### 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

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

#### 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

**Excellent**

**Very Good**

**Good**

**Acceptable**