DOCSLIB.ORG

Fundamental Journal of Mathematics and Applications VOLUME IV ISSUE I

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fundamental Journal of Mathematics and Applications VOLUME IV ISSUE I fundamental journal of mathematics and applications VOLUME IV ISSUE I www.dergipark.org.tr/en/pub/fujma ISSN 2645-8845 VOLUME 4 ISSUE 1 March 2021 ISSN 2645-8845 www.dergipark.org.tr/en/pub/fujma FUNDAMENTAL JOURNAL OF MATHEMATICS AND APPLICATIONS Editors in Chief Mahmut Akyi˘git Merve Ilkhan˙ Kara Department of Mathematics, Department of Mathematics, Faculty of Science and Arts, Sakarya University, Faculty of Science and Arts, D¨uzce University, Sakarya-TURK¨ IYE˙ D¨uzce-TURK¨ IYE˙ [email protected] [email protected] Soley Ersoy Department of Mathematics, Faculty of Science and Arts, Sakarya University, Sakarya-TURK¨ IYE˙ [email protected] Managing Editor Fuat Usta Department of Mathematics, Faculty of Science and Arts, D¨uzce University, D¨uzce-TURK¨ IYE˙ [email protected] Editorial Board Stoil I. Ivanov Slavica Iveli´cBradanovi´c University of Plovdiv Paisii Hilendarski University of Split BULGARIA CROATIA Murat Tosun Mohammad Saeed KHAN Sakarya University Sultan Qaboos University TURK¨ IYE˙ OMAN Vladimir Vladicic Emrah Evren Kara University East Sarajevo D¨uzce University BOSNIA and HERZEGOVINA TURK¨ IYE˙ T¨ulay Kesemen Mohammad Mursaleen Karadeniz Technical University Aligarh Muslim University TURK¨ IYE˙ INDIA i Zlatko Pavi´c Sidney Allen Morris University of Osijek Federation University CROATIA AUSTRALIA Ramazan Kama Naoyuki Ishimura Siirt University Chuo University TURK¨ IYE˙ JAPAN Syed Abdul Mohiuddine Figen Oke¨ King Abdulaziz University Trakya University SAUDI ARABIA TURK¨ IYE˙ Ismet˙ Altınta¸s Ay¸se Yılmaz Ceylan Kyrgyz-Turkish Manas University Akdeniz University KYRGYZSTAN TURK¨ IYE˙ Editorial Secretariat Editorial Secretariat Bahar Do˘ganYazıcı Department of Mathematics Bilecik S¸eyh Edebali University Bilecik-TURK¨ IYE˙ ii Contents 1 Unrestricted Fibonacci and Lucas Quaternions Ahmet Da¸sdemir, G¨oksal Bilgici 1-9 2 Fibonacci Elliptic Biquaternions Kahraman Esen Ozen,¨ Murat Tosun 10-16 3 Faber Polynomial Expansion for a New Subclass of Bi-univalent Functions Endowed with (p,q) Calculus Operators Om P. Ahuja, Asena C¸etinkaya 17-24 4 Construction of Degenerate q-Daehee Polynomials with Weight α and its Applications Serkan Aracı 25-32 5 Discrete Networked Dynamic Systems with Eigen-Spectrum Gap: Analysis and Performance Magdi S. Mahmoud, Bilal J. Karaki 33-44 6 I-Limit and I-Cluster Points for Functions Defined on Amenable Semigroups U˘gur Ulusu, Fatih Nuray, Erdin¸cD¨undar 45-48 7 Circulant m-Diagonal Matrices Associated With Chebyshev Polynomials Ahmet Otele¸s¨ 49-58 8 Recurrence Relations for Knot Polynomials of Twist Knots Kemal Ta¸sk¨opr¨u, Zekiye S¸evval Sinan 59-66 iii Fundamental Journal of Mathematics and Applications, 4 (1) (2021) 1-9 Research Article Fundamental Journal of Mathematics and Applications Journal Homepage: www.dergipark.gov.tr/fujma ISSN: 2645-8845 doi: 10.33401/fujma.752758 Unrestricted Fibonacci and Lucas Quaternions Ahmet Das¸demir1* and Goksal¨ Bilgici2 1Department of Mathematics, Faculty of Arts and Sciences, Kastamonu University, Kastamonu, Turkey 2Department of Elementary Mathematics Education, Faculty of Education, Kastamonu University, Kastamonu, Turkey *Corresponding author Article Info Abstract Keywords: Binet’s formula, Catalan’s Many quaternion numbers associated with Fibonacci and Lucas numbers or even their identity, Fibonacci quaternion, Gener- generalizations have been defined and widely discussed so far. In all the studies, the ating function, Lucas quaternion coefficients of these quaternions have been selected from consecutive terms of these numbers. 2010 AMS: 11B39, 15B36, 11R52 In this study, we define other generalizations for the usual Fibonacci and Lucas quaternions. Received: 14 June 2020 We also present some properties, including the Binet’s formulas and d’Ocagne’s identities, Accepted: 20 January 2021 for these types of quaternions. Available online: 29 January 2021 1. Introduction Since Sir William Rowan Hamilton introduced the quaternion algebra in 1843, their usage areas have developed rapidly. Due to the fact that the quaternions are encountered in many problems from elastodynamics, quantum mechanics, elasticity theory and many other fields of modern sciences, they have been studied widely. A quaternion q can be regarded as a quadruple of real numbers and is formally defined by q = q0 + iq1 + jq2 + kq3, where q0,q1,q2, and q3 are any real numbers and the standard basis 1,i, j,k satisfies { } i2 = j2 = k2 = 1,i j = ji = k, jk = k j = i,ki = ik = j. − − − − The conjugate of q is q∗ = q0 iq1 jq2 kq3 − − − and, the norm of q is Email addresses and ORCID numbers: [email protected], https://orcid.org/0000-0001-8352-2020 (A. Das¸demir), [email protected], https://orcid.org/0000-0001-9964-5578 (G. Bilgici) 2 Fundamental Journal of Mathematics and Applications 2 2 2 2 N(q)= qq∗ = q0 + q1 + q2 + q3. Fibonacci numbers are recursively defined as Fn = Fn 1 + Fn 2, − − with the initial terms F0 = 0 and F1 = 1. The Lucas numbers satisfy the same recurrence relation but with the initial terms L0 = 2 and L1 = 1. Binet’s formulas for the Fibonacci and Lucas numbers are αn β n F = − and L = αn + β n, n α β n − 1+√5 1 √5 2 respectively. Here, α = and β = − are the positive and negative roots of x x 1 = 0, respectively. 2 2 − − Horadam [1] defined Fibonacci quaternions as Qn := Fn + iFn+1 + jFn+2 + kFn+3, (1.1) where Fn is the nth term of the Fibonacci sequence. Iyer [2] gave a similar definition for Lucas quaternions by the relation Tn := Ln + iLn+1 + jLn+2 + kLn+3, (1.2) and gave their many properties, where Ln is the nth Lucas number. Halici [3] gave Binet’s formulas for the Fibonacci and Lucas quaternions as follows: αα ββ Q = − and T = αα + ββ, n α β n − where α = 1 + iα + jα2 + kα3 and β = 1 + iβ + jβ 2 + kβ 3. There exist many papers devoted to generalizations of the quaternion sequences in (1.1) and (1.2) today. For example, the references in [4]–[11] can be investigated. Note that all authors have used a known generalization of Fibonacci and Lucas numbers and have placed these numbers in coefficients of the basis vectors similar to the format given by Horadam and Iyer. In this study, we present an another perspective to the Fibonacci and Lucas quaternions. According to our approach, we define a new classes of quaternions whose the coefficients are arbitrarily selected from these splendid integers. The outline of this paper is as follows: In Section 2, we introduce the unrestricted Fibonacci and Lucas quaternions and give Binet’s formulas and the generating functions for these quaternions. Furthermore, we present certain special identities such as d’Ocagne’s identity and Catalan’s identity; in Section 3, we display many fundamental properties and some sum formulas for these quaternion families. 2. Main results Here, we present our definitions, some concepts and results. First of all, we give a definition in the following. Definition 2.1. Let p, r and s be arbitrary integers. Hence, nth unrestricted Fibonacci and Lucas quaternions are given by the relations (p,r,s) Fn := Fn + iFn+p + jFn+r + kFn+s (2.1) and (p,r,s) Ln := Ln + iLn+p + jLn+r + kLn+s, (2.2) respectively. Fundamental Journal of Mathematics and Applications 3 According to our definitions, we have the following special cases: • For p = r = s = n, the usual Fibonacci numbers are obtained: − ( n, n, n) Fn− − − = Fn • For p = 1 and r = s = n, the Gaussian Fibonacci numbers are obtained: − (1, n, n) Fn − − = Fn + iFn+1 • For p = 1, r = 2 and s = 3, the well-known Fibonacci and Lucas quaternions are obtained: (1,2,3) Fn = Qn = Fn + iFn+1 + jFn+2 + kFn+3 (1,2,3) Ln = Tn = Ln + iLn+1 + jLn+2 + kLn+3 Taking Eqs. (2.1) and (2.2) into account, we directly obtain F (p,r,s) F (p,r,s) F (p,r,s) n = n 1 + n 2 (2.3) − − and L (p,r,s) L (p,r,s) L (p,r,s) n = n 1 + n 2 , (2.4) − − by the recurrence relations of Fibonacci and Lucas numbers respectively. To make it easier to present and prove the results of the rest of paper, we now present next theorem that states Binet’s formulas for the unrestricted Fibonacci and Lucas quaternions. Theorem 2.2. (Binet’s Formula) Let n be an integer. Then the Binet’s formulas of the unrestricted Fibonacci and Lucas quaternions are n ˘ n p,r,s αα˘ ββ F ( ) = − (2.5) n α β − and (p,r,s) n n Ln = αα˘ + ββ˘ , (2.6) where α˘ = 1 + iα p + jαr + kαs and β˘ = 1 + iβ p + jβ r + kβ s. Proof. Applying Binet’s formulas of the classic Fibonacci numbers to the definition of the unrestricted Fibonacci quaternions, we obtain (p,r,s) Fn = Fn + iFn+p + jFn+r + kFn+s 1 = αn β n + i(αn+p β n+p)+ j(αn+r β n+r)+ k(αn+s β n+s) α β − − − − − 1 = αn(1 + iα p + jαr + kαs) β n(1 + iβ p + jβ r + kβ s) , α β − − and the last equation gives Eq. (2.5). The other can be proved similarly. 4 Fundamental Journal of Mathematics and Applications α˘ β˘ α˘ β˘ · ∗ ∗ α˘ 2α˘ N(α˘ ) B + √5C N(α˘ ) A √5C + 2γ − − β˘ B √5C 2β˘ N β˘ A + √5C N β˘ − − − α˘ N(α˘ ) A √5C 2α˘ N(α˘ ) B + √5C 2γ + 4 ∗ − − ∗ − − − β˘ A + √5C + 2γ N β˘ B √5C 2γ + 4 2β˘ N β˘ ∗ − − − ∗ − Table 1: The multiplicative properties of α˘ and β˘ Table 1 displays important features corresponding to the multiplication of α˘ and β˘. They will undertake important tasks in the process of proofing the next theorems.
Load more

Recommended publications
  • Implementation and Performance Analysis of a Parallel Oil Reservoir Simulator Tool Using a CG Method on a GPU-Based System
    2014 UKSim-AMSS 16th International Conference on Computer Modelling and Simulation Implementation and Performance Analysis of a Parallel Oil Reservoir Simulator Tool using a CG Method on a GPU-Based System Leila Ismail Jamal Abou-Kassem Bibrak Qamar Computer and Software Engineering/ Department of Chemical and Petroleum HPGCL Research Lab HPGCL Research Lab Engineering College of IT College of IT College of Engineering UAE University UAE University UAE University Al-Ain, UAE Al-Ain, UAE Al-Ain, UAE Email: [email protected] (Correspondence Email: [email protected] Author) 978-1-4799-4923-6/14 $31.00 © 2014 IEEE 374 DOI 10.1109/UKSim.2014.113 described in section IV. Our experiments and analysis of The above formula is the transmissibility in the x-direction results are presented in section V. Finally, VI concludes our where μ is the viscosity, B is the formation volume factor, work. Δx is the dimension of a grid block in the x-direction, Ax is the area of a grid block (y-dimension * height), Kx is the II. OVERALL IMPLEMENTATION OF THE OIL RESERVOIR permeability in the x-direction and βc is the transmissibility SIMULATOR conversion factor. Based on the values computed for the transmissibility and the production rates of the wells, the simulator then generates the system Ax = d, where A is the coefficient matrix, x is the solution vector for the value of pressure in each grid block of the reservoir and d is is a known vector which represents the right hand side of the generated system of equations.
    [Show full text]
  • Spectral Properties of Anti-Heptadiagonal Persymmetric
    Spectral properties of anti-heptadiagonal persymmetric Hankel matrices Jo˜ao Lita da Silva1 Department of Mathematics and GeoBioTec Faculty of Sciences and Technology NOVA University of Lisbon Quinta da Torre, 2829-516 Caparica, Portugal Abstract In this paper we express the eigenvalues of anti-heptadiagonal persymmetric Hankel matrices as the zeros of explicit polynomials giving also a representation of its eigenvectors. We present also an expression depending on localizable parameters to compute its integer powers. In particular, an explicit formula not depending on any unknown parameter for the inverse of anti-heptadiagonal persymmetric Hankel matrices is provided. Key words: Anti-heptadiagonal matrix, Hankel matrix, eigenvalue, eigenvector, diagonalization 2010 Mathematics Subject Classification: 15A18, 15B05 1 Introduction The importance of Hankel matrices in computational mathematics and engineering is well-known. As a matter of fact, these type of matrices have not only a varied and numerous relations with polyno- mial computations (see, for instance, [2]) but also applications in engineering problems of system and control theory (see [4], [12] or [14] and the references therein). Recently, several authors have studied particular cases of these matrices in order to derive explicit expressions for its powers (see [7], [11], [16], [17], [19], [20] among others). The aim of this paper is to explore spectral properties of general anti-heptadiagonal persymmetric Hankel matrices, namely locating its eigenvalues and getting an explicit representation of its eigenvec- tors. Additionally, it is our purpose announce formulae for the computation of its integer powers using, thereunto, a diagonalization for the referred matrices. Particularly, an expression free of any unknown parameter to calculate the inverse of any anti-heptadiagonal persymmetric Hankel matrix (assuming arXiv:1907.00260v1 [math.RA] 29 Jun 2019 its nonsingularity) is made available.
    [Show full text]
  • Recursion Formulae for the Characteristic Polynomial of Symmetric Banded Matrices
    Recursion Formulae for the Characteristic Polynomial of Symmetric Banded Matrices Werner Kratz und Markus Tentler Preprint Series: 2007-15 Fakult¨at fur¨ Mathematik und Wirtschaftswissenschaften UNIVERSITAT¨ ULM Recursion Formulae for the Characteristic Polynomial of Symmetric Banded Matrices Werne Kratz und Markus Tentler Preprint Series: 2007-15 Fakult¨at fur¨ Mathematik und Wirtschaftswissenschaften UNIVERSITAT¨ ULM Recursion formulae for the characteristic polynomial of symmetric banded matrices Werner Kratz and Markus Tentler Abstract . In this article we treat the algebraic eigenvalue problem for real, symmetric, and banded matrices of size N × N , say. For symmetric, tridi- agonal matrices, there is a well-known two-term recursion to evaluate the characteristic polynomials of its principal submatrices. This recursion is of complexity O(N) and it requires additions and multiplications only. More- over, it is used as the basis for a numerical algorithm to compute particular eigenvalues of the matrix via bisection. We derive similar recursion formu- lae with the same complexity O(N) for symmetric matrices with arbitrary bandwidth, containing divisions. The main results are divisionfree recursions for penta- and heptadiagonal symmetric matrices. These recursions yield sim- ilarly as in the tridiagonal case effective (with complexity O(N) ), fast, and stable algorithms to compute their eigenvalues. Running head: Recursion formulae for banded matrices Key words: Banded matrix, eigenvalue problem, Sturm-Liouville equation, pentadiagonal matrix, heptadiagonal matrix, bisection method AMS subject classi¯cation: 15A18; 65F15, 15A15, 39A10, 15A24 W. Kratz and M. Tentler Universit¨atUlm Institut f¨urAngewandte Analysis D - 89069 Ulm, Germany e-mail: [email protected] e-mail: [email protected] Recursion formulae for the characteristic polynomial of symmetric banded matrices Werner Kratz and Markus Tentler 1 Introduction In this article we consider the algebraic eigenvalue problem for real, sym- metric, and banded matrices of size N × N , say.
    [Show full text]
  • RELATIONSHIPS BETWEEN the PERMANENTS of a CERTAIN TYPE of K-TRIDIAGONAL SYMMETRIC TOEPLITZ MATRIX and the CHEBYSHEV POLYNOMIALS
    Journal of Applied Mathematics and Computational Mechanics 2017, 16(1), 75-86 www.amcm.pcz.pl p-ISSN 2299-9965 DOI: 10.17512/jamcm.2017.1.07 e-ISSN 2353-0588 RELATIONSHIPS BETWEEN THE PERMANENTS OF A CERTAIN TYPE OF k-TRIDIAGONAL SYMMETRIC TOEPLITZ MATRIX AND THE CHEBYSHEV POLYNOMIALS 1 2 Ahmet Zahid Küçük , Murat Düz 1 Eskipazar Vocational School Turkey 2 Department of Mathematics Faculty of Science, Karabuk University Turkey [email protected], [email protected] Received: 16 December 2016; accepted: 13 March 2017 Abstract. In this study, the recursive relations between the permanents of a certain type of the k-tridiagonal symmetric Toeplitz matrix with complex entries and the Chebyshev polynomials of the second kind are presented. MSC 2010: 33C45 Keywords: permanent, Toeplitz matrix, k-tridiagonal matrix, Chebyshev polynomials, recurrence relation 1. Introduction A k-tridiagonal symmetric Toeplitz matrix can be written in the form (k ) T= t n ij n× n where a, i= j tij = bijk, − = . (1) 0 ,otherwise . The Toeplitz structured tridiagonal matrix family can be seen with many studies in different areas. Some of them include the solution of ordinary and partial differ- ential equations [1] in the applied mathematics, including the equilibrium problem [2] in the statistical physic, wireless and sensor network applications [3, 4] about control theory in computer sciences and the sensitivity of MRNA [5] in molecular biology. 76 A.Z. Küçük, M. Düz The aim of this study is to point out some relationships between the permanents of the certain type of k-tridiagonal symmetric Toeplitz matrix and the Chebyshev polynomials of the second kind.
    [Show full text]
  • Variational Reconstruction and GPU Ray-Casting of Non-Uniform Point Sets Using B-Spline Pyramids
    Variational Reconstruction and GPU Ray-Casting of Non-Uniform Point Sets using B-Spline Pyramids DIPLOMARBEIT zur Erlangung des akademischen Grades Diplom-Ingenieur im Rahmen des Studiums Computergraphik/Digitale Bildverarbeitung eingereicht von Martin Kinkelin Matrikelnummer 0326997 an der Fakultät für Informatik der Technischen Universität Wien Betreuung Betreuer: Prof. Dr. Eduard Gröller Mitwirkung: Dr. techn. Erald Vuçini Wien, 03.05.2011 (Unterschrift Verfasser) (Unterschrift Betreuer) Technische Universität Wien A-1040 Wien Karlsplatz 13 Tel. +43-1-58801-0 www.tuwien.ac.at i Abstract In this work, we focus on the problem of reconstructing a volume (scalar 3D field) based on non-uniform point samples and then rendering the volume by exploiting the processing power of GPUs. In the first part involving the reconstruction, we motivate our choice of tensor-product uniform B-splines for the discretized representation of the continuous volume. They al- low for highly efficient, scalable and accurate reconstruction at multiple scales (resolution levels) at once. By subdividing the volume into blocks and reconstructing them indepen- dently, current desktop PCs are able to reconstruct large volumes and multiple CPU cores can be efficiently exploited. We focus on linear and cubic B-splines and on how to eliminate otherwise resulting block discontinuities. Once we have reconstructed the volume at multiple scales, we can derive different Levels of Detail (LoDs) by subdividing the volume into blocks and selecting a suitable scale for each block. We present a fusion scheme which guarantees global C0 continuity for linear LoDs and C2 continuity for cubic ones. The challenge here is to minimize visual block interscale discontinuities.
    [Show full text]
  • Symbolic Algorithm for Solving Slaes with Heptadiagonal Coefficient
    http://mmg.tversu.ru ISSN 2311-1275 Mathematical Modelling M G M and Geometry Volume 6, No 3, pp. 22 { 29 (2018) doi:10.26456/mmg/2018-633 Symbolic Algorithm for Solving SLAEs with Heptadiagonal Coefficient Matrices Milena Veneva1;a and Alexander Ayriyan1;b 1 Joint Institute for Nuclear Research, Laboratory of Information Technologies, Joliot-Curie 6, 141980 Dubna, Moscow region, Russia e-mail: a [email protected], b [email protected] Received 27 November 2018. Published 19 December 2018. Abstract. This paper presents a symbolic algorithm for solving band matrix sy- stems of linear algebraic equations with heptadiagonal coefficient matrices. The algorithm is given in pseudocode. A theorem which gives the condition for the algorithm to be stable is formulated and proven. Keywords: systems of linear algebraic equations, heptadiagonal matrix, symbolic algorithm, LU factorization c The author(s) 2018. Published by Tver State University, Tver, Russia Symbolic Algorithm for Solving SLAEs 23 1. Introduction Systems of linear algebraic equations (SLAEs) with heptadiagonal coefficient ma- trices may arise after many different scientific and engineering problems, as well as problems of the computational linear algebra where finding the solution of a SLAE is considered to be one of the most important problems. For instance, a semi-implicit formulation for the discretization of the transient terms of the system of partial differential equations (PDEs) which models a multiphase fluid flow in porous media yields to a heptadiagonal system of pressure equations for each time step (see [1]). On the other hand, in [2] the 3D problem, simulating the incompres- sible blood flow in arteries with a structured mesh domain leads to a heptadiagonal SLAE.
    [Show full text]
  • On the Inverting of a General Heptadiagonal Matrix
    British Journal of Applied Science & Technology 18(5): 1-12, 2016; Article no.BJAST.31213 ISSN: 2231-0843, NLM ID: 101664541 SCIENCEDOMAIN international www.sciencedomain.org On the Inverting of a General Heptadiagonal Matrix ∗ A. A. Karawia1;2 1Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. 2Computer Science Unit, Deanship of Educational Services, Qassim University, P.O.Box 6595, Buraidah 51452, Saudi Arabia. Author’s contribution The sole author designed, analyzed and interpreted and prepared the manuscript. Article Information DOI: 10.9734/BJAST/2016/31213 Editor(s): (1) Orlando Manuel da Costa Gomes, Professor of Economics, Lisbon Accounting and Business School (ISCAL), Lisbon Polytechnic Institute, Portugal. Reviewers: (1) Jia Liu, University of West Florida, USA. (2) Hammad Khalil, University of Education (Attock Campus), Lahore, Pakistan. (3) Gerald Bourgeois, University of French Polynesia and University of Aix-Marseille, France. Complete Peer review History: http://www.sciencedomain.org/review-history/17657 Received: 26th December 2016 Accepted: 21st January 2017 Original Research Article Published: 28th January 2017 ABSTRACT In this paper, the author presents a new algorithm computing the inverse of any nonsingular heptadiagonal matrix. The computational cost of our algorithms is O(n2) operations in C. The algorithms are suitable for implementation using computer algebra system such as MAPLE, MATLAB and MATHEMATICA. Examples are given to illustrate the efficiency of the algorithms. Keywords: Heptadiagonal matrices; LU factorization; Determinants; Computer algebra systems(CAS). 2010 Mathematics Subject Classification: 15A15, 15A23, 68W30, 11Y05, 33F10, F.2.1, G.1.0. *Corresponding author: E-mail: [email protected]; Karawia; BJAST, 18(5), 1-12, 2016; Article no.BJAST.31213 1 INTRODUCTION The n × n general heptadiagonal matrices take the form: 0 1 d1 e1 f1 g1 B C B c2 d2 e2 f2 g2 C B C B b3 c3 d3 e3 f3 g3 0 C B C B a4 b4 c4 d4 e4 f4 g4 C B C B C B C B .
    [Show full text]
  • Preconditioned Iterative Solvers on GPU and an In-Situ Combustion Simulator
    University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2020-01 Preconditioned Iterative Solvers on GPU and an In-Situ Combustion Simulator Yang, Bo Yang, B. (2020). Preconditioned Iterative Solvers on GPU and an In-Situ Combustion Simulator (Unpublished doctoral thesis). University of Calgary, Calgary, AB. http://hdl.handle.net/1880/111505 doctoral thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY Preconditioned Iterative Solvers on GPU and an In-Situ Combustion Simulator by Bo Yang A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN CHEMICAL AND PETROLEUM ENGINEERING CALGARY, ALBERTA JANUARY, 2020 c Bo Yang 2020 Abstract This thesis consists of two parts: preconditioned iterative solvers on GPU and an in-situ combus- tion simulator. The purpose of the first research is to develop a new parallel solution platform based on GPU features. An application of HPC (high-performance computing) technology to reservoir simulation has become an inevitable trend. As a platform for HPC, GPU can provide an effective solution for per- sonal computers and workstations. In this research, not only a series of special CPR (constrained pressure residual) preconditioned solvers are developed for black oil models, but also a variety of other preconditioned solvers are completed as contrast solvers.
    [Show full text]
  • T.C. Selçuk Ünġversġtesġ Fen Bġlġmlerġ Enstġtüsü
    T.C. SELÇUK ÜNĠVERSĠTESĠ FEN BĠLĠMLERĠ ENSTĠTÜSÜ HEPTADĠAGONAL MATRĠSLERĠN POZĠTĠF TAMSAYI KUVVETLERĠNĠN HESAPLANMASI Murat GÜBEġ YÜKSEK LĠSANS TEZĠ MATEMATĠK ANABĠLĠM DALI MART-2011 KONYA Her Hakkı Saklıdır TEZ KABUL VE ONAYI Murat GÜBEŞ tarafından hazırlanan ―Heptadiagonal Matrislerin Pozitif Tamsayı Kuvvetlerinin Hesaplanması‖ adlı tez çalışması 04/03/2011 tarihinde aşağıdaki jüri tarafından oy birliği ile Selçuk Üniversitesi Fen Bilimleri Enstitüsü Matematik Anabilim Dalı’nda YÜKSEK LİSANS TEZİ olarak kabul edilmiştir. Jüri Üyeleri Ġmza BaĢkan Doç. Dr. Galip OTURANÇ ………………….. DanıĢman Prof. Dr. Durmuş BOZKURT ………………….. Üye Doç. Dr. Ramazan TÜRKMEN ………………….. Yukarıdaki sonucu onaylarım. Prof. Dr. Bayram SADE FBE Müdürü TEZ BĠLDĠRĠMĠ Bu tezdeki bütün bilgilerin etik davranış ve akademik kurallar çerçevesinde elde edildiğini ve tez yazım kurallarına uygun olarak hazırlanan bu çalışmada bana ait olmayan her türlü ifade ve bilginin kaynağına eksiksiz atıf yapıldığını bildiririm. DECLARATION PAGE I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Murat GÜBEŞ Tarih:04/03/2011 ÖZET YÜKSEK LĠSANS TEZĠ HEPTADĠAGONAL MATRĠSLERĠN POZĠTĠF TAMSAYI KUVVETLERĠNĠN HESAPLANMASI Murat GÜBEġ Selçuk Üniversitesi Fen Bilimleri Enstitüsü Matematik Anabilim Dalı DanıĢman: Prof. Dr. DurmuĢ BOZKURT 2011, 38 Sayfa Jüri Prof. Dr. DurmuĢ BOZKURT Doç. Dr. Ramazan TÜRKMEN Doç. Dr. Galip OTURANÇ Bu çalışmada, genel bir Heptadiagonal matris tanımı verildi. Buna bağlı olarak belli tipteki heptadiagonal bir matrisin inci kuvvetinin genel ifadesi elde edildi. Yine genel ifade içinde bu tip matrislerin Jordan formu, dönüşüm matrisi ve tersi elde edilmiştir.
    [Show full text]
  • On the Inverting of a General Heptadiagonal Matrix
    On the Inverting of A General Heptadiagonal Matrix A. A. KARAWIA∗ Computer science unit, Deanship of educational services, Qassim University, P.O.Box 6595, Buraidah 51452, Saudi Arabia. E-mail: [email protected] Abstract In this paper, we developed new numeric and symbolic algorithms to find the inverse of any nonsin- gular heptadiagonal matrix. Symbolic algorithm will not break and it is without setting any restrictive conditions. The computational cost of our algorithms is O(n). The algorithms are suitable for imple- mentation using computer algebra system such as MAPLE, MATLAB and MATHEMATICA. Examples are given to illustrate the efficiency of the algorithms. Keywords:Heptadiagonal matrices; LU factorization; Determinants; Computer algebra systems(CAS). AMS Subject Classification:15A15; 15A23; 68W30; 11Y05; 33F10; F.2.1; G.1.0. 1 Introduction Then × n general heptadiagonal matrices take the form: d1 e1 f1 g1 c2 d2 e2 f2 g2 b3 c3 d3 e3 f3 g3 0 a4 b4 c4 d4 e4 f4 g4 . H = .. .. .. .. .. .. .. ,n> 4. (1.1) a 3 b 3 c 3 d 3 e 3 f 3 g 3 n− n− n− n− n− n− n− a −2 b −2 c −2 d −2 e −2 f −2 0 n n n n n n arXiv:1412.4155v2 [math.NA] 18 Dec 2014 a −1 b −1 c −1 d −1 e −1 n n n n n a b c d n n n n where {ai}4≤i≤n, {bi}3≤i≤n, {ci}2≤i≤n, {di}1≤i≤n, {ei}1≤i≤n, {fi}1≤i≤n, and {gi}1≤i≤n are sequences of numbers such that gi 6=0, gn−2 = gn−1 = gn = 1 and fn−1 = fn = en = 0.
    [Show full text]
  • A New Algorithm for Inverting General Cyclic Heptadiagonal Matrices Recursively
    A New Algorithm for Inverting General Cyclic Heptadiagonal Matrices Recursively A.A. KARAWIA1 Computer Science Unit, Deanship of Educational Services, Qassim University, Buraidah 51452, Saudi Arabia. [email protected] ABSTRACT In this paper, we describe a reliable symbolic computational algorithm for inverting general cyclic heptadiagonal matrices by using parallel computing along with recursion. The algorithm is implementable to the Computer Algebra System(CAS) such as MAPLE, MATLAB and MATHEMATICA. An example is presented for the sake of illustration. Key Words: Cyclic heptadiagonal matrices; LU factorization; Determinants; Inverse matrix; Linear systems; Computer Algebra System(CAS). 1. INTRODUCTION The n × n general periodic heptadiagonal matrices are takes the form: d1 a 1 A 1 C 100 B 1 b 1 b d a A C00 B 2 2 2 2 2 2 B3 b 3 d 3 a 3 A 3 C 3 00 D3 B 3 b 3 d 3 a 3 A 3 C 3 0 0 H (1.1) 0 0 D B b d a A C n3 n 3 n 3 n 3 n 3 n 3 n 3 00Dn2 B n 2 b n 2 d n 2 a n 2 A n 2 A00 D B b d a n1 n 1 n 1 n 1 n 1 n 1 an A n0 0 0 D n B n b n d n Where n 8. The inverses of cyclic heptadiagonal matrices are usually required in science and engineering applications, for more details (see special cases, [1-9]). The motivation of the current paper is to establish efficient algorithms for inverting cyclic heptadiagonal matrices of the form (1.1) and for solving linear systems of the form: Hx r (1.2) 1 Home address: Mathematics Department, Faculty of Science, Mansoura University, Mansoura, 35516, Egypt.
    [Show full text]
  • THESIS Contributions to Parametric Image Registration and 3D Surface
    THESIS Presented at Universite´ d’Auvergne For the degree of Doctor of the Universite´ d’Auvergne Speciality Computer Vision Defended by Florent BRUNET on 2010, November 30 Title Contributions to Parametric Image Registration and 3D Surface Reconstruction Jury President Laurent Sarry Co-Directors Adrien Bartoli Remy´ Malgouyres Nassir Navab Reviewers Lourdes Agapito Joachim Hornegger Etienne´ Memin´ 2 3 Abstract This thesis deals with the modelling and the estimation of parametric functions in Computer Vision. It focuses on three main topics: range surface fitting, image registration, and 3D reconstruction of smooth surfaces. In addition to these three main topics, we consider other transversal elements. In particular, we focus on the general problem caused by the hyperparameters in parametric model estimation, which includes regularization problems. All these topics are related by a single objective which is nowadays one of the most important goals in Computer Vision: the reconstruction of an arbitrary surface from images taken in an arbitrarily deforming environment. This thesis can be divided into four main parts. The first part deals with the basics. It includes background on optimization and on parameter estimation. The problems related to the hyperparameters are also explained and illustrated. The rest of the thesis is centred on our original contributions. The second part of this thesis deals with the problem of fitting a surface to range data. This problem consists in finding a parametric smooth surface that approximates accurately a sparse set of 3D points. We consider two main problems. First, we propose methods to automatically tune the hyperparameters such as a regularization weight. Second, we show how heteroskedastic noise may be handled.
    [Show full text]