# Computing

## Numerical Linear Algebra

- Class 45
- Practice 13
- Independent work 92

### Course title

Numerical Linear Algebra

### Lecture type

Elective

### Course code

183464

### Semester

6

### ECTS

5

### Lecturers and associates

- Associate Professor PhD Tomislav Šikić
- Assistant Professor PhD Ana Žgaljić Keko
- Full Professor PhD Darko Žubrinić

### 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.

Quadratic forms and their diagonalization. Definite and indefinite quadratic forms. Signature. Minimum, maximum and saddle points of a quadratic form.

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.

### Required reading

(.), 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.,

#### Online education during epidemiological measures

- Study program duration
- 6 semesters (3 years)
- Semester duration
- 15 weeks of active teaching + 5 examination weeks
- Total number of ECTS points
- 180
- Title
- Bacc.ing.comp (Bachelor of Science in Computing)

**Academic calendar**

#### 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