Study

# Computing

## Numerical Linear Algebra

• Class 45
• Practice 13
• Independent work 92
Total 150

### Course title

Numerical Linear Algebra

Elective

183464

6

5

### Course objectives

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

(.), 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.,

#### Minimal learning outcomes

• Define and describe fundamental concepts such as matrix norms, singular and eigen values and vectors
• Recognize types of matrices, such as ortogonal, unitary, symetric, Hermitian, normal and positive definite matrices
• Describe Jordan and Schure form of the matrix and a notion of a diagonalizable matrix
• Apply matrix transformation in order to transform a matrix into triangular, Hessenberg and tridiagonal form
• Derive and utilize SVD and QR factorization of the matrix for efficiently solving problems in practice
• Analyze and employ iterative methods for linear algebraic systems
• Explain and employ iterative algorithms for computing eigenvalues
• Relate the quality of obtained numerical solution to derived theoretical results