments
III. Linear
systems of algebraic equations: iterative methods
Iterative methods in linear algebra
Introduction
Simple iteration method
Gauss-Seidel's method
Matrix inversion by iterative methods
Program realization
Assignments
IV.
Linear systems of algebraic equations:
Ortogonalisation, sparse systems, gradient methods, relaxation methods
Method using orthogonalization
Relaxation methods
Gradient methods
Assignments
V. Eigenvalue problem, computations in dynamical
systems
Localization of Eigenvalues
Characteristic Polynomial
Dominant Eigenvalues
Sub-dominant Eigenvalues
Jacobi Method
Givens and Hauseholder’s Method
Eigenvalues of Symmetric Tridiagonal
Matrices
LR and QR algorithm
Assignments
VI. Solution of
nonlinear equations and systems of equations
VI.1.
Nonlinear Equations
Introduction
Newton’s
Method
Newton’s
Method for Multiple Zeros
Secant
Method, False Position Method, and Ridders’ Method
Bisection
Method
Schröder
Development
Methods of
Higher Order
VI.2. Systems
of Nonlinear Equations
Introduction
Newton-Kantorowitch
(Raphson) Method
Gradient
Method
Globally
Convergent Methods
Assignments
VII. Finite Differences
Calculus. Interpolation of Functions
Chebyshev’s Systems
Lagrange’s Interpolation
Newton’s Interpolation with Divided
Differences
Finite Differences Calculus
Newton’s Interpolation Formulas
Interpolation Formulas with Central
Differences
Convergence of Interpolation Processes
Hermite Interpolation
Trigonometric Interpolation
Algorithms for Calculation Trigonometric
Sums
Spline Functions and Interpolation by
Splines
Cubic Spline Interpolation
Extremal and Approximate Characteristics
of Cubic Spline
Prony Interpolation
Assignments
VIII.
Approximations of Functions
Problem of Best Approximations
Least Squares Approximation
Least Squares Approximation with
Boundaries
Discrete Least Squares Approximation
Chebyshev min-max Approximation
Assignments
IX.
Numerical Differentiations
and Integration
IX.1.
Numerical differentiation
Introductory
notes
Formulas for
numerical differentiation
IX.2.
Quadrature formulas
Introductory notes
Newton-Cotes formulas
Generalized quadrature formulas
Romberg integration
Gauss-Christoffel quadrature formulas
Gauss-Christoffel quadrature formulas for classical weight
functions
General method for construction of Gauss-Christoffel formulas
Modified Gauss formulas
Formulas of Cronrod-type
Chebyshev formulas
Integration of fast-oscillating functions
Multidimensional integrals
Numerical evaluation of a class of double integrals
Convergence of quadrature processes
Program realization
Assignments
X.
Ordinary Differential Equations – ODE
Introduction
Euler's method
General linear multi-step method
Choice of starting values
Predictor-corrector methods
Program realization of multi-step methods
Runge-Kutta's methods
Program realization of Runge-Kutta's methods
Solution of system of equations and equations of higher order
Contour problems
Assignments
XI. Partial Differential Equations –
PDE
Introduction
Fourier method
Method of grid
Approximation of difference operator
Approximation of contour conditions
Stability of difference scheme
Difference scheme with separation
Difference scheme for equation of hyperbolic type
Assignments
XII.
Integral Equations
Introduction
Method of successive approximations
Application of quadrature formulas to the solution
Program
realization
Assignments
Appendices:
A.
Areas of Special Interest
A.1.
Equations of technical physics
A.2. Special
functions (Orthogonal functions)
A.3.
Numerical Methods in FEM
A.4.
Numerical Methods in Informatics
A.5.
Programming Numerical Methods using Object Oriented Technique
A.6. Elements
of Interval Calculus (Interval Arithmetic)
B. Mathematical Modeling
B.1.
Principles of Mathematical Modeling
B.2. Modeling
of Data
B.3. Modeling
Steps
B.4. Analysis
of Influenced Parameters
