Numerical methods 1

Completed courses Analysis 1 and Algebra 1.

Introduction to numerical computation. Sources of inexactness in numerical computation. Sensitivity of a problem, convergence of a method, stability of computation. Error analysis.
Nonlinear equations. Bisection. Fixed-point iteration. Newton's and Secant method. Methods for algebraic equations.
Systems of nonlinear equations. Fixed-point iteration. Newton's metod.
Systems of linear equations. Vector and matrix norms. Condition number. Error bounds. Gaussian elimination. Pivoting. Special types of linear systems.
Linear least square problems. Overdetermined systems. Normal equations. Orthogonal decomposition. Givens rotations and Householder transformations. Singular values decomposition. Pseudoinverse.
Iterative methods for linear equations.
Jacobi, Gauss-Seidel, and SOR iteration.

J. W. Demmel: Uporabna numerična linearna algebra, DMFA-založništvo, Ljubljana, 2000.
B. N. Datta: Numerical Linear Algebra and Applications, Brooks/Cole, Pacific Grove, 1995.
Z. Bohte: Numerično reševanje sistemov linearnih enačb, DMFA-založništvo, Ljubljana, 1994.
Z. Bohte: Numerične metode, DMFA-založništvo, Ljubljana, 1991.
L. N. Trefethen, D. Bau: Numerical Linear Algebra, SIAM, Philadelphia, 1997.

Students learn fundamentals of numerical computation and extend the knowledge on analytical methods for nonlinear equations with some well-known numerical methods. 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 the linear least squares problem. 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. This course is required for the course Numerical methods 2.
Reflection: Understanding of the theory from the applications.
Transferable skills: The ability to select an appropriate method, solve a problem, and analize the obtained results. 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, midterm exams, project work)
Final (written and oral exam)
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Marjetka Krajnc:
KRAJNC, Marjetka. Interpolation scheme for planar cubic G [sup] 2 spline curves. Acta applicandae mathematicae, ISSN 0167-8019, 2011, vol. 113, no. 2, str. 129-143. [COBISS-SI-ID 16215385]
KRAJNC, Marjetka, VITRIH, Vito. Motion design with Euler-Rodrigues frames of quintic Pythagorean-hodograph curves. Mathematics and computers in simulation, ISSN 0378-4754. [Print ed.], 2012, vol. 82, iss. 9, str. 1696-1711. [COBISS-SI-ID 1024447572]
KOZAK, Jernej, KRAJNC, Marjetka. Geometric interpolation by planar cubic polynomial curves. Computer Aided Geometric Design, ISSN 0167-8396, 2007, vol. 24, no. 2, str. 67-78. [COBISS-SI-ID 14227545]
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]
MUHIČ, Andrej, PLESTENJAK, Bor. On the quadratic two-parameter eigenvalue problem and its linearization. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2010, vol. 432, iss. 10, str. 2529-2542. [COBISS-SI-ID 15469913]
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]