Kvadratično programiranje i linearna zadaća komplementarnosti

Linearna zadaća komplementarnosti je sjajan kontekst u kojem se mogu prikazati pojmovi iz linearne algebre i teorije matrica pa smo se na početku rada dotakli osnovnih pojmova i rezultata vezanih za matrice, konkretno pozitivno definitne i semidefinitne matrice. Osim toga, prisjetili smo se još pokojeg rezultata o konveksnim kvadratičnim funkcijama koji su nam pomogli u razumijevanju zadaće kvadratičnog programiranja. Prije uspostavljanja veze između zadaće kvadratičnog programiranja i linearne zadaće komplementarnosti, pozabavili smo se uvjetima optimalnosti prvog reda, odnosno izveli smo Karush-Kuhn-Tuckerove uvjete koji su ključno vezivo za izgradnju mosta među spomenutim zadaćama. Kako mnogo toga u linearnoj zadaći komplementarnosti počiva na ideji komplementarnog konusa, bilo je neizbježno zastati i prokomentirati zadaću u terminima konusa te demonstrirati to primjerom. Glavni dio poglavlja o linearnoj zadaći komplementarnosti je svakako rasprava o egzistenciji i broju rješenja. Isprva smo se bazirali na klasu pozitivno definitnih i semidefinitnih matrica te pokazali u slučaju pozitivno semidefinitne matrice M da je zadaća rješiva ako je dopustiva, a u slučaju pozitivno definitnh matrica je k tome rješenje i jedinstveno za svaki vektor q. Osim te dvije klase, govorili smo i o klasama S-matrica i P-matrica. Ova posljednja nam je bila posebno zanimljiva jer smo za tu klasu mogli dati teorem koji govori da je jedinstveno rješenje linearne zadaće komplementarnosti također jedinstveno rješenje zadaće kvadratičnog programiranja. Na samom kraju rada pozabavili smo se algoritmima za rješavanje linearne zadaće komplementarnosti direktnim metodama te pokazali na primjerima kako možemo riješiti zadaću linearne komplementarnosti te zadaću kvadratičnog programiranja koristeći Lemkeov algoritam.The linear complementarity problem is an excellent context to illustrate concepts of linear algebra and matrix theory. At the beginning we introduced some basic terms and results regarding matrix theory, especially positive definite and semi-definite matrices. Moreover, we mentioned some results concerning convex quadratic functions as they are essential for understanding of quadratic program. Before establishing the connection between quadratic program and linear complementarity problem, we defined first-order optimality conditions. More precisely, we derived Karush-Kuhn-Tucker conditions that are integral part of building the connection between aforementioned problems. In the third chapter, we introduced the concept of complementarity cones as linear complementarity problem rests on that idea. We demonstrated that by the example. Main part of the chapter is based on presenting the results pertaining to the existence and multiplicity of solutions to the linear complementarity problem. At first, we were based only on the class of positive definite and positive semi-definite matrices. In the case of positive semidefinite matrices we showed an important result: if the linear complementarity problem is feasible, then it’s solvable. In the case of positive definite matrices, the linear complementarity problem has a unique solution. Moreover, we mentioned the classes of S-matrices and P-matrices. Class of P-matrices has an interesting property as the unique solution of linear complementarity problem characterized by P-matrix is also the unique solution of the quadratic program. At the end, we developed Lemke’s algorithm for solving the linear complementarity problem and made some examples with linear complementarity problem and quadratic program

Positive Definite Matrices

U radu ćemo se baviti posebnom vrstom hermitskih matrica zvanih pozitivno definitne matrice. Definirat ćemo pojmove i iskazati tvrdnje koje ćemo koristiti u daljnjem radu. Također ćemo iskazati i dokazati poznati Sylvesterov kriterij te važnu Cholesky dekompoziciju koja je učinkovita u numeričkim rješavanjima linearnih jednadžbi te ćemo pokazati koja posebna svojstva vrijede za pozitivno definitne matrice.This paper will deal with the special type of Hermitian matrices, so called positive definite matrices. The terms and claims which are going to be used further in this paper will also be defined and stated. The well-known Sylvester’s criterion will be determined and proved as well as the important Cholesky decomposition, which is effective in numerical solving of linear equations. The special properties which are valid for positive definite matrices will also be shown

Computing interior eigenvalues and corresponding eigenvectors of definite matrix pairs

U prvom dijelu ove disertacije predstavljamo nove algoritme koji za dani hermitski matrični par $(A, B)$ ispituju je li on pozitivno definitan, u smislu da postoji realan broj $\lambda_0$ takav da je matrica $A-\lambda_0B$ pozitivno definitna. Skup svih takvih $\lambda_0$ čini otvoreni interval koji zovemo definitan interval, a bilo koji takav $\lambda_0$ zovemo definitan pomak. Najjednostavniji algoritmi ispitivanja koje predlažemo temelje se na ispitivanju glavnih podmatrica reda 1 ili 2. Također razvijamo efikasniji algoritam ispitivanja potprostora uz pretpostavku indefinitnosti matrice B. Taj se algoritam temelji na iterativnom ispitivanju malih gusto popunjenih komprimiranih parova koji nastaju korištenjem test-potprostora malih dimenzija, a predlažemo i ubrzanje samog algoritma. Algoritam ispitivanja potprostora posebno je pogodan za velike rijetko popunjene vrpčaste matrične parove, a može se primijeniti u ispitivanju hiperbolnosti kvadratnog svojstvenog problema. U drugom dijelu ove disertacije za dani pozitivno definitni matrični par $(A, B)$ reda $n$ s indefinitnom matricom $B$ konstruiramo nove algoritme minimizacije traga funkcije $f(X)=X^HAX$ uz uvjet $X^HBX=diag(I_{k_+}, -I_{k_-})$ gdje je $X \in \mathbb{C}^{n \times (k_++k_-)}, 1 \leq k_+ \leq n_+, 1 \leq k_- \leq n_-$ i $(n_+, n_-, n_0)$ inercija matrice $B$. Predlažemo opći indefinitni algoritam, te razvijamo efikasne algoritme prekondicioniranih gradijentnih iteracija koje smo nazvali indefinitna $m$-shema. Stoga metode indefinitne $m$-sheme za dani pozitivno definitni par i jedan ili dva definitna pomaka (koji se mogu dobiti algoritmom ispitivanja potprostora) istovremeno računaju manji broj unutarnjih svojstvenih vrijednosti oko definitnog intervala i pridružene svojstvene vektore. Također, dajemo ideje kako računati manji broj svojstvenih vrijednosti oko bilo kojeg broja unutar rubova spektra, a izvan definitnog intervala, i pridruženih svojstvenih vektora, danog pozitivno definitnog matričnog para koristeći pozitivno definitnu matricu prekondicioniranja. Algoritmi su posebno pogodni za velike rijetko popunjene matrične parove. Nizom numeričkih eksperimenata pokazujemo efikasnost samih algoritama ispitivanja i algoritama računanja unutarnjih svojstvenih vrijednosti i pridruženih svojstvenih vektora. Efikasnost naših metoda uspoređujemo s nekim postojećim metodama.The generalized eigenvalue problem (GEP) for given matrices $A, B \in \mathbb{C}^{n \times n}$ is to find scalars $\lambda$ and nonzero vectors $x \in \mathbb{C}^n$ such that $Ax = \lambda Bx$ (1). The pair $(\lambda, x)$ is called an eigenpair, $\lambda$ is an eigenvalue and $x$ corresponding eigenvector. GEP (1) where A and B are both Hermitian, or real symmetric, occurs in many applications of mathematics. Very important case is when B (and A) is positive definite (appearing, e.g., in the finite element discretization of self-adjoint and elliptic PDE-eigenvalue problem [25]). Another very important case is when B (and A) is indefinite, but the matrix pair (A, B) is definite, meaning, there exist real numbers $\alpha, \beta$ such that the matrix $\alpha A + \beta B$ is positive definite (appearing, e.g., in mechanics [83] and computational quantum chemistry [4]). Many theoretical properties (variational principles, perturbation theory, etc.) and eigenvalue solvers for Hermitian matrix are extended to definite matrix pairs [64, 79, 83]. A Hermitian matrix pair (A, B) is called positive (negative) definite if there exists a real $\lambda_0$ such that $A- \lambda_0 B$ is positive ( negative) definite. The set of all such $\lambda_0$ is an open interval called the definiteness interval [83], and any such $\lambda_0$ will be called definitizing shift. In the first part of this thesis we propose new algorithms for detecting definite Hermitian matrix pairs (A, B). The most simple algorithms we propose are based on testing the main submatrices of order 1 or 2. These algorithms do not have to give a final answer about (in)definiteness of the given pair, so we develop a more efficient subspace algorithm assuming B is indefinite. Our subspace algorithm for detecting definiteness is based on iterative testing of small full compressed matrix pairs formed using test-subspaces of small dimensions. It is generalization of the method of coordinate relaxation proposed in [36, Section 3.6]. We also propose acceleration of the subspace algorithm in a way that certain linear systems must be solved in every or in some iteration steps. If the matrix pair is definite, the subspace algorithm detects if it is positive or negative definite and returns one definitizing shift. The subspace algorithm is particulary suited for large, sparse and banded matrix pairs, and can be used in testing hyperbolicity of a Hermitian quadratic matrix polynomial. Numerical experiments are given which illustrate efficiency of several variants of our subspace algorithm and comparison is made with an arc algorithm [19, 17, 29]. In the second part of this thesis we are interested in solving partial positive definite GEP (1) where B (and A) is indefinite (both A and B can be singular). Specifically, we are interested in iterative algorithms which will compute a small number of eigenvalues closest to the definiteness interval and corresponding eigenvectors. These algorithms are based on trace minimization property [41, 49]: find the minimum of the trace of the function: $f(X)=X^HAX$ such that $X^HBX=diag(I_{k_+}, -I_{k_-})$ where $X \in \mathbb{C}^{n \times (k_++k_-)}, 1 \leq k_+ \leq n_+, 1 \leq k_- \leq n_-$ and $(n_+, n_-, n_0)$ is the inertia of B. The class of algorithms we propose will be preconditioned gradient type iteration, suitable for large and sparse matrices, previously studied for the case with A and/or B are known to be positive definite (for a survey of preconditioned iterations see [3, 39]). In the recent paper [42] an indefinite variants of LOBPCG algorithm [40] were suggested. The authors of [42] were not aware of any other preconditioned eigenvalue solver tailored to definite matrix pairs with indefinite matrices. In this thesis we propose some new preconditioned eigenvalue solvers suitable for this case, which include truncated and extended versions of indefinite LOBPCG from [42]. Our algorithms use one or two definitizing shifts. For the truncated versions of indefinite LOBPCG, which we call indefinite BPSD/A, we derive a sharp convergence estimates. Since for the LOBPCG type algorithms there are still no sharp convergence estimates, estimates derived for BPSD/A type methods serve as an upper (non-sharp) convergence estimates. We also devise some possibilities of using our algorithms to compute a modest number of eigenvalues around any spectral gap of a definite matrix pair (A, B). Numerical experiments are given which illustrate efficiency and some limitations of our algorithms

Computing interior eigenvalues and corresponding eigenvectors of definite matrix pairs

U prvom dijelu ove disertacije predstavljamo nove algoritme koji za dani hermitski matrični par $(A, B)$ ispituju je li on pozitivno definitan, u smislu da postoji realan broj $\lambda_0$ takav da je matrica $A-\lambda_0B$ pozitivno definitna. Skup svih takvih $\lambda_0$ čini otvoreni interval koji zovemo definitan interval, a bilo koji takav $\lambda_0$ zovemo definitan pomak. Najjednostavniji algoritmi ispitivanja koje predlažemo temelje se na ispitivanju glavnih podmatrica reda 1 ili 2. Također razvijamo efikasniji algoritam ispitivanja potprostora uz pretpostavku indefinitnosti matrice B. Taj se algoritam temelji na iterativnom ispitivanju malih gusto popunjenih komprimiranih parova koji nastaju korištenjem test-potprostora malih dimenzija, a predlažemo i ubrzanje samog algoritma. Algoritam ispitivanja potprostora posebno je pogodan za velike rijetko popunjene vrpčaste matrične parove, a može se primijeniti u ispitivanju hiperbolnosti kvadratnog svojstvenog problema. U drugom dijelu ove disertacije za dani pozitivno definitni matrični par $(A, B)$ reda $n$ s indefinitnom matricom $B$ konstruiramo nove algoritme minimizacije traga funkcije $f(X)=X^HAX$ uz uvjet $X^HBX=diag(I_{k_+}, -I_{k_-})$ gdje je $X \in \mathbb{C}^{n \times (k_++k_-)}, 1 \leq k_+ \leq n_+, 1 \leq k_- \leq n_-$ i $(n_+, n_-, n_0)$ inercija matrice $B$. Predlažemo opći indefinitni algoritam, te razvijamo efikasne algoritme prekondicioniranih gradijentnih iteracija koje smo nazvali indefinitna $m$-shema. Stoga metode indefinitne $m$-sheme za dani pozitivno definitni par i jedan ili dva definitna pomaka (koji se mogu dobiti algoritmom ispitivanja potprostora) istovremeno računaju manji broj unutarnjih svojstvenih vrijednosti oko definitnog intervala i pridružene svojstvene vektore. Također, dajemo ideje kako računati manji broj svojstvenih vrijednosti oko bilo kojeg broja unutar rubova spektra, a izvan definitnog intervala, i pridruženih svojstvenih vektora, danog pozitivno definitnog matričnog para koristeći pozitivno definitnu matricu prekondicioniranja. Algoritmi su posebno pogodni za velike rijetko popunjene matrične parove. Nizom numeričkih eksperimenata pokazujemo efikasnost samih algoritama ispitivanja i algoritama računanja unutarnjih svojstvenih vrijednosti i pridruženih svojstvenih vektora. Efikasnost naših metoda uspoređujemo s nekim postojećim metodama.The generalized eigenvalue problem (GEP) for given matrices $A, B \in \mathbb{C}^{n \times n}$ is to find scalars $\lambda$ and nonzero vectors $x \in \mathbb{C}^n$ such that $Ax = \lambda Bx$ (1). The pair $(\lambda, x)$ is called an eigenpair, $\lambda$ is an eigenvalue and $x$ corresponding eigenvector. GEP (1) where A and B are both Hermitian, or real symmetric, occurs in many applications of mathematics. Very important case is when B (and A) is positive definite (appearing, e.g., in the finite element discretization of self-adjoint and elliptic PDE-eigenvalue problem [25]). Another very important case is when B (and A) is indefinite, but the matrix pair (A, B) is definite, meaning, there exist real numbers $\alpha, \beta$ such that the matrix $\alpha A + \beta B$ is positive definite (appearing, e.g., in mechanics [83] and computational quantum chemistry [4]). Many theoretical properties (variational principles, perturbation theory, etc.) and eigenvalue solvers for Hermitian matrix are extended to definite matrix pairs [64, 79, 83]. A Hermitian matrix pair (A, B) is called positive (negative) definite if there exists a real $\lambda_0$ such that $A- \lambda_0 B$ is positive ( negative) definite. The set of all such $\lambda_0$ is an open interval called the definiteness interval [83], and any such $\lambda_0$ will be called definitizing shift. In the first part of this thesis we propose new algorithms for detecting definite Hermitian matrix pairs (A, B). The most simple algorithms we propose are based on testing the main submatrices of order 1 or 2. These algorithms do not have to give a final answer about (in)definiteness of the given pair, so we develop a more efficient subspace algorithm assuming B is indefinite. Our subspace algorithm for detecting definiteness is based on iterative testing of small full compressed matrix pairs formed using test-subspaces of small dimensions. It is generalization of the method of coordinate relaxation proposed in [36, Section 3.6]. We also propose acceleration of the subspace algorithm in a way that certain linear systems must be solved in every or in some iteration steps. If the matrix pair is definite, the subspace algorithm detects if it is positive or negative definite and returns one definitizing shift. The subspace algorithm is particulary suited for large, sparse and banded matrix pairs, and can be used in testing hyperbolicity of a Hermitian quadratic matrix polynomial. Numerical experiments are given which illustrate efficiency of several variants of our subspace algorithm and comparison is made with an arc algorithm [19, 17, 29]. In the second part of this thesis we are interested in solving partial positive definite GEP (1) where B (and A) is indefinite (both A and B can be singular). Specifically, we are interested in iterative algorithms which will compute a small number of eigenvalues closest to the definiteness interval and corresponding eigenvectors. These algorithms are based on trace minimization property [41, 49]: find the minimum of the trace of the function: $f(X)=X^HAX$ such that $X^HBX=diag(I_{k_+}, -I_{k_-})$ where $X \in \mathbb{C}^{n \times (k_++k_-)}, 1 \leq k_+ \leq n_+, 1 \leq k_- \leq n_-$ and $(n_+, n_-, n_0)$ is the inertia of B. The class of algorithms we propose will be preconditioned gradient type iteration, suitable for large and sparse matrices, previously studied for the case with A and/or B are known to be positive definite (for a survey of preconditioned iterations see [3, 39]). In the recent paper [42] an indefinite variants of LOBPCG algorithm [40] were suggested. The authors of [42] were not aware of any other preconditioned eigenvalue solver tailored to definite matrix pairs with indefinite matrices. In this thesis we propose some new preconditioned eigenvalue solvers suitable for this case, which include truncated and extended versions of indefinite LOBPCG from [42]. Our algorithms use one or two definitizing shifts. For the truncated versions of indefinite LOBPCG, which we call indefinite BPSD/A, we derive a sharp convergence estimates. Since for the LOBPCG type algorithms there are still no sharp convergence estimates, estimates derived for BPSD/A type methods serve as an upper (non-sharp) convergence estimates. We also devise some possibilities of using our algorithms to compute a modest number of eigenvalues around any spectral gap of a definite matrix pair (A, B). Numerical experiments are given which illustrate efficiency and some limitations of our algorithms

Kvadratično programiranje i linearna zadaća komplementarnosti

Linearna zadaća komplementarnosti je sjajan kontekst u kojem se mogu prikazati pojmovi iz linearne algebre i teorije matrica pa smo se na početku rada dotakli osnovnih pojmova i rezultata vezanih za matrice, konkretno pozitivno definitne i semidefinitne matrice. Osim toga, prisjetili smo se još pokojeg rezultata o konveksnim kvadratičnim funkcijama koji su nam pomogli u razumijevanju zadaće kvadratičnog programiranja. Prije uspostavljanja veze između zadaće kvadratičnog programiranja i linearne zadaće komplementarnosti, pozabavili smo se uvjetima optimalnosti prvog reda, odnosno izveli smo Karush-Kuhn-Tuckerove uvjete koji su ključno vezivo za izgradnju mosta među spomenutim zadaćama. Kako mnogo toga u linearnoj zadaći komplementarnosti počiva na ideji komplementarnog konusa, bilo je neizbježno zastati i prokomentirati zadaću u terminima konusa te demonstrirati to primjerom. Glavni dio poglavlja o linearnoj zadaći komplementarnosti je svakako rasprava o egzistenciji i broju rješenja. Isprva smo se bazirali na klasu pozitivno definitnih i semidefinitnih matrica te pokazali u slučaju pozitivno semidefinitne matrice M da je zadaća rješiva ako je dopustiva, a u slučaju pozitivno definitnh matrica je k tome rješenje i jedinstveno za svaki vektor q. Osim te dvije klase, govorili smo i o klasama S-matrica i P-matrica. Ova posljednja nam je bila posebno zanimljiva jer smo za tu klasu mogli dati teorem koji govori da je jedinstveno rješenje linearne zadaće komplementarnosti također jedinstveno rješenje zadaće kvadratičnog programiranja. Na samom kraju rada pozabavili smo se algoritmima za rješavanje linearne zadaće komplementarnosti direktnim metodama te pokazali na primjerima kako možemo riješiti zadaću linearne komplementarnosti te zadaću kvadratičnog programiranja koristeći Lemkeov algoritam.The linear complementarity problem is an excellent context to illustrate concepts of linear algebra and matrix theory. At the beginning we introduced some basic terms and results regarding matrix theory, especially positive definite and semi-definite matrices. Moreover, we mentioned some results concerning convex quadratic functions as they are essential for understanding of quadratic program. Before establishing the connection between quadratic program and linear complementarity problem, we defined first-order optimality conditions. More precisely, we derived Karush-Kuhn-Tucker conditions that are integral part of building the connection between aforementioned problems. In the third chapter, we introduced the concept of complementarity cones as linear complementarity problem rests on that idea. We demonstrated that by the example. Main part of the chapter is based on presenting the results pertaining to the existence and multiplicity of solutions to the linear complementarity problem. At first, we were based only on the class of positive definite and positive semi-definite matrices. In the case of positive semidefinite matrices we showed an important result: if the linear complementarity problem is feasible, then it’s solvable. In the case of positive definite matrices, the linear complementarity problem has a unique solution. Moreover, we mentioned the classes of S-matrices and P-matrices. Class of P-matrices has an interesting property as the unique solution of linear complementarity problem characterized by P-matrix is also the unique solution of the quadratic program. At the end, we developed Lemke’s algorithm for solving the linear complementarity problem and made some examples with linear complementarity problem and quadratic program

Kvadratično programiranje i linearna zadaća komplementarnosti

Linearna zadaća komplementarnosti je sjajan kontekst u kojem se mogu prikazati pojmovi iz linearne algebre i teorije matrica pa smo se na početku rada dotakli osnovnih pojmova i rezultata vezanih za matrice, konkretno pozitivno definitne i semidefinitne matrice. Osim toga, prisjetili smo se još pokojeg rezultata o konveksnim kvadratičnim funkcijama koji su nam pomogli u razumijevanju zadaće kvadratičnog programiranja. Prije uspostavljanja veze između zadaće kvadratičnog programiranja i linearne zadaće komplementarnosti, pozabavili smo se uvjetima optimalnosti prvog reda, odnosno izveli smo Karush-Kuhn-Tuckerove uvjete koji su ključno vezivo za izgradnju mosta među spomenutim zadaćama. Kako mnogo toga u linearnoj zadaći komplementarnosti počiva na ideji komplementarnog konusa, bilo je neizbježno zastati i prokomentirati zadaću u terminima konusa te demonstrirati to primjerom. Glavni dio poglavlja o linearnoj zadaći komplementarnosti je svakako rasprava o egzistenciji i broju rješenja. Isprva smo se bazirali na klasu pozitivno definitnih i semidefinitnih matrica te pokazali u slučaju pozitivno semidefinitne matrice M da je zadaća rješiva ako je dopustiva, a u slučaju pozitivno definitnh matrica je k tome rješenje i jedinstveno za svaki vektor q. Osim te dvije klase, govorili smo i o klasama S-matrica i P-matrica. Ova posljednja nam je bila posebno zanimljiva jer smo za tu klasu mogli dati teorem koji govori da je jedinstveno rješenje linearne zadaće komplementarnosti također jedinstveno rješenje zadaće kvadratičnog programiranja. Na samom kraju rada pozabavili smo se algoritmima za rješavanje linearne zadaće komplementarnosti direktnim metodama te pokazali na primjerima kako možemo riješiti zadaću linearne komplementarnosti te zadaću kvadratičnog programiranja koristeći Lemkeov algoritam.The linear complementarity problem is an excellent context to illustrate concepts of linear algebra and matrix theory. At the beginning we introduced some basic terms and results regarding matrix theory, especially positive definite and semi-definite matrices. Moreover, we mentioned some results concerning convex quadratic functions as they are essential for understanding of quadratic program. Before establishing the connection between quadratic program and linear complementarity problem, we defined first-order optimality conditions. More precisely, we derived Karush-Kuhn-Tucker conditions that are integral part of building the connection between aforementioned problems. In the third chapter, we introduced the concept of complementarity cones as linear complementarity problem rests on that idea. We demonstrated that by the example. Main part of the chapter is based on presenting the results pertaining to the existence and multiplicity of solutions to the linear complementarity problem. At first, we were based only on the class of positive definite and positive semi-definite matrices. In the case of positive semidefinite matrices we showed an important result: if the linear complementarity problem is feasible, then it’s solvable. In the case of positive definite matrices, the linear complementarity problem has a unique solution. Moreover, we mentioned the classes of S-matrices and P-matrices. Class of P-matrices has an interesting property as the unique solution of linear complementarity problem characterized by P-matrix is also the unique solution of the quadratic program. At the end, we developed Lemke’s algorithm for solving the linear complementarity problem and made some examples with linear complementarity problem and quadratic program

Matrix Decomposition

Matrice se dijele u različite klase, ovisno o formi i određenim svojstvima. Matrične faktorizacije ovise o svojstvima određene klase matrica pa su faktorizacije matrica od velikog značaja u teoriji matrica, pri analizi numeričkih algoritama i uopće u numeričkoj linearnoj algebri. Faktorizacija matrice A je prikaz matrice A kao produkta "jednostavnijih" matrica, što omogućuje jednostavnije rješavanje nekog problema. U teoriji matrica značajne su faktorizacije onih matrica kod kojih je moguća transformacija sličnost, kod što su Schurova dekompozicija, spektralna dekompozicija, singularna dekompozicija. Nadalje, osnovni alat za rješavanje sustava linearnih jednadžbi, kao jednog od osnovnih problema numeričke linearne algebre, je LU faktorizacija. Također, bitno je spomenuti i QR faktorizaciju i njeno računanje preko rotacija i reflektora.Matrices are divided into different classes, depending on the form and specific properties of the matrix. Matrix factorizations depend on the properties of certain class of matrices, hence matrix factorization are of great importance in the matrix theory, in the analysis of numerical algorithms and even in numerical linear algebra. A factorization of the matrix A is a representation of A as a product of several "simpler" matrices, which makes the problem at hand easier to solve. Factorizations of matrices into some special sorts of matrices with similarity are of fundamental importance in matrix theory, like Schur decomposition, spectral decomposition and the singular value decomposition. Furthermore, the basic tool for solving systems of linear equations, as one of the basic problems of numerical linear algebra, is the LU factorization. Also, it is important to mention QR factorization and its calculation through rotation and reflectors

Direct methods for linear systems

Rjesavanje linearnih sustava pomocu LU faktorizacije,gaussove eliminacije te pivotiranja.Implementacija algoritama u programskom jeziku python

Direct methods for linear systems

Rjesavanje linearnih sustava pomocu LU faktorizacije,gaussove eliminacije te pivotiranja.Implementacija algoritama u programskom jeziku python

Analiza stabilnosti nelinearnih sustava vođenih analitičkim neizrazitim regulatorom

Tema ove disertacije je analiza stabilnosti nelinearnih mehaničkih sustava vođenih analitičkim neizrazitim regulatorom. Analitički neizraziti regulator je nekonvencionalni pristup koji koristi analitičke funkcije za određivanje centara izlaznih neizrazitih skupova umjesto baze pravila ponašanja. Analiza stabilnosti je zasnovana na Lyapunovljevoj izravnoj metodi i ne zahtijeva prikaz dinamike sustava upravljanja u obliku Takagi- Sugeno neizrazitog modela. Analiza stabilnosti je podijeljena na četiri osnovna dijela. Prvo se formiraju jednadžbe pogreške zatvorenog sustava upravljanja. Drugo, formira se kandidat za Lyapunovljevu funkciju. Zatim se izvode uvjeti stabilnosti koji garantiraju pozitivnu definitnost Lypunovljeve funkcije i negativnu definitnost njene vremenske derivacije. Na kraju, primjenjuje se LaSalleov princip invarijantnosti koji garantira asimptotsku stabilnost. Time su dobiveni kriteriji lokalne stabilnosti koji uključuju svega nekoliko parametara upravljačkog sustava. Na osnovu dobivenih rezultata razmatrane su neke modifikacije analitičkog neizrazitog regulatora koje osiguravaju globalnu asimptotsku stabilnost. Nadalje, Lyapunovljeve funkcije analiziranih regulatora su iskorištene za evaluaciju performansi i određivanje optimalnih vrijednosti parametara regulatora. \Navedeni pristup zasnovan je na konstrukciji parametarski ovisne Lyapunovljeve funkcije. Odgovarajućim izborom slobodnog parametra dobivena je ocjena integralnog indeksa performanse. Indeks performanse ovisi samo o nekoliko parametara regulatora i nekoliko parametara koji karakteriziraju dinamiku robota. Optimalne vrijednosti parametara regulatora dobivene su minimizacijom indeksa performanse. Procedura podešavanja parametara demonstrirana je na simulacijskom modelu dva različita tipa robota