Introduction to numerical methods

2022/2023
Programme:
Mathematics, First Cycle
Year:
3 year
Semester:
first
Kind:
mandatory
ECTS:
6
Language:
slovenian
Hours per week – 1. semester:
Lectures
3
Seminar
0
Tutorial
3
Lab
0
Content (Syllabus outline)

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.

Readings
  • J. W. Demmel: Uporabna numerična linearna algebra, DMFA-založništvo, Ljubljana, 2000.
  • Z. Bohte: Numerične metode, DMFA-založništvo, Ljubljana, 1991.
  • Z. Bohte: Numerično reševanje nelinearnih enačb, DMFA-založništvo, Ljubljana, 1993.
  • Z. Bohte: Numerično reševanje sistemov linearnih enačb, DMFA-založništvo, Ljubljana, 1994.
  • G. H. Golub, C. F. Van Loan: Matrix Computations, 3rd edition, Johns Hopkins Univ. Press, Baltimore, 1996.
  • B. N. Datta: Numerical Linear Algebra and Applications, Brooks/Cole, Pacific Grove, 1995.
  • L. N. Trefethen, D. Bau: Numerical Linear Algebra, SIAM, Philadelphia, 1997.
  • R. L. Burden, J. D. Faires: Numerical Analysis, 8th edition, Brooks/Cole, Pacific Grove, 2005.
  • D. R. Kincaid, E. W. Cheney: Numerical Analysis : Mathematics of Scientific Computing, 3rd edition, Brooks/Cole, Pacific Grove, 2002.
Objectives and competences

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.

Intended learning outcomes

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.

Learning and teaching methods

Lectures, exercises, homework, consultations

Assessment

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

Lecturer's references

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]