DOCSLIB.ORG

On the Inverting of a General Heptadiagonal Matrix

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read 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. Heptadiagonal matrices are frequently arise from boundary value problems. So a good technique for compute the inverse of such matrices is required. Also, These kind of matrices appear in many areas of science and engineering[1-9]. To the best of our knowledge, the inversion of a general heptadiagonal matrix of the form (1.1) has not been considered. In [10], Karawia described a reliable symbolic computational algorithm for inverting general cyclic heptadiagonal matrices by using parallel computing along with recursion. An explicit formula for the de- terminant of a heptadiagonal symmetric matrix is given in[6]. Many researchers are studied special cases of heptadiagonal matrix. In [11], the authors presented a symbolic algorithm for finding the inverse of any general nonsingular tridiagonal matrix. A new efficient computational algorithm to find the inverse of a general tridiagonal matrix is presented in [12] based on the Doolittle LU factorization. In [13], the authors introduced a computationally efficient algorithm for obtaining the inverse of a tridiagonal matrix and a pentadiogonal matrix and they assumed a few conditions to avoid failure in their own algorithm. The ∗Home Address: Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt. E- mail:[email protected] 1 motivation of the current paper is to establish efficient algorithms for inverting heptadiagonal matrix. We generalized the algorithm[13] for finding the inverse of a general heptadiagonal matrix and we presented an efficient symbolic algorithm for finding the inverse of such matrices. The development of a symbolic algorithm is considered in order to remove all cases where the numeric algorithm fails. The paper is organized as follows: In Section 2, Main result is presented. New numeric and symbolic algorithms are given in Section 3. In Section 4, Illustrative examples are presented. Conclusions of the work are given in Section 5. 2 Main Result In this section, we present recurrence formulas for the columns of the inverse of a heptadiagonal matrix H. When the matrix H is nonsingular, its inversion is computed as follows. Let −1 H = [Sij ]1≤i,j≤n = [C1, C2, ..., Cn] −1 where Ck is the kth column of the inverse matrix H . −1 By using the fact HH = In, where In is the identity matrix, the first (n − 3) columns can be obtain by relations 1 Cn−3 = (En − dnCn − en−1Cn−1 − fn−2Cn−2), gn−3 1 C E c C d C e C f C , n−4 = ( n−1 − n n − n−1 n−1 − n−2 n−2 − n−3 n−3) gn−4 1 (2.1) Cn−5 = (En−2 − bnCn − cn−1Cn−1 − dn−2Cn−2 − en−3Cn−3 − fn−4Cn−4), gn−5 1 C = (E +3 − a +6C +6 − b +5C +5 − c +4C +4 − d +3C +3 − e +2C +2 − f +1C +1), j g j j j j j j j j j j j j j j j = n − 6,n − 7, ..., 1, where Ek is the kth unit vector. From (2.1), we note that if we knowing the last three columns Cn, Cn−1, and Cn−2 then we can recur- sively compute the remaining (n − 3) columns Cn−3, Cn−4, ..., C1. At this point it is convenient to give recurrence formulas for computing Cn, Cn−1, and Cn−2. Consider the sequence of numbers {Ai}1≤i≤n+3, {Bi}1≤i≤n+3,and {Ci}1≤i≤n+3 characterized by a term recurrence relations A1 =0, A2 =0, A3 =1, d1A1 + e1A2 + f1A3 + g1A4 =0, (2.2) c2A1 + d2A2 + e2A3 + f2A4 + g2A5 =0, b3A1 + c3A2 + d3A3 + e3A4 + f3A5 + g3A6 =0, aiAi−3 + biAi−2 + ciAi−1 + diAi + eiAi+1 + fiAi+2 + giAi+3 =0, i ≥ 4, 2 B1 =0, B2 =1, B3 =0, d1B1 + e1B2 + f1B3 + g1B4 =0, (2.3) c2B1 + d2B2 + e2B3 + f2B4 + g2B5 =0, b3B1 + c3B2 + d3B3 + e3B4 + f3B5 + g3B6 =0, aiBi−3 + biBi−2 + ciBi−1 + diBi + eiBi+1 + fiBi+2 + giBi+3 =0, i ≥ 4, and C1 =1, C2 =0, C3 =0, d1C1 + e1C2 + f1C3 + g1C4 =0, (2.4) c2C1 + d2C2 + e2C3 + f2C4 + g2C5 =0, b3C1 + c3C2 + d3C3 + e3C4 + f3C5 + g3C6 =0, aiCi−3 + biCi−2 + ciCi−1 + diCi + eiCi+1 + fiCi+2 + giCi+3 =0, i ≥ 4. Now, we can give matrix forms for term recurrences (2.2), (2.3) and (2.4) HA = −An+1En−2 − An+2En−1 − An+3En, (2.5) HB = −Bn+1En−2 − Bn+2En−1 − Bn+3En, (2.6) HC = −Cn+1En−2 − Cn+2En−1 − Cn+3En, (2.7) t t t where A = [A1, A2, ..., An] , B = [B1,B2, ..., Bn] , and C = [C1, C2, ..., Cn] . Let’s define the following determinants: Ai An+2 An+3 Xi = Bi Bn+2 Bn+3 , i =1, 2, ..., n +1, (2.8) Ci Cn+2 Cn+3 Ai An+1 An+3 Yi = Bi Bn+1 Bn+3 , i =1, 2, ..., n +2, (2.9) Ci Cn+1 Cn+3 Ai An+1 An+2 Zi = Bi Bn+1 Bn+2 , i =1, 2, ..., n +3. (2.10) Ci Cn+1 Cn+2 By simple calculations, we have HX = −Xn+1En−2, (2.11) HY = −Yn+2En−1, (2.12) HZ = −Zn+3En, (2.13) t t t where X = [X1,X2, ..., Xn] , Y = [Y1, Y2, ..., Yn] , and Z = [Z1,Z2, ..., Zn] . Remark 2.1. Xn+1 = −Yn+2 = Zn+3. Lemma 2.1.(generalization version of Lemma 3.1 in [13]) If Xn+1 = 0, then the matrix H is singu- lar. Proof. The proof is simple. 3 Theorem 2.1.(generalization version of theorem 3.1 in [13]) Suppose that Xn+1 6= 0, then H is invertible and t −Z1 −Z2 −Zn Cn = , , ..., , (2.14) Zn+3 Zn+3 Zn+3 t −Y1 −Y2 −Yn Cn−1 = , , ..., , (2.15) Yn+2 Yn+2 Yn+2 t −X1 −X2 −Xn Cn−2 = , , ..., . (2.16) Xn+1 Xn+1 Xn+1 n−3 Proof. Since det(H) = − i=1 gi Xn+1 6= 0, then H is invertible. From (2.11), (2.12) and (2.13) we obtain Cn, Cn−1, and Cn−2.Q The proof is completed. 3 New numeric and symbolic algorithms for the inverse of hep- tadiagonal matrix In this section, we formulate the result in the previous section . It is a numerical algorithm to compute the inverse of a general heptadiagonal matrix of the form (1.1) when it exists. Algorithm 3.1 To find the inverse of heptadiagonal matrix (1.1). let fn−1 = fn = en =0 and gn−2 = gn−1 = gn. INPUT: Order of the matrix n and the components ai,bj ,ck, dl,el,fl, and gl for i =4, 5, ..., n, j =3, 4, ..., n, k =2, 3, ..., n, and l =1, 2, ..., n, OUTPUT: The inverse of heptadiagonal matrix H−1. Step 1: Compute the sequence of numbers Ai,Bi, and Ci for i =1, 2, ..., n +3 using (2.2), (2.3) and (2.4) respectively. Step 2: Compute Xi, i =1, 2, ..., n +1 using (2.8), Yi, i =1, 2, ..., n +2 using (2.9) and Zi, i =1, 2, ..., n +3 using (2.10). Step 3: Compute the last three columns Cn, Cn−1, and Cn−2 using (2.14), (2.15), and (2.16) respectively. Step 4: Compute the remaining (n-3)-columns Cj , j = n − 3,n − 4, ..., 1 using (2.1). −1 Step 5: Set H = [C1, C2, ..., Cn]. The numeric algorithm 3.1 will be referred to as NINVHEPTA algorithm. The computational cost of NINVHEPTA algorithm is 103n + 69 operations. As can be easily seen, it breaks down unless the conditions gi 6= 0 are satisfied for all i =1, 2, ..., n − 3. So the following symbolic algorithm is developed in order to remove the cases where the numeric algorithm fails. Algorithm 3.2 To find the inverse of heptadiagonal matrix (1.1).
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]
  • Fast Singular Value Thresholding Without Singular Value Decomposition∗
    METHODS AND APPLICATIONS OF ANALYSIS. c 2013 International Press Vol. 20, No. 4, pp. 335–352, December 2013 002 FAST SINGULAR VALUE THRESHOLDING WITHOUT SINGULAR VALUE DECOMPOSITION∗ JIAN-FENG CAI† AND STANLEY OSHER‡ Abstract. Singular value thresholding (SVT) is a basic subroutine in many popular numerical schemes for solving nuclear norm minimization that arises from low-rank matrix recovery prob- lems such as matrix completion. The conventional approach for SVT is first to find the singular value decomposition (SVD) and then to shrink the singular values. However, such an approach is time-consuming under some circumstances, especially when the rank of the resulting matrix is not significantly low compared to its dimension. In this paper, we propose a fast algorithm for directly computing SVT for general dense matrices without using SVDs. Our algorithm is based on matrix Newton iteration for matrix functions, and the convergence is theoretically guaranteed. Numerical experiments show that our proposed algorithm is more efficient than the SVD-based approaches for general dense matrices. Key words. Low rank matrix, nuclear norm minimization, matrix Newton iteration. AMS subject classifications. 65F30, 65K99. 1. Introduction. Singular value thresholding (SVT) introduced in [7] is a key subroutine in many popular numerical schemes (e.g. [7, 12, 13, 52, 54, 66]) for solving nuclear norm minimization that arises from low-rank matrix recovery problems such as matrix completion [13–15,60]. Let Y Rm×n be a given matrix, and Y = UΣV T be its singular value decomposition (SVD),∈ where U and V are orthonormal matrices and Σ = diag(σ1, σ2,...,σs) is the diagonal matrix with diagonals being the singular values of Y .
    [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]
  • A Simplex-Type Voronoi Algorithm Based on Short Vector Computations of Copositive Quadratic Forms
    Simons workshop Lattices Geometry, Algorithms and Hardness Berkeley, February 21st 2020 A simplex-type Voronoi algorithm based on short vector computations of copositive quadratic forms Achill Schürmann (Universität Rostock) based on work with Mathieu Dutour Sikirić and Frank Vallentin ( for Q n positive definite ) Perfect Forms 2 S>0 DEF: min(Q)= min Q[x] is the arithmetical minimum • x n 0 2Z \{ } Q is uniquely determined by min(Q) and Q perfect • , n MinQ = x Z : Q[x]=min(Q) { 2 } V(Q)=cone xxt : x MinQ is Voronoi cone of Q • { 2 } (Voronoi cones are full dimensional if and only if Q is perfect!) THM: Voronoi cones give a polyhedral tessellation of n S>0 and there are only finitely many up to GL n ( Z ) -equivalence. Voronoi’s Reduction Theory n t GLn(Z) acts on >0 by Q U QU S 7! Georgy Voronoi (1868 – 1908) Task of a reduction theory is to provide a fundamental domain Voronoi’s algorithm gives a recipe for the construction of a complete list of such polyhedral cones up to GL n ( Z ) -equivalence Ryshkov Polyhedron The set of all positive definite quadratic forms / matrices with arithmeticalRyshkov minimum Polyhedra at least 1 is called Ryshkov polyhedron = Q n : Q[x] 1 for all x Zn 0 R 2 S>0 ≥ 2 \{ } is a locally finite polyhedron • R is a locally finite polyhedron • R Vertices of are perfect forms Vertices of are• perfect R • R 1 n n ↵ (det(Q + ↵Q0)) is strictly concave on • 7! S>0 Voronoi’s algorithm Voronoi’s Algorithm Start with a perfect form Q 1.
    [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]
  • Combinatorial Optimization, Packing and Covering
    Combinatorial Optimization: Packing and Covering G´erard Cornu´ejols Carnegie Mellon University July 2000 1 Preface The integer programming models known as set packing and set cov- ering have a wide range of applications, such as pattern recognition, plant location and airline crew scheduling. Sometimes, due to the spe- cial structure of the constraint matrix, the natural linear programming relaxation yields an optimal solution that is integer, thus solving the problem. Sometimes, both the linear programming relaxation and its dual have integer optimal solutions. Under which conditions do such integrality properties hold? This question is of both theoretical and practical interest. Min-max theorems, polyhedral combinatorics and graph theory all come together in this rich area of discrete mathemat- ics. In addition to min-max and polyhedral results, some of the deepest results in this area come in two flavors: “excluded minor” results and “decomposition” results. In these notes, we present several of these beautiful results. Three chapters cover min-max and polyhedral re- sults. The next four cover excluded minor results. In the last three, we present decomposition results. We hope that these notes will encourage research on the many intriguing open questions that still remain. In particular, we state 18 conjectures. For each of these conjectures, we offer $5000 as an incentive for the first correct solution or refutation before December 2020. 2 Contents 1Clutters 7 1.1 MFMC Property and Idealness . 9 1.2 Blocker . 13 1.3 Examples .......................... 15 1.3.1 st-Cuts and st-Paths . 15 1.3.2 Two-Commodity Flows . 17 1.3.3 r-Cuts and r-Arborescences .
    [Show full text]
  • Quadratic Forms and Their Applications
    Quadratic Forms and Their Applications Proceedings of the Conference on Quadratic Forms and Their Applications July 5{9, 1999 University College Dublin Eva Bayer-Fluckiger David Lewis Andrew Ranicki Editors Published as Contemporary Mathematics 272, A.M.S. (2000) vii Contents Preface ix Conference lectures x Conference participants xii Conference photo xiv Galois cohomology of the classical groups Eva Bayer-Fluckiger 1 Symplectic lattices Anne-Marie Berge¶ 9 Universal quadratic forms and the ¯fteen theorem J.H. Conway 23 On the Conway-Schneeberger ¯fteen theorem Manjul Bhargava 27 On trace forms and the Burnside ring Martin Epkenhans 39 Equivariant Brauer groups A. FrohlichÄ and C.T.C. Wall 57 Isotropy of quadratic forms and ¯eld invariants Detlev W. Hoffmann 73 Quadratic forms with absolutely maximal splitting Oleg Izhboldin and Alexander Vishik 103 2-regularity and reversibility of quadratic mappings Alexey F. Izmailov 127 Quadratic forms in knot theory C. Kearton 135 Biography of Ernst Witt (1911{1991) Ina Kersten 155 viii Generic splitting towers and generic splitting preparation of quadratic forms Manfred Knebusch and Ulf Rehmann 173 Local densities of hermitian forms Maurice Mischler 201 Notes towards a constructive proof of Hilbert's theorem on ternary quartics Victoria Powers and Bruce Reznick 209 On the history of the algebraic theory of quadratic forms Winfried Scharlau 229 Local fundamental classes derived from higher K-groups: III Victor P. Snaith 261 Hilbert's theorem on positive ternary quartics Richard G. Swan 287 Quadratic forms and normal surface singularities C.T.C. Wall 293 ix Preface These are the proceedings of the conference on \Quadratic Forms And Their Applications" which was held at University College Dublin from 5th to 9th July, 1999.
    [Show full text]
  • A Framework for Efficient Execution of Matrix Computations
    A Framework for Efficient Execution of Matrix Computations Doctoral Thesis May 2006 Jos´e Ram´on Herrero Advisor: Prof. Juan J. Navarro UNIVERSITAT POLITECNICA` DE CATALUNYA Departament D'Arquitectura de Computadors To Joan and Albert my children To Eug`enia my wife To Ram´on and Gloria my parents Stillicidi casus lapidem cavat Lucretius (c. 99 B.C.-c. 55 B.C.) De Rerum Natura1 1Continual dropping wears away a stone. Titus Lucretius Carus. On the nature of things Abstract Matrix computations lie at the heart of most scientific computational tasks. The solution of linear systems of equations is a very frequent operation in many fields in science, engineering, surveying, physics and others. Other matrix op- erations occur frequently in many other fields such as pattern recognition and classification, or multimedia applications. Therefore, it is important to perform matrix operations efficiently. The work in this thesis focuses on the efficient execution on commodity processors of matrix operations which arise frequently in different fields. We study some important operations which appear in the solution of real world problems: some sparse and dense linear algebra codes and a classification algorithm. In particular, we focus our attention on the efficient execution of the following operations: sparse Cholesky factorization; dense matrix multipli- cation; dense Cholesky factorization; and Nearest Neighbor Classification. A lot of research has been conducted on the efficient parallelization of nu- merical algorithms. However, the efficiency of a parallel algorithm depends ultimately on the performance obtained from the computations performed on each node. The work presented in this thesis focuses on the sequential execution on a single processor.
    [Show full text]
  • Zero-Error Slepian-Wolf Coding of Confined Correlated Sources With
    Zero-error Slepian-Wolf Coding of Confined- Correlated Sources with Deviation Symmetry Rick Ma∗and Samuel Cheng† October 31, 2018 Abstract In this paper, we use linear codes to study zero-error Slepian-Wolf coding of a set of sources with deviation symmetry, where the sources are generalization of the Hamming sources over an arbitrary field. We extend our previous codes, Generalized Hamming Codes for Multiple Sources, to Matrix Partition Codes and use the latter to efficiently compress the target sources. We further show that every perfect or linear-optimal code is a Matrix Partition Code. We also present some conditions when Matrix Partition Codes are perfect and/or linear-optimal. Detail discussions of Matrix Partition Codes on Hamming sources are given at last as examples. 1 Introduction Slepian-Wolf (SW) Coding or Slepian-Wolf problem refers to separate encod- ing of multiple correlated sources but joint lossless decoding of the sources [1]. Since then, many researchers have looked into ways to implement SW cod- ing efficiently. Noticeably, Wyner was the first who realized that linear coset codes can be used to tackle the problem [2]. Essentially, considering the source from each terminal as a column vector, the encoding output will simply be the multiple of a “fat”1 coding matrix and the input vector. The approach was popularized by Pradhan et al. more than two decades later [3]. Practical syndrome-based schemes for S-W coding using channel codes have been further studied in [4, 5, 6, 7, 8, 9, 10, 11, 12]. arXiv:1308.0632v1 [cs.IT] 2 Aug 2013 Unlike many prior works focusing on near-lossless compression, in this work we consider true lossless compression (zero-error reconstruction) in which sources are always recovered losslessly [13, 14, 15, 16, 17].
    [Show full text]
  • Contents September 1
    Contents September 1 : Examples 2 September 3 : The Gessel-Lindstr¨om-ViennotLemma 3 September 15 : Parametrizing totally positive unipotent matrices { Part 1 6 September 17 : Parametrizing totally positive unipotent matrices { part 2 7 September 22 : Introduction to the 0-Hecke monoid 9 September 24 : The 0-Hecke monoid continued 11 September 29 : The product has the correct ranks 12 October 1 : Bruhat decomposition 14 October 6 : A unique Bruhat decomposition 17 October 8: B−wB− \ N − + as a manifold 18 October 13: Grassmannians 21 October 15: The flag manifold, and starting to invert the unipotent product 23 October 20: Continuing to invert the unipotent product 28 October 22 { Continuing to invert the unipotent product 30 October 27 { Finishing the inversion of the unipotent product 32 October 29 { Chevalley generators in GLn(R) 34 November 5 { LDU decomposition, double Bruhat cells, and total positivity for GLn(R) 36 November 10 { Kasteleyn labelings 39 November 12 { Kasteleyn labellings continued 42 November 17 { Using graphs in a disc to parametrize the totally nonnegative Grassmannian 45 November 19 { cyclic rank matrices 45 December 1 { More on cyclic rank matrices 48 December 3 { Bridges and Chevalley generators 51 December 8 { finishing the proof 55 1 2 September 1 : Examples We write [m] = f1; 2; : : : ; mg. Given an m × n matrix M, and subsets I =⊆ [m] and I J ⊆ [n] with #(I) = #(J), we define ∆J (M) = det(Mij)i2I; j2J , where we keep the elements of I and the elements of J in the same order. For example, 13 M12 M15 ∆25(M) = det : M32 M35 I The ∆J (M) are called the minors of M.
    [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]