1.
Mathematics and Computer Science
Calculus
Number representation
Error, accuracy, and stability
Programming
Numerical software
Case study: errors, round-off, and
stability
Bibliography
Assignment-I
2. 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
Bibliography
3. Linear
systems of algebraic equations: iterative methods
Introduction
Simple iteration method
Gauss-Seidel method
Program realization
Bibliography
4.
Linear systems of algebraic
equations: Ortogonalisation, sparse systems, gradient methods, relaxation
methods
Method using orthogonalization
Relaxation methods
Gradient methods
Packages for systems of linear algebraic
equations
Bibliography
Assignment-II-IV
V.
Eigenvalue
Introduction
Methods for determination of
characteristic polynomial
Methods for dominant eigenvalues
Methods for subdominant eigenvalues
Jacobi Method
Givens and Hauseholder’s Method
Eigenvalues of Symmetric Tridiagonal
Matrices
LR and QR algorithm
Software eigenpackages
Generalized and nonlinear eigenvalue
problems
Bibliography
Assignment-V
6.
Nonlinear equations and systems of equations
6.1.
Nonlinear Equations
Introduction
Newton’s
Method
Newton’s
Method for Multiple Zeros
Secant Method
Bisection
Method
Schröder
Development
Methods of
Higher Order
6.2. Systems
of Nonlinear Equations
Introduction
Newton-Kantorowitch
(Raphson) Method
Gradient
Method
Globally
Convergent Methods
Bibliography
Assignment-VI
VII.
Finite Differences Calculus. Interpolation of
Functions
Introduction
Chebyshev Systems
Lagrange’s Interpolation
Newton’s Interpolation with Divided
Differences
Finite Differences Calculus
Newton’s Interpolation Formulas
Interpolation Formulas with Central
Differences
Spline Functions and Interpolation by
Splines
Prony's Interpolation
Packages for interpolation of functions
Bibliography
Assignment-VII
VIII.
Approximations of Functions
Introduction
Mean-Square Approximation
Mean-Square Approximation with
Boundaries
Economiztion of power series
Discrete Mean-Square Approximation
Chebyshev min-max Approximation
Packages for approximation of functions
Bibliography
Assignment-VIII
9.
Numerical
Differentiations and Integration
9.1.
Numerical differentiation
Introductory
notes
Formulas for
numerical differentiation
9.2.
Numerical integration - Quadrature formulas
Introductory notes
Newton-Cotes formulas
Generalized quadrature formulas
Romberg integration
Program realisation
On numerical evaluation of a class of double integrals
Packages for numerical integration
Bibliography
Assignment-IX
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
Packages for ODEs
Bibliography
Assignment-X
XI.
Partial
Differential Equations – PDE
Introduction
Method of grid
Laplace equation
Wave equation
Packages for PDEs
Bibliography
Assignment-XI
XII.
Integral Equations
Introduction
Method of successive approximations
Application of quadrature formulas
Program
realization
Bibliography
Assignment-XII
Appendices:
A.1.
Equations of technical physics
A.2.
Special
functions (Orthogonal functions)
A.3.
Numerical Methods in FEM
A.4.
Numerical Methods in
Informatics
