Numerical Linear Algebra

Learning Outcomes

  1. define and describe fundamental concepts such as matrix norms, singular and eigen values and vectors
  2. recognize types of matrices, such as ortogonal, unitary, symetric, Hermitian, normal and positive definite matrices
  3. describe Jordan and Schure form of the matrix and a notion of a diagonalizable matrix
  4. apply matrix transformation in order to transform a matrix into triangular, Hessenberg and tridiagonal form
  5. derive and utilize SVD and QR factorization of the matrix for efficiently solving problems in practice
  6. analyze and employ iterative methods for linear algebraic systems
  7. explain and employ iterative algorithms for computing eigenvalues
  8. Relate the quality of obtained numerical solution to derived theoretical results

Forms of Teaching

Lectures

Partial e-learning

Laboratory

Week by Week Schedule

  1. Symmetric and i Hermitian (self-adjoint) matrices; Gramm-Schmidt's orthogonalization; Orthogonal and unitary matrices.
  2. Diagonalization of a matrix; Spectral theorem for symmetric matrices; Nilpotent matrices; Jordan cells; Minimal polynomial.
  3. Nilpotent matrices; Jordan cells; Minimal polynomial; Positively (negatively) semidefinite matrices; Positively (negatively) definite matrices; Indefinite matrices.
  4. Quadratic forms and their diagonalization; Definite and indefinite quadratic forms; Signature; Minimum, maximum and saddle points of a quadratic form.
  5. Operator norms; The space L(X,Y); Matrix norm; Convergence of matrices; Series of matrices; Neumann series; Spectral radius and spectral norm.
  6. Projectors; QR factorization; HouseHolder triangularization and its stability.
  7. Singular Value Decomposition and its application; Algorithms for the SVD.
  8. Midterm exam.
  9. Jacobi, Gauss-Seidel and Relaxation Methods; Convergence Results for Jacobi, Gauss-Seidel and Relaxation Methods.
  10. Symmetric Form of the Gauss-Seidel and SOR Methods; Implementation Issues.
  11. Conjugate Gradient Method and other Krylov Subspace Iterations; Applications: Analysis of an Electric Network; Finite Difference Analysis of Beam Bending.
  12. Geometrical Location of the Eigenvalues; Stability and Conditioning Analysis; The Power Method; Inverse Iteration.
  13. The QR Method; Hessenberg Reduction; Tridiagonal and Bidiagonal Reductions; QR Iterations with Implicit Shifts.
  14. Methods for Eigenvalues of Symmetric Matrices (The Jacobi's Method; Tridiagonal QR Iteration, Rayleigh Quotient Iteration); Applications: Free Dynamic Vibration of a Bridge.
  15. Final exam.

Study Programmes

University undergraduate
Computing (study)
Elective Courses (6. semester)
Electrical Engineering and Information Technology (study)
Elective Courses (6. semester)

Literature

(.), A. Aglić Aljinović, N. Elezović, D. Žubrinić, Linearna algebra, Element, Zagreb, 2011.,
(.), Z. Drmač i ostali, Numerička analiza (predavanja i vježbe), Zagreb, 2003. https://web.math.pmf.unizg.hr/~rogina/2001096/num_anal.pdf),
(.), N. Truhar, Numerička linearna algebra, Osijek, 2012. http://www.mathos.unios.hr/nla/NLA.pdf,
(.), G. H. Golub i C. F. Van Loan: Matrix Computations, 3rd Edition, John Hopkins University Press, Baltimore, Maryland, 1996.,
(.), L.N. Trefethen, D. Bau: Numerical Linear Algebra, SIAM, 1997.,

General

ID 183464
  Summer semester
5 ECTS
L3 English Level
L2 e-Learning
45 Lectures
0 Exercises
13 Laboratory exercises
0 Project laboratory