### Numerical Mathematics

#### Learning Outcomes

1. distinguish and identify types of errors in numerical computing
2. apply direct methods in solving systems of linear equations
3. identify ill conditioned system of linear equations
4. compute interpolation polynomial and cubic spline for given data and estimate the error of the approximation
5. compute approximation value of a definite integral and estimate error of the approximation
6. describe and apply important methods in solving problems of linear least squares
7. apply derived numerical methods for solving nonlinear equations
8. use fundamental methods of unconstrained optimization
9. use choosen numerical software for solving problems using numerical methods
10. 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

Continuous Assessment Exam
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

1. Sources of Error in Computational Models; Floating point precision and error propagation ; Numerical Differentiation.
2. The Gaussian Elimination Method (GEM); LU Factorization; Pivoting Strategies; PLU Factorization.
3. Symmetric and Positive Definite Matrices; The Cholesky Factorization; Applications: Nodal Analysis of a Structured Frame; Regularization of a Triangular Grid.
4. Stability Analysis of Linear Systems; Matrix Norms; The Condition Number of the Matrix; Improving the Accuracy of GEM; Scaling; Iterative refinement.
5. Lagrangee Form of the Interpolation Polynomial; The Interpolation Error; Divided Differences; Newton Form of the Interpolation Polynomial.
6. Approximation by splines; Cubic Splines; Applications: Geometric reconstruction based on computer Tomographies.
7. 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.
8. Midterm exam.
9. Sensitivity and conditioning; Rank-Deficient Least Squares Problem.
10. 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.
11. The Bisection Method; The Newton's Method; The Secant Method.
12. Fixed-Point Iterations Method; Applications: Analysis of the State Equation for a Real Gas; Analysis of a Nonlinear Electrical Circuit.
13. Direct Search Algorithms (the Hooke-Jeeves method); Gradient Methods (the steepest descent); ; Trusted-Region Methods; Conjugate Gradient Methods.
14. Quasi-Newton Methods; Large-Scale Unconstrained Optimization; Nonlinear Least-Squares Problems; The Gauss-Newton Method; The Levenberg-Marquardt Method.
15. Final exam.

#### Study Programmes

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.,

#### General

ID 183445
Winter semester
5 ECTS
L1 English Level
L1 e-Learning
45 Lectures
15 Exercises
10 Laboratory exercises
0 Project laboratory