Numerical methods

Enrollment into the program.

Introduction to numerical computation. Floating-point arithmetic. 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. Condition number. 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. Systems of nonlinear equations. Fixed-point iteration. Newton's metod.

Linear least square problems. Overdetermined systems. Normal equations. Orthogonal decomposition.

Eigenvalue problem. Schur form. Power iteration. Inverse iteration. QR iteration.

Polynomial interpolation. Lagrange interpolation. Divided differences. Newton interpolation. 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. Multi-step methods. Systems of differential equations and initial value problems of higher order.

Z. Bohte, Numerične metode. DMFA, Ljubljana 1991.
Z. Bohte, Numerično reševanje nelinearnih enačb. DMFA, Ljubljana 1993.
Z. Bohte, Numerično reševanje sistemov linearnih enačb. DMFA, Ljubljana 1995.
E. Zakrajšek: Uvod v numerične metode, DMFA-založništvo, Ljubljana, 2000.
R. L. Burden, J. D. Faires: Numerical Analysis, 8th edition, Brooks/Cole, Pacific Grove, 2005.
D. Kincaid, W. Cheney, Numerical Analysis : Mathematics of Scientific Computing, 3rd edition, Brooks/Cole, Pacific Grove, 2002.

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

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

Application: 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.

Lectures, exercises, homeworks, midterm exams, written exams, and consultations

Continuing (homework, midterm exams)
Final (written and oral exam)
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.
Appl. math. comput., 2012, vol. 218, iss. 24, str. 11990-12000.
· PLESTENJAK, Bor, BAREL, Marc van, CAMP, Ellen van. A Cholesky LR
algorithm for the positive definite symmetric diagonal-plus-semiseparable
eigenproblem. Linear algebra appl., 2008, vol. 428, iss. 2-3, str. 586-599.
· PLESTENJAK, Bor. Numerical methods for the tridiagonal hyperbolic quadratic
eigenvalue problem. SIAM j. matrix anal. appl., 2006, vol. 28, no. 4, str. 1157-
1172.
Emil Žagar:
· JAKLIČ, Gašper, ŽAGAR, Emil. Curvature variation minimizing cubic Hermite
interpolants. Appl. math. comput., 2011, vol. 218, iss. 7, str. 3918-3924.
· JAKLIČ, Gašper, KOZAK, Jernej, KRAJNC, Marjetka, VITRIH, Vito, ŽAGAR, Emil.
Hermite geometric interpolation by rational Bézier spatial curves. SIAM j.
numer. anal., 2012, vol. 50, no. 5, str. 2695-2715.
105
· KOZAK, Jernej, ŽAGAR, Emil. On geometric interpolation by polynomial
curves. SIAM j. numer. anal., 2004, vol. 42, no. 3, str. 953-967