### Numerical Mathematics

#### Learning Outcomes

- distinguish and identify types of errors in numerical computing
- apply direct methods in solving systems of linear equations
- identify ill conditioned system of linear equations
- compute interpolation polynomial and cubic spline for given data and estimate the error of the approximation
- compute approximation value of a definite integral and estimate error of the approximation
- describe and apply important methods in solving problems of linear least squares
- apply derived numerical methods for solving nonlinear equations
- use fundamental methods of unconstrained optimization
- use choosen numerical software for solving problems using numerical methods
- analyze accuracy of obtained results and visualization of numerical solutions

#### Forms of Teaching

**Lectures**Lectures are held in two cycles, 3 hours per week

**Exercises**Exercises are held in two cycles, 1 hour per week

**Partial e-learning**Students will solve online quizzes

**Laboratory**During the semester students will work on program tasks

#### Grading Method

Continuous Assessment | Exam | |||||
---|---|---|---|---|---|---|

Type | Threshold | Percent of Grade | Threshold | Percent of Grade | ||

Laboratory Exercises | 0 % | 20 % | 0 % | 20 % | ||

Quizzes | 0 % | 10 % | 0 % | 10 % | ||

Mid Term Exam: Written | 0 % | 35 % | 0 % | |||

Final Exam: Written | 0 % | 35 % | ||||

Exam: Written | 70 % | 0 % |

#### Week by Week Schedule

- Sources of Error in Computational Models; Floating point precision and error propagation ; Numerical Differentiation.
- The Gaussian Elimination Method (GEM); LU Factorization; Pivoting Strategies; PLU Factorization.
- Symmetric and Positive Definite Matrices; The Cholesky Factorization; Applications: Nodal Analysis of a Structured Frame; Regularization of a Triangular Grid.
- Stability Analysis of Linear Systems; Matrix Norms; The Condition Number of the Matrix; Improving the Accuracy of GEM; Scaling; Iterative refinement.
- Lagrangee Form of the Interpolation Polynomial; The Interpolation Error; Divided Differences; Newton Form of the Interpolation Polynomial.
- Approximation by splines; Cubic Splines; Applications: Geometric reconstruction based on computer Tomographies.
- Midpoint, Trapezoidal and Simpson rule; Composite Newton-Cotes Formulae; Richardson Extrapolation; Romberg integration; Applications: Computation of an Ellipsoid Surface; Computation of the Wind Action on a Sailboat Mast.
- Midterm exam.
- Sensitivity and conditioning; Rank-Deficient Least Squares Problem.
- Matrix Factorization that Solve the Linear Least Square Problem; Normal Equations; QR Decomposition; Applications of the SVD and QR decomposition in solving linear least squares problem.
- The Bisection Method; The Newton's Method; The Secant Method.
- Fixed-Point Iterations Method; Applications: Analysis of the State Equation for a Real Gas; Analysis of a Nonlinear Electrical Circuit.
- Direct Search Algorithms (the Hooke-Jeeves method); Gradient Methods (the steepest descent); ; Trusted-Region Methods; Conjugate Gradient Methods.
- Quasi-Newton Methods; Large-Scale Unconstrained Optimization; Nonlinear Least-Squares Problems; The Gauss-Newton Method; The Levenberg-Marquardt Method.
- Final exam.

#### Study Programmes

##### University undergraduate

Computing (study)

Elective Courses
(5. semester)
Electrical Engineering and Information Technology (study)

Elective Courses
(5. semester)
#### Literature

(.),

*Milišić, Josipa Pina; Žgaljić Keko, Ana, Uvod u numeričku matematiku za inženjere, Zagreb, Element, 2013. (Sveučilišni udžbenik)*,
(.),

*Z. Drmač i ostali, Numerička analiza (predavanja i vježbe), Zagreb, 2003. https://web.math.pmf.unizg.hr/~rogina/2001096/num_anal.pdf)*,
(.),

*I. Ivanšić: Numerička matematike, Element, Zagreb, 1998.*,
(.),

*M. T. Heath: Scientific Computing: An Introductory Survey, McGrawHill, New York, 2002.*,
(.),

*A. Quarteroni, R. Sacco, F.Saleri, Numerical Mathematics, Text in Applied Mathematics, Springer, Berlin, Heildeberg, 2007.*,
(.),

*R. Plato, Concise Numerical Mathematics, American Mathematical Society, Graduate Studies in Mathematics 57, 2000.*,#### Lecturers

#### General

**ID**183445

Winter semester

**5**ECTS

**L1**English Level

**L1**e-Learning

**45**Lectures

**15**Exercises

**10**Laboratory exercises

**0**Project laboratory

#### Grading System

**85**Excellent

**70**Very Good

**55**Good

**45**Acceptable