Primjena hiperdualnih brojeva za računanje derivacija prvog i drugog reda funkcija više varijabli s primjenama u MATLAB-u

master's thesis

• Title (english): Application of hyper-dual numbers to compute first and second order derivatives of multivariate functions with applications in MATLAB
• Author: Jakov Jonjić
• Mentor: Vladimir Milić (mentor)
• Granter: University of Zagreb; Faculty of Mechanical Engineering and Naval Architecture; Zagreb
• Defense date and country: 2024-03-28, Croatia
• Scientific / art field, discipline and subdiscipline: TECHNICAL SCIENCES; Mechanical Engineering; General Mechanical Engineering (Construction)

Abstract

U ovom diplomskom radu ostvarena je implementacija aritmetike hiperdualnih brojeva u programskom jeziku MATLAB za računanje prve i druge derivacije funkcije. Pošto je naglasak u radu na praktičnoj primjeni hiperdualnih brojeva, implementacija je ostvarena objektnim programiranjem i uvođenjem algoritma za računanje derivacija. Točnost algoritma je pokazana detaljnim matematičkim izvodom, te su opisane sve tehnike programiranja korištene u izradi klase objekata. Točnost metode je uspoređena s metodom centralne razlike i s koračno kompleksnom metodom i rezultati su prikazani na grafovima. Numerički eksperimenti uključuju klasu hiperdualnih brojeva u računanju Jacobijana i Hessijana u Newtonovoj metodi traženja nultočke, te u ugrađenim optimizacijskim algoritmima u MATLAB-u za računanje prve derivacije funkcije cilja.

Abstract (english)

In this master’s thesis, an implementation of the arithmetic of hyperdual numbers is realized in MATLAB programming language for computing the first and second derivatives of a function. Since the emphasis in the work is on the practical application of hyperdual numbers, the implementation is realised using object-orianted programming and introducing an algorithm for computing the derivatives. The correctness of the method is shown by a detailed mathematical derivation and all programming techniques used in the class of objects are described. The accuracy of the algorithm is compared to the central difference method and the complex-step method and the results are shown on graphs. The numerical experiments include the class of hyperdual numbers in the evaluation of the Jacobian and Hessian in Newton’s method of finding the zero and in MATLAB’s built-in optimization methods for evaluating the first derivative of the cost function.

• Keywords: hiperdualni brojevi; objektno orijentirano programiranje; automatsko računanje derivacija; metoda konačnih diferencija; koračno kompleksna metoda; Newtonova metoda; ugrađeni optimizacijski algoritmi
• Keywords (english): hyper-dual numbers; object oriented programming; automatic differentiation; finite difference method; complex-step method; Newton’s method; built-in optimization algorithms
• Language: croatian
• URN:NBN: urn:nbn:hr:235:142030
• Study programme: Title: Mechanical Engineering; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra inženjer/inženjerka strojarstva (magistar/magistra inženjer/inženjerka strojarstva)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-03-21 13:26:41