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The Schur Complement and H-Matrix Theory

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The Schur Complement and H-Matrix Theory UNIVERSITY OF NOVI SAD FACULTY OF TECHNICAL SCIENCES Maja Nedović THE SCHUR COMPLEMENT AND H-MATRIX THEORY DOCTORAL DISSERTATION Novi Sad, 2016 УНИВЕРЗИТЕТ У НОВОМ САДУ ФАКУЛТЕТ ТЕХНИЧКИХ НАУКА 21000 НОВИ САД, Трг Доситеја Обрадовића 6 КЉУЧНА ДОКУМЕНТАЦИЈСКА ИНФОРМАЦИЈА Accession number, ANO: Identification number, INO: Document type, DT: Monographic publication Type of record, TR: Textual printed material Contents code, CC: PhD thesis Author, AU: Maja Nedović Mentor, MN: Professor Ljiljana Cvetković, PhD Title, TI: The Schur complement and H-matrix theory Language of text, LT: English Language of abstract, LA: Serbian, English Country of publication, CP: Republic of Serbia Locality of publication, LP: Province of Vojvodina Publication year, PY: 2016 Publisher, PB: Author’s reprint Publication place, PP: Novi Sad, Faculty of Technical Sciences, Trg Dositeja Obradovića 6 Physical description, PD: 6/161/85/1/0/12/0 (chapters/pages/ref./tables/pictures/graphs/appendixes) Scientific field, SF: Applied Mathematics Scientific discipline, SD: Numerical Mathematics Subject/Key words, S/KW: H-matrices, Schur complement, Eigenvalue localization, Maximum norm bounds for the inverse matrix, Closure of matrix classes UC Holding data, HD: Library of the Faculty of Technical Sciences, Trg Dositeja Obradovića 6, Novi Sad Note, N: Abstract, AB: This thesis studies subclasses of the class of H-matrices and their applications, with emphasis on the investigation of the Schur complement properties. The contributions of the thesis are new nonsingularity results, bounds for the maximum norm of the inverse matrix, closure properties of some matrix classes under taking Schur complements, as well as results on localization and separation of the eigenvalues of the Schur complement based on the entries of the original matrix. Accepted by the Scientific Board on, ASB: 20.11.2014 Defended on, DE: Defended Board, DB: President: Mila Stojaković, PhD, full professor Member: Vladimir Kostić, PhD, associated professor Member: Tomasz Szulc, PhD, full professor Menthor's sign Member: Ksenija Doroslovački, assistant profesor Member, Mentor: Ljiljana Cvetković, PhD, full professor Obrazac Q2.НА.06-05- Izdanje 1 УНИВЕРЗИТЕТ У НОВОМ САДУ ФАКУЛТЕТ ТЕХНИЧКИХ НАУКА 21000 НОВИ САД, Трг Доситеја Обрадовића 6 КЉУЧНА ДОКУМЕНТАЦИЈСКА ИНФОРМАЦИЈА Редни број, РБР: Идентификациони број, ИБР: Тип документације, ТД: Монографска документација Тип записа, ТЗ: Текстуални штампани материјал Врста рада, ВР: Докторска дисертација Аутор, АУ: Маја Недовић Ментор, МН: Проф. др Љиљана Цветковић Наслов рада, НР: Шуров комплемент и теорија Х-матрица Језик публикације, ЈП: Енглески Језик извода, ЈИ: Српски, Енглески Земља публиковања, ЗП: Република Србија Уже географско подручје, УГП: Војводина Година, ГО: 2016 Издавач, ИЗ: Ауторски репринт Место и адреса, МА: Нови Сад, Факултет техничких наука, Трг Доситеја Обрадовића 6 Физички опис рада, ФО: 6/161/85/1/0/12/0 (поглавља/страна/цитата/табела/слика/графика/прилога) Научна област, НО: Примењена математика Научна дисциплина, НД: Нумеричка математика Предметна одредница/Кључне речи, ПО: Х-матрице, Шуров комплемент, локализација карактеристичних корена, оцена максимум норме инверзне матрице, затвореност класа матрица УДК Чува се, ЧУ: Библиотека Факултета техничких наука, Трг Доситеја Обрадовића 6, Нови Сад Важна напомена, ВН: Извод, ИЗ: Докторска дисертација изучава поткласе класе Х-матрица и њихове примене, првенствено у истраживању својстава Шуровог комплемента. Оригиналан допринос тезе представљају нови услови за регуларност матрица, оцене максимум норме инверзне матрице, резултати о затворености појединих класа матрица на Шуров комплемент, као и резултати о локализацији и сепарацији карактеристичних корена Шуровог комплемента на основу елемената полазне матрице. Датум прихватања теме, ДП: 20.11.2014 Датум одбране, ДО: Чланови комисије, КО: Председник: др Мила Стојаковић, редовни професор Члан: др Владимир Костић, ванредни професор Члан: др Тоmаsz Szulc, редовни професор Потпис ментора Члан: др Ксенија Дорословачки, доцент др Љиљана Цветковић, редовни професор Члан, ментор: Образац Q2.НА.06-05- Издање 1 Acknowledgements My deep, sincere gratitude, I owe to my mentor, Professor Ljiljana Cvetkovic,´ who inspired me to work in this research area, for the infinite energy, support, motivation and creativity. I am truly thankful for the opportunity to work with her and to experience her unique gift of scaling, both matrices and people, into a better form. I am grateful to my colleagues from Faculty of Technical Sciences in Novi Sad, to all the members of SC:ALA group and matrix community and to all of my teachers and students, with whom I have shared the mathematical side of life. I owe my warmest thanks to my extended family, above all, to my parents, Anica and Vladimir, for all their love, understanding and support along the way and for infecting me with their enthusiasm for mathematics and teaching. My sincere thanks are due to Marko and his family and to all my friends, for bringing joy to these busy years. My deepest, heartfelt thanks go to my favourite people in Rn; Maša, Bogdan and Ljubo. For everything. Maja Nedovic´ To my family Mojoj porodici i Apstrakt U teoriji matrica, od devetnaestog veka pa do danas, pitanje regularnosti date kvadratne kompleksne matrice ima posebnu ulogu. Prepoznata su brojna svojstva matrica i strukture koje garantuju da je determinanta posmatrane matrice razlicitaˇ od nule. Tokom dvadesetog veka, lista uslova dovoljnih za regularnost matrica postala je veoma bogata. Neki od ovih uslova poseduju i dodatne kvalitete, koji ih cineˇ pogodnim za prakticnuˇ primenu. Naime, u poslednjih nekoliko decenija, paralelno sa razvojem racunara,ˇ raste i potreba za takvim matematickimˇ rezulta- tima i takvim uslovima koji su elegantni i lako proverljivi (u smislu da njihova provera nije racunskiˇ zahtevna). Ovo narocitoˇ dolazi do izražaja u matematickimˇ modelima velikih dimenzija, koji se pojavljuju sve cešˇ ce´ u inženjerstvu, ekonomiji, molekularnoj biologiji. Jednostavnost i elegancija u formulacijama matematickihˇ rezultata, iako oduvek na ceni, danas imaju i dodatnu vrednost, jer omogucavaju´ smislenu prakticnuˇ primenu. Ideja sa kojom zapocinjemoˇ ima sve navedene kvalitete. To je ideja (stroge) dijagonalne dominacije, formulisana za kvadratne (isprva realne, a zatim i kom- pleksne) matrice. Lévy i Desplanques (1881), dokazali su da stroga dominacija dijagonalnog elementa po modulu u odnosu na sumu modula vandijagonalnih el- emenata u svakoj vrsti garantuje regularnost matrice. Iako jednostavna, oduvek prisutna u teoriji matrica i isprva formulisana sa, verovatno, drugacijimˇ motivima, ova klasicnaˇ ideja je i danas prisutna u aktuelnim istraživanjima. Teorije koje su nastale njenim uopštavanjem, (prvenstveno teorija M- i H-matrica, vidi [4]), pokazale su se fundamentalnim u numerickojˇ linearnoj algebri. Osim teorijskog znacaja,ˇ ove ideje imaju i veliku prakticnuˇ vrednost i primenu u raznim oblastima istraživanja, u inženjerstvu, robotici, ekonomiji, molekularnoj i populacionoj bi- ologiji. n;n Kažemo da je data matrica, A = [ai j] 2 C ; strogo dijagonalno dominantna (SDD) ako jaiij > ri(A); za sve i 2 N = f1;2;:::;ng; gde je ri(A) = ∑ jai jj: j2Nnfig Poslednju velicinu,ˇ ri(A); nazivamo (brisanom) sumom i−te vrste. Ispostavilo se da su mnoge znacajneˇ teme u primenjenoj linearnoj algebri na neki nacinˇ povezane sa pricomˇ o SDD svojstvu. Najpoznatiji takav primer je, svakako, veza SDD svojstva sa problemima lokalizacije karakteristicnihˇ korena ii kvadratnih kompleksnih matrica. Danas smo svesni da se pricaˇ o regularnosti SDD matrica, kao i nekih drugih tpova matrica koji predstavljaju uopštenja SDD matrica, može ekvivalentno ispricatiˇ u terminima lokalizacije karakteristicnihˇ korena. Iako implicitno prisutna u radovima s pocetkaˇ dvadesetog veka, ova veza je precizno formulisana i u potpunosti iskorišcena´ tek u knjizi „Geršgorin i njegovi krugovi”, cijiˇ je autor Richard Varga, vidi [82]. Rezultat Geršgorina (1931) objavljen u radu [35] omogucio´ je jednostavnu lokalizaciju karakteristicnihˇ korena date kvadratne kompleksne matrice. Svakoj vrsti matrice pridružen je jedan krug u kompleksnoj ravni sa centrom u dijagonal- nom elementu i poluprecnikaˇ koji je jednak brisanoj sumi vrste. Unija ovih n kru- gova u kompleksnoj ravni sadrži sve karakteristicneˇ korene date matrice. Drugim recima,ˇ ako je i−ti Geršgorinov krug definisan na sledeci´ nacin,ˇ Gi(A) = fz 2 Cjjz − aiij ≤ ri(A)g; tada je Geršgorinov skup dat sa [ G(A) = Gi(A): i2N Ako sa s(A) oznacimoˇ spektar matrice A; koji predstavlja skup svih karakteris- ticnihˇ korena matrice A; s(A) = fl 2 C j det(lI − A) = 0g; gde I oznacavaˇ jedinicnuˇ matricu reda n, tada, Geršgorinova teorema tvrdi s(A) ⊆ G(A): Olga Tausski Tod, u radovima [78, 79, 80] razvila je i dalje promovisala ovu ideju, kao i Ostrowski [65, 66], Brauer [5] i Brualdi [10]. Ljiljana Cvetkovic´ i Vladimir Kostic,´ [16, 17, 18, 49], definisali su pojam matricneˇ klase DD-tipa, koji objedinjuje razliciteˇ klase matrica definisane uslovima zasnovanim na dijagonal- noj dominaciji, kao i odgovarajuce´ rezultate u oblasti lokalizacije karakteristicnihˇ korena. Teorija M- i H-matrica, koja je poslužila kao osnova za brojna istraživanja u raznim oblastima numerickeˇ i linearne algebre, narocitoˇ u ispitivanju konvergencije iterativnih postupaka i stabilnosti dinamickihˇ sistema, takode¯ je u tesnoj vezi sa strogom dijagonalnom dominacijom. n;n Kažemo da je data matrica, A = [ai j] 2 R ; M-matrica,
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