Introduction to numerical methods

There are no prerequisites.

Introduction to numerical computation. Floating-point arithmetic, IEEE standard. Sources of inexactness in numerical computation. Sensitivity of a problem, convergence of a method, stability of computation. Error analysis. Software for numerical computation.
Systems of linear equations. Matrix norms and condition numbers. Error bounds. Gaussian elimination. Error analysis. Pivoting. Special types of linear systems.
Nonlinear equations. Bisection. Fixed-point iteration. Newton's and Secant method. Algebraic equations. Laguerre's method, Root reduction. System of nonlinear equations.
Linear least square problems. Overdetermined systems. Normal equations. Orthogonal decomposition. Givens rotations and Householder transformations.
Eigenvalue problems. Schur form. Power iteration. Inverse iteration. QR iteration.
Polynomial interpolation. Lagrange interpolation. Divided differences. Newton form. Numerical differentiation.
Numerical integration. Newton-Cotes rules. Composite rules. Romberg extrapolation. Gaussian quadrature.
Numerical methods for ordinary differential equations. Methods for initial value problems. One-step methods. Runge-Kutta methods. Systems of differential equations and initial problems of higher order.

1. Z. Bohte: Numerične metode, Ljubljana : Društvo matematikov, fizikov in astronomov SRS : Zveza organizacij za tehnično kulturo Slovenije, 1985, 1987.
2. Z. Bohte: Numerično reševanje nelinearnih enačb, Ljubljana : Društvo matematikov, fizikov in astronomov Slovenije, 1993.
3. Z. Bohte: Numerično reševanje sistemov linearnih enačb, Ljubljana : Društvo matematikov, fizikov in astronomov Slovenije, 1994.
4. R. L. Burden, J. D. Faires: Numerical analysis, 6th ed., Pacific Grove (Canada) : Brooks/Cole Publ. : ITP An International Thompson Publishing Company, cop. 1997.
5. B. N. Datta: Numerical linear algebra and applications, Pacific Grove : Brooks/Cole : International Thomson Publ., cop. 1994.
6. J. W. Demmel (prevod in priredba E. Zakrajšek): Uporabna numerična linearna algebra, Ljubljana : DMFA - založništvo, 2000.
7. G. H. Golub, C. F. Van Loan: Matrix Computations, 3rd ed., Baltimore : The Johns Hopkins University Press, cop. 1996.
8. D. Kincaid, W. Cheney: Numerical analysis : mathematics of scientific computing, 2nd ed., Pacific Grove (California) : Brooks/Cole Publishing Company, 1996.
9. B. Plestenjak: Razširjen uvod v numerične metode, DMFA-založništvo, Ljubljana, 2015.
10. L. N. Trefethen, D. Bau: Numerical linear algebra, Philadelphia : SIAM, cop. 1997.

Students learn fundamentals of numerical computation. They learn in detail the fixed-point arithmetic and methods for system of linear and nonlinear equations. They learn basics of eigenvalue computation, polynomial interpolation, numerical quadrature, and methods for the ordinary differential problems. The acquired knowledge is consolidated by homework assignements and solving problems using software for numerical computation.

Knowledge and understanding: Understanding of floating-point arithmetic and sources of errors in numerical computation. Proficiency in basic numerical methods for linear and nonlinear systems. Knowledge of basic numerical algorihtms for computing eigenvalues, interpolation, integration, and solving differential equations. Knowledge of computer programming and Matlab or other similar software for solving such problems.

Applications: Economical and accurate numerical solution of various mathematical problems. In addition to mathematics, numerical methods are used in many other fields when the problem can be described by a mathematical model and a result in a numerical form is required. Many problems can not be solved analytically but only numerically. Also, in some cases, the numerical solution is much more economical than the analytical one.

Reflection: Understanding of the theory from the applications.

Transferable skills: The ability to solve mathematical problems using a computer. Understanding the differences between the exact and the numerical computation. The subject enriches constructively the knowledge of algebra and analysis.

Lectures, exercises, homework, consultations

Continuing (homework and written exam): 50 %
Final (oral exam): 50 %
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Bor Plestenjak:

• GHEORGHIU, C. I., HOCHSTENBACH, Michiel E., PLESTENJAK, Bor, ROMMES, Joost. Spectral collocation solutions to multiparameter Mathieu's system. Applied mathematics and computation, ISSN 0096-3003. [Print ed.], 2012, vol. 218, iss. 24, str. 11990-12000. [COBISS-SI-ID 16484185]
• PLESTENJAK, Bor, BAREL, Marc van, CAMP, Ellen van. A Cholesky LR algorithm for the positive definite symmetric diagonal-plus-semiseparable eigenproblem. V: CHING, Wai-Ki (ur.). Second international conference on structured matrices : Hong Kong Baptist University, 08-11 June 2006, (Linear algebra and its applications, ISSN 0024-3795, Vol. 428, Issues 2-3, 2008). New York: North Holland, 2008, vol. 428, iss. 2-3, str. 586-599. [COBISS-SI-ID 14475097]
• PLESTENJAK, Bor. Numerical methods for the tridiagonal hyperbolic quadratic eigenvalue problem. V: Fifth international workshop on accurate solution in eigenvalue problems : hagen, Germany from June 29 to July 1, 2004. Philadelphia: SIAM, 2006, vol. 28, no. 4, str. 1157-1172. [COBISS-SI-ID 14367833]

Emil Žagar:

• JAKLIČ, Gašper, ŽAGAR, Emil. Curvature variation minimizing cubic Hermite interpolants. Applied mathematics and computation, ISSN 0096-3003. [Print ed.], 2011, vol. 218, iss. 7, str. 3918-3924. [COBISS-SI-ID 16049241]
• JAKLIČ, Gašper, KOZAK, Jernej, KRAJNC, Marjetka, VITRIH, Vito, ŽAGAR, Emil. Hermite geometric interpolation by rational Bézier spatial curves. SIAM journal on numerical analysis, ISSN 0036-1429, 2012, vol. 50, no. 5, str. 2695-2715. [COBISS-SI-ID 16449369]
• KOZAK, Jernej, ŽAGAR, Emil. On geometric interpolation by polynomial curves. SIAM journal on numerical analysis, ISSN 0036-1429, 2004, vol. 42, no. 3, str. 953-967. [COBISS-SI-ID 13398617]