|
1. |
Name of discipline |
Numerical method |
|
2. |
Training course |
3 |
|
3. |
Semester of study |
5 |
|
4. |
Number of credits |
2 |
|
5. |
Full name of lecturer |
doctor of physical and mathematical Sciences, associate Professor Vasily Volkov |
|
6. |
Objectives of the discipline |
– study of the principles of construction and software implementation of numerical algorithms of linear algebra; – – study of iterative methods for solving nonlinear equations and systems; – acquisition of skills to choose adequate numerical methods for solving a specific problem; – study of methods for estimating the correctness of numerical results and error of the solution; – study of applied aspects of numerical methods of linear algebra. introduction to numerical methods implemented in simulation software. |
|
7. |
Prerequisites |
– algebra and number theory; – functional analysis; – programming method; – computer mathematics systems |
|
8. |
Content of the discipline |
Numerical methods for solving systems of LAU. Norms of vectors and matrices. Error estimate for solving systems of LAU. Condition number. Direct method. Gauss method. Select the master item. LU factorization. Cholesky Decomposition. Iterative methods for solving systems of LAU. A simple iteration method. Convergence of iterative methods. Estimate the number of iterations. Selection of the optimal parameter. Concept of a precontitioner. Implicit iterative methods. Methods of Jacobi, Seidel, successive over-relaxation. Calculation of eigenvalues and eigenvectors of matrices. Properties of eigenvectors and eigenvalues of matrices. Similarity transformation. Canonical form of Frobenius. Danilevsky’s Method. Power method of finding the maximum module of the eigenvalues. Method of rotation. The concept of QR algorithm. Solution of nonlinear equations and systems. Separation of roots. The method of dichotomy. Multiple roots. Roots of polynomials. A simple iteration method. Convergence condition and convergence rate. Quadratic convergence. Modifications of the Newton method. The concept of nonlinear optimization methods. Gradient methods. |
|
9. |
Recommended reading |
1. Bakhvalov N. S., Zhidkov N. P., Kobelkov G. M. Numerical methods. M. : BINOM. Lab. knowledge, 2003. 636 p. 2. Samarskiy A. A., Gulin A.V. Numerical methods. M.: Science, 1989. 432 p. 3. Krylov V. I., Bobkov V. V., Monastic P. I. The beginning of the theory of computational methods. Linear algebra and nonlinear equations. Minsk: Science and technology, 1985. 279 p. 4. Fadeev D. K., Fadeeva V. N. Computational methods of linear algebra. SPT. 2002. 5. Voevodin V. V., Kuznetsov Yu. a. Matrices and calculations. M., Science, 1984. 320 p. 6. Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P.. Numerical Recipes: the art of scientific computing. New York. 1997. 973p 7. Numerical methods in 2 parts. Part I. / V. M. Volkov. – Minsk: BSU, 2016. |
|
10. |
Teaching method |
passive, active, interactive, verbal, visual, problem |
|
11. |
Language of instruction |
Russian |
|
12. |
Conditions (requirements), current control |
– laboratory work report; – test; – colloquium. |
|
13. |
Current certification form |
credit |