Rudolf Scitovski 

Full Professor
Department of Mathematics
Josip Juraj Strossmayer University of Osijek
Trg Ljudevita Gaja 6
Osijek, HR-31000, Croatia¸
phone: +385-31-224-800
fax: +385-31-224-801
email:  scitowsk@
office:  23 (first floor)


Research Interests

Numerical and applied mathematics – especially least squares and least absolute deviationsons problems, clustering and applications, global optimization

Aspects: existence of solutions, numerical methods for solving

Applications: solving parameter identification problems in mathematical problems (agriculture, economy, marketing, electrical engineering, medicine, food industry), smoothing the data (electrical engineering, medicine), surface generating on the basis of experimental data (electrical engineering, selection in livestock industry, civil engineering), clustering (earthquake zoning, short-term and long-term energy products prediction, short-term and long-term water level prediction, acceptable definition of constituencies, image and signal analysis) 




PhD in mathematics, Department of Mathematics, University of Zagreb, 1984.
MSc in mathematics, Department of Mathematics, University of Zagreb, 1977.
BSc in mathematics, Department of Mathematics, University of Zagreb, 1974.


Journal Publications

  1. Antonio Morales-Esteban, Francisco Martínez-Álvarez, Sanja Scitovski, Rudolf Scitovski, A fast partitioning algorithm using adaptive Mahalanobis clustering with application to seismic zoning, Computers & Geosciences 73 (2014), 132-141
    In this paper we construct an efficient adaptive Mahalanobis k-means algorithm. In addition, we propose a new efficient algorithm to search for a globally optimal partition obtained by using the adoptive Mahalanobis distance-like function. The algorithm is a generalization of the previously proposed incremental algorithm [36]. It successively finds optimal partitions with k = 2, 3, . . . clusters. Therefore, it can also be used for the estimation of the most appropriate number of clusters in a partition by using various validity indexes. The algorithm has been applied to the seismic catalogues of Croatia and the Iberian Peninsula. Both regions are characterized by a moderate seismic activity. One of the main advantages of the algorithm is its ability to discover not only circular but also elliptical shapes, whose geometry fits the faults better. Three seismogenic zonings are proposed for Croatia and two for the Iberian Peninsula and adjacent areas, according to the clusters discovered by the algorithm.
  2. Rudolf Scitovski, Kristian Sabo, Analysis of the k-means algorithm in the case of data points occurring on the border of two or more clusters, Knowledge-Based Systems 57 (2014), 1-7
    In this paper, the well-known $k$-means algorithm for searching for a locally optimal partition of the set $A subset R^n$ is analyzed in the case if some data points occur on the border of two or more clusters. For this special case, a useful strategy by implementation of the $k$-means algorithm is proposed.
  3. Ivan Vidović, Rudolf Scitovski, Center-based clustering for line detection and application to crop rows detection, Computers and Electronics in Agriculture (2014)
    This paper proposes a new efficient method for line detection based on 12 known incremental methods of searching for an approximate globally optimal partition 13 of a set of data points A and on the DIRECT algorithm for global optimization. The pro14 posed method was modified for solving the problem of detecting crop rows in agricultural 15 production. This modification can recognize crop rows with a high accuracy, and the 16 corresponding CPU-time is very acceptable. The method has been tested and compared 17 on synthetic data sets with the method based on Hough transformation. The efficiency of 18 this method might be significantly improved in direct application. The proposed method 19 has been used in this paper for the case of two or three crop rows. The generalization to 20 several crop rows is also given in the paper, but was not implemented. Also, the method 21 could be expanded in case when the number of crop rows is not known in advance.
  4. Kristian Sabo, Rudolf Scitovski, Interpretation and optimization of the k-means algorithm, Applications of Mathematics 59/4 (2014), 391-406
    The paper gives a new interpretation and a possible optimization of the well-known $k$-means algorithm for searching for the locally optimal partition of the set $mathcal{A}={a_iinR^n:i=1,dots,m}subset R^n$ which consists of $k$ disjoint nonempty subsets $pi_1,dots,pi_k$, $1leq kleq m$. For this purpose, a new Divided $k$-means Algorithm was constructed as a limit case of the well-known Smoothed k-means Algorithm. It is shown that the algorithm constructed in such way coincides with the $k$-means algorithm if during the iterative procedure no data points appear in the Voronoi diagram. If in the partition obtained by applying the Divided $k$-means Algorithm there are data points lying in the Voronoi diagram, it is shown that the obtained result can be improved further.
  5. Rudolf Scitovski, Tomislav Marošević, Multiple circle detection based on center-based clustering, Pattern Recognition Letters 52 (2014), 9-16
    The multiple circle detection problem has been considered in the paper on the basis of given data point set $mathcal{A}subset  Rn$. It is supposed that all data points from the set $mathcal{A}$ come from $k$ circles that should be reconstructed or detected. The problem has been solved by the application of center-based clustering of the set $mathcal{A}$, i.e. an optimal $k$-partition is searched for, whose clusters are determined by corresponding circle-centers. Thereby, the algebraic distance from a point to the circle is used. First, an adaptation of the well-known $k$-means algorithm is given in the paper. Also, the incremental algorithm for searching for an approximate globally optimal $k$-partition is proposed. The algorithm locates either a globally optimal $k$-partition or a locally optimal k-partition close to the global one. Since optimal partitions with 2, 3, ... clusters are determined successively in the algorithm, several well-known indexes for determining an appropriate number of clusters in a partition are adopted for this case. Thereby, the Hausdorff distance between two circles is used and adopted. The proposed method and algorithm are illustrated and tested on several numerical examples.

Refereed Proceedings

  1. Sanja Scitovski, Rudolf Scitovski, Cluster analysis of the data on unit circle, 1st Virtual International Conference in Advanced Research in Scientific Areas, Slovakia, 2012, 1574-1577
    The problem of data clustering on the unit circle is considered in this paper. For that purpose, the metric on the unit circle is introduced and the problem is reduced to a one-dimensional center-based clustering problem. For solving this problem, an efficient method is proposed. The method is illustrated on seismic activity data from a wider area of the town of Dubrovnik in the Republic of Croatia since 1973. In this way, the moments in the year with most intensive seismic activity in this area are detected.
  2. Gordana Kralik, Kristian Sabo, Rudolf Scitovski, Ivan Vazler, Solving parameter identification problem by the moving least absolute deviations method, 12th International Conference on Operational Research, Pula, Croatia, 2010, 297-307
    On the basis of measured data, among which a significant number of outliers might appear, we introduce one method for parameter identification in a mathematical model given by the ordinary differential equation of the first order. The method consists of two steps. In the first step, we construct a smooth function by applying the moving least absolute deviations method. In the second step, by applying the least absolute deviations method we estimate unknown parameters of mathematical models. The method is applied to and tested on the problem of estimating saturation level and asymmetry coefficient in the mathematical model with saturation. The mathematical model described by a generalized Verhulst differential equation [frac{;dy(t)};{;dt};= c, y(t)left(1-left(frac{;y(t)};{;A};right)^gammaright), quad c, , gamma, , A > 0, ] is considered especially. In this case the parameter estimation problem is reduced to the nonlinear least absolute deviations problem for a 3-parametric exponential regression model. For solving this problem an efficient method is developed. The method is tested on real measurement data of weights of 60 pigs in the period of 26 weeks.
  3. Ivana Kuzmanović, Goran Kušec, Kristian Sabo, Rudolf Scitovski, A new method for searching an L_1 solution of an overdetermined system of linear equations and applications, 12th International Conference on Operational Research, Pula, Croatia, 2008, 309-319
  4. Ivana Kuzmanović, Rudolf Scitovski, Kristian Sabo, Ivan Vazler, The least absolute deviation linear regression: properties and two efficient methods, Aplimat 2008, Bratislava, 2008, 227-240
  5. Branimir Dukić, Rudolf Scitovski, Analiza učinaka jednostavnog i složenog ispodgodišnjeg ukamaćivanja kod obračuna zakonskih zateznih kamata u RH, 10th International Conference on Operational Research KOI 2004, Trogir, 2005, 249-259
    Potaknuta problemima velikih potraživanja vjerovnika od dužnika po sudskim presudama u ovršnom postupku Vlada RH je 30. travnja 2004. godine donijela Vjerodostojno tumačenje čl. 3. st. 1. Zakona o zateznim kamatama objavljenog u NN br. 28/96 kojim je praktično aplicirala primjenu metode jednostavnog ukamaćivanja na potraživanja dulja od godinu dana. S ovakvom mjerom, uz ZOO-om propisani nominalistički pristup, kada se radi o potraživanjima koja su podložna monetarnim promjenama koje su se zbivale krajem 80-tih i početkom 90-tih godina prošlog stoljeća, što je analizom i utvrđeno, Vlada je praktično dovela vjerovnike u vrlo težak položaj obezvrjeđujući njihova potraživanja u ovršnom postupku. Sagledavajući i tumačeći efekte primjene metoda jednostavnog i složenog ispodgodišnjeg ukamaćivanja, u ovom istraživanju pokušat će se ukazati na bit problema, te sugerirati rješenje za otklanjanje problema koje kumulira navedeno Vjerodostojno tumačenje. Za potrebe analize načinjeni su algoritmi koji su aplicirani u vidu računalnih programa kojima je testiran učinak jednostavnog i složenog ispodgodišnjeg ukamaćivanja.


  1. Kristian Sabo, Rudolf Scitovski, Ivan Vazler, Grupiranje podataka: klasteri, Osječki matematički list 10 (2010), 149-176
    U ovom radu razmatramo problem grupiranja elemenata skupa A u disjunktne neprazne podskupove - klastere, pri cemu pretpostavljamo da su elementi skupa A odreženi s jednim ili dva obilježja. Za rješavanje problema koristi se kriterij najmanjih kvadrata te kriterij najmanjih apsolutnih udaljenosti. Naveden je niz primjera koji ilustriraju razlike mežu tim kriterijima. Izražena je odgovarajuca programska podrška s ciljem da zainteresirani strucnjaci u svom znanstvenom ili strucnom radu mogu olakšano koristiti ovu metodologiju i pristup.
  2. Kristian Sabo, Rudolf Scitovski, Prosti brojevi, Osječki matematički list 3 (2003), 13-20
    U članku se opisuju neka važna svojstva prostih brojeva. Pored danih primjera i zadataka, navodi se i nekoliko neriješenih problema vezanih uz proste brojeve.
  3. Drago Vukojević, Rudolf Scitovski, Matematika na šahovskoj ploči, Matka 4 (1996), 17-25
    U članku se navode neke zanimljive pravilnosti i matematička svojstva vezana uz šahovsku ploču i samu igru, kao primjerice magični kvadrati na šahovskoj ploči, dokaz Pitagorinog teorema na šahovskoj ploči, Pascalov trokut itd.
  4. Dario Galić, Rudolf Scitovski, Neka geometrijska svojstva balističkih parabola u vakuumu, Matematičko fizički list 46 (1996), 129-133
    Promatra se najjednostavniji problem eksterne balistike za vakuum, bez utjecaja specijalnih efekata (promjena sile teže o visini, rotacija Zemlje, Coriolisova sila, utjecaj Zemljinog magnetskog polja itd.). Daju se neka svojstva sustava balističkih parabola u slučaju ako je početna brzina konstantna a varira se izlazni kut ili ako je izlazni kut konstantan, a varira se početna brzina.
  5. Rudolf Scitovski, Klaudija Scitovski, Nemoguće figure, Matka 3 (1995)
    U radu se opisuju i daju konstrukcije nekih osnovnih nemogućih figura, primjerice nemogućeg trokuta.


  1. Rudolf Scitovski, Ninoslav Truhar, Zoran Tomljanović, Metode optimizacije, Svučilište Josipa Jurja Strossmayera u Osijeku, Odjel za matematiku., Osijek, 2014.
    Namjena ovog teksta je upoznati čitatelja s glavnim metodama jednodimenzionalne i višedimenzionalne minimizacije sa i bez ograničnja. Posebna je pozornost posvećena metodama minimizacije nediferencijabilnih funkcija. Pri tome je izbjegavano dokazivanje zahtjevnih teorema, osim u slučajevima konstruktivnih dokaza koji sami po sebi upućuju na izgradnju ideja ili metoda. Navedeni optimizacijski problemi imaju veliku primjenu u raznim dijelovima života. Na primjer, često se javljaju problemi poput optimalnog oblikovanja odredenih mehaničkih sustava (oblikovanje dijelova automobilskih motora, nosivih konstrukcija u gradjevinarstvu, . . .), problem modeliranja ponašsanja tržišta, problemi iz teorije upravljanja (smirivanje sustava, optimalno upravljanje, . . . ) i mnogi drugi. Upravo činjenica da se razni problemi optimizicije pojavljuju u raznim dijelovima ljudske djelatnosti osigurava ovom tekstu široku primjenu.
  2. Rudolf Scitovski, Numerička matematika, Odjel za matematiku Sveučilišta u Osijeku i Elektrotehnički f., Osijek, 1999.
    Sadržaj: 1. Pogreške. 2. Interpolacija. Spline interpolacija. 3. Rješavanje sustava linearnih jednadžbi. 4. Rješavanje nelinearnih jednadžbi. 5. Aproksimacija funkcija. 6. Problemi najmanjih kvadrata. 7. Numerička integracija. 8. Numeričko rješavanje običnih diferencijalnih jednadžbi. 9. Numeričko rješavanje parcijalnih diferencijalnih jednadžbi.
  3. Dragan Jukić, Rudolf Scitovski, Matematika I, Odjel za matematiku, Osijek, 1998.
  4. Miljenko Crnjac, Dragan Jukić, Rudolf Scitovski, Matematika, Ekonomski fakultet, Osijek, 1994.
  5. Rudolf Scitovski, Radoslav Galić, Mirta Šilac-Benšić, Numerička analiza. Vjerojatnost i statistika, Elektrotehnički fakultet, Osijek, 1993.
    Sažetak: Greške. Rješavanje jednadžbe f(x)=0. Interpolacija. Spline aproksimacije. Rješavanje sustava linearnih jednadžbi. Problemi najmanjih kvaddrata. Numerička integracija. Numeričko rješavanje običnih diferencijalnih jednadžbi. Kombinatorika. Vjerojatnost događaja. Slučajne varijable. Diskretna slučajna varijabla. Kontinuirana slučajna varijabla. Višedimenzionalna slučajna varijabla. Granični teoremi slučajne vjerojatnosti. Uzorak. Procjenjivanje parametara. Testiranje statističkih hipoteza.


  • 1986 - 1990 - head of project task ( „Operationalization of categories and relationships of value laws“ that was carried out within project (2.08.01) „Fundamental research in economy“ (Ministry of Science, Technology and Computing)

  •  1991-1995 - principal investigator of scientific project (1-01-129) „Application of numerical and finite mathematics“ (Ministry of Science, Technology and Computing)

  •  1996 - 2000 - principal investigator of scientific project (165021) „Parameter identification problems in mathematical models“ (Department of Mathematics, University of Osijek - Ministry of Science and Technology)

  •  2002 - 2006 - principal investigator of scientific project (023501) "Parameter estimation in mathematical models“ (Department of Mathematics, University of Osijek - Ministry of Science and Technology)

  •  2007 - 2013 scientific project (235-2352818-1034)  "Nonlinear parameter estimation problems in mathematical models“ (Ministry of Science, Education and Sports), investigator


Professional Activities

Editorial Boards

Mathematical Communications (since 1996)

Osječki matematički list (since 2003)  

Croatian Operational Research Review (since 2013)


Committee Memberships
  • Chairman of the Organizing Committee of the VII Conference on Applied Mathematics, Osijek, September 13-15, 1989
  • Member of the Programming Committee and chairman or deputy chairman of the Organizing Committee of the 6th-10th International Conference on Operational Research, which were organized by the Croatian Operational Research Society

  • Member of the Scientific Committee of the  "International Conference on Operational Research", Croatian Operational Research Society (since 1996 - )

  • Since 1999 - 2011: member of the Scientific or Organizing Committee of the International Conference  on Applied Mathematics and Scientific Computing

  • Member of the Scientific Committee of the scientific-professional conference PrimMath, 2001

  • Member of the Scientific Committee of the 2nd (Zagreb, 2000), 3rd (Split, 2004), 4th (Osijek, 2008), 5th (Rijeka, 2012), Croatian Congress of Mathematics

  • President of the Scientific Committee of the 4th Croatian Congress of Mathematics, Osijek, June 17-20, 2008





Mathematical Communications 

Osječki matematički list

Applied Mathematics and Computation

European Journal of Operational Research

Central European Journal of Operations Research

Expert Systems With Applications

Knowledge-Based Systems


Communications in Statistics - Theory and Methods



Service Activities
  • since 2013 - Vice-Rector for Science, Technology, Projects and International Cooperation, University of  Osijek
  • 1999 – 2003 and  2009 - 2013 Head of the Department of Mathematics,   University of  Osijek
  • since 2008 - chairman of the Seminar for optimization and applications
  • 1994 – 2000 - Head of the Mathematical Colloquium in Osijek
  • 2003 - 2010 - member of the Managing Board of The National Foundation for Science, Higher Education and Technological Development of Republic of Croatia
  • 2001 – 2008 - member of the National Council for Higher Education of the Republic of Croatia
  • 2006. – 2009 - member of the National Scientific Field Committee for Natural Sciences
  • 1997 – 1998 and 2001 – 2005 - member of the   National  Commission for Mathematics  
  • 2003 – 2007 -   vice-head of the Department of Mathematics, University of Osijek
  • 1998 – 1999 - dean of the  Faculty of Electrical Engineering in Osijek
  • 1997 - vice-dean for science at the  Faculty of Electrical Engineering in Osijek
  • since 2008 - chairman of the Osijek Students Center Council
  • 2002 - 2008 - chairman of the Osijek Students Center Recovery Council
  • 1998 – 2001 – member and chairman of the Managing Board of the City and University Library in Osijek


Nastavne aktivnosti u zimskom semestru Akademske 2014./2015.




Nastavne aktivnosti u ljetnom semestru Akademske 2014./2015.





Konzultacije (Office Hours): Srijeda (Wed) 10:00am.



Here goes the private stuff.