DiVA Digitala Vetenskapliga Arkivet


 Silvia Pitts
 1 years ago
 Views:
Transcription
1 DVA Dgtala Vetenskaplga Arkvet Ths s a book chapter publshed n Hghperformance scentfc computng: algorthms and applcatons (ed Berry, MW; Gallvan, KA; Gallopoulos, E; Grama, A; Phlppe, B; Saad, Y; Saed, F) Ctaton for the publshed paper: Kjelgaard Mkkelsen, Carl Chrstan 0 The explct Spke algorthm: Iteratve soluton of the reduced system In Hghperformance scentfc computng: algorthms and applcatons, Berry, MW; Gallvan, KA; Gallopoulos, E; Grama, A; Phlppe, B; Saad, Y; Saed, F (ed), p  London: Sprnger The orgnal publcaton s avalable at wwwsprngerlnkcom
2 The explct SPIKE algorthm: Iteratve soluton of the reduced system Carl Chrstan Kjelgaard Mkkelsen Department of Computng Scence and HPCN Umeå Unversty Sweden Dedcated to Ahmed Sameh on the occason of hs 0th brthday Summary The explct SPIKE algorthm apples to narrow banded lnear systems whch are strctly dagonally domnant by rows The parallel bottleneck s the soluton of the socalled reduced system whch s block trdagonal and strctly dagonally domnant by rows The reduced system can be solved teratvely usng the truncated reduced system matrx as a precondtoner In ths paper we derve a tght estmate for the ualty of ths precondtoner Introducton A matrx A = [a j] R n n s dagonally domnant by rows f : X a j a j If the neualty s sharp, then A s strctly dagonally domnant by rows If A s nonsngular and dagonally domnant by rows or f A s strctly dagonally domnant by rows, then a 0, and the domnance factor ɛ gven by ɛ = max ( P j aj a s well defned The matrx A has lower bandwdth b l f a j = 0 for > j + b l and upper bandwdth b u f a j = 0 for j > + b u If b = max{b l, b u} n, then we say that A s narrow banded Every suare banded matrx can be parttoned as a block trdagonal matrx wth suare dagonal blocks, e )
3 Carl Chrstan Kjelgaard Mkkelsen A C A = B Cm, () B m A m only the dmenson of each dagonal block must be bounded from below by b In partcular, we do not have to choose the same dmenson for each dagonal block, even n the exceptonal case where b dvdes n The SPIKE algorthms are desgned to solve banded systems on a parallel machne The central dea was ntroduced by Sameh and Kuck [] who consdered the trdagonal case and Chen, Kuck and Sameh [] who studed the trangular case Lawre and Sameh [] appled the algorthm to the symmetrc postve defnte case whle Dongarra and Sameh [] consdered the dagonally domnant case Polzz and Sameh [8, 9] ntroduced the truncated SPIKE algorthm for systems whch are strctly dagonally domnant by rows Recently, Manguoglu, Sameh, and Schenk [] have combned PARDISO wth SPIKE n the PSPIKE package The explct SPIKE algorthm by Dongarra and Sameh [] can be used to solve narrow banded lnear systems whch are strctly dagonally domnant by rows The algorthm extends naturally to systems whch are block trdagonal Moreover, the analyss s smplfed f we focus on the number of dagonal blocks, rather than the bandwdth of the matrx In Secton we state the explct SPIKE algorthm for systems whch are block trdagonal and strctly dagonally domnant by rows The parallel bottleneck s the soluton of a reduced system whch s block trdagonal and strctly dagonally domnant by rows The reduced system can be solved teratvely usng the man block dagonal as a precondtoner We derve a tght estmate for the ualty of ths precondtoner n Secton Ths s a specal case of a more general theorem by Mkkelsen [] The explct SPIKE algorthm In ths secton we state the explct SPIKE algorthm for systems whch are block trdagonal and strctly dagonally domnant by rows The valdty and the basc analyss of the algorthm hnges on the followng lemma Lemma Let G = ˆE, D, F be a matrx such that ˆD, E, F s strctly dagonally domnant by rows wth domnance factor ɛ Then G s row euvalent to a unue matrx K = ˆU, I, V Moreover, the matrx ˆU, V satsfes ˆU, V ɛ Proof Mkkelsen and Manguoglu [] contans an elementary proof Now consder the soluton of a block trdagonal lnear system Ax = f on a parallel machne wth p processors Gven a small tolerance δ > 0, we shall now seek an approxmaton y, such that the forward error satsfes
4 The explct SPIKE algorthm: Iteratve soluton of the reduced system ˆ A f = A () C () f () B () () C B () A () C () f () f () B () A () C () B () () C B () A () C () f () f () B () A () C () B () C () B () A () f () Fg The SPIKE parttonng for p = processors x y δ x We assume that A has m = p dagonal blocks and we assgn consecutve block rows to each processor The case of p = s llustrated n Fgure If A s strctly dagonally domnant by rows, then we can predvde wth the man block dagonal n order to obtan an euvalent lnear system Sx = g The case of p = s dsplayed n Fgure It s from the narrow columns or spkes protrudng from the man dagonal that the orgnal algorthm has derved t name The matrx S s called the SPIKE matrx; the vector g s called the modfed rght hand sde By Lemma, S I ɛ <, so S s strctly dagonally domnant by rows The euatons wthn a sngle block row of each the man parttons lnes form a reduced system Rx r = g r whch can be solved ndependently The general structure of the reduced system s gven n Fgure Once the reduced system has been solved, the soluton of the orgnal system can be retreved by backsubsttuton Specfcally, we have x (j) = g (j) U (j) x (j ) V (j) x (j+),, j p, () where U (), V (p), x (0), and x (p+) are undefned and should be taken as zero Suppose for the moment that we have somehow solved the reduced system wth a small normwse relatve forward error, say,
5 Carl Chrstan Kjelgaard Mkkelsen I V () g () I V () g () U () I V () g () ˆ S g = U () I V () g () U () I g () U () I g () Fg The SPIKE matrx correspondng to p = processors ˆ R gr = I V () g () U () I V () g () U () I V () U () I g () () g V (p ) U (p ) I V (p ) g (p ) U (p) I g (p) Fg The general structure of the reduced system x r y r δ x r In vew of euaton () t s natural to partton y r conformally wth x r, e y r = (x () and defne a vector y R n usng Then y (j) = g (j) x (j) U (j) y (j ) y (j) and t follows mmedately that T () T, x,, x (p ) h = T, x (p) T ) T V (j) y (j+),, j p U (j) ", V (j) x (j ) y (j ) x (j+) y (j+) #,
6 The explct SPIKE algorthm: Iteratve soluton of the reduced system ˆ T gr = I V () g () U () I g () I V () g () U () I g () I U (p) I g (p) V (p ) g (p ) Fg The structure of the truncated reduced system x y ɛ x r y r ɛδ x r δ x It s clear that we must solve the reduced system accurately n order to acheve a small forward normwse relatve error We now consder the soluton of the reduced system The reduced system matrx R s block trdagonal and strctly dagonally domnant by rows The neualty R I ɛ < s nherted from the SPIKE matrx S Freuently, but not unversally, the off dagonal blocks are nsgnfcant and can be dropped Ths phenomenon s exploted heavly n the truncated SPIKE algorthm by Polzz and Sameh [8, 9] Let T denote the man block dagonal of R, see Fgure Mkkelsen and Manguoglu [] showed that T R ɛ when A s banded and strctly dagonally domnant by rows In ths paper we consder the sgnfcance of the off dagonal blocks relatve to the man block dagonal To ths end we defne an auxlary matrx B by B = T (T R) Now, let x tr be the soluton of the truncated reduced system Then T x tr = g r T (x r x tr) = (R (T R))x r g r = (Rx r g r) (T R))x r = (T R)x r from whch t mmedately follows, that f x r 0, then x tr x r x r B We have already understood the need to solve the reduced system wth a forward normwse relatve error of at most δ If B δ, then we smply drop the off dagonal blocks and approxmate x r wth x tr If B > δ, then we can solve the
7 Carl Chrstan Kjelgaard Mkkelsen reduced system teratvely usng the man block dagonal as a precondtoner If we use the statonary teraton where x (0) r = 0, then T x () r and we can stop the teraton whenever = (T R)x ( ) r + g r, =,,, x r x () r B x r B δ In the next secton we establsh a tght upper bound on the central parameter B The SPIKE and the PSPIKE packages both apply BCG, rather than the statonary teraton Nevertheless, the sze of B remans an nterestng ueston The man result Our purpose s to establsh Theorem Theorem The auxlary matrx B satsfes B ɛ, where s the number of dagonal blocks assgned to each processor and eualty s possble We shall reduce the problem of provng Theorem to a sngle applcaton of the followng theorem Theorem (Mkkelsen []) Let G k be a representaton of k consecutve block rows of a block trdagonal matrx A whch s strctly dagonally domnant by rows wth domnance factor ɛ, e G k = B k A k C k B A C B 0 A 0 C 0 B A C Then G k s row euvalent to a unue matrx K k of the form B k A k C k
8 The explct SPIKE algorthm: Iteratve soluton of the reduced system K k = k I V(k) k I V (k) 0 I V (k) 0 I V (k) k Z (k) = I V (k) k where the spkes decay exponentally as we move towards the man block row Specfcally, f we defne " # and then V (k) V (k), 0 < < k, h Z (k) 0 = 0, V (k) 0 Z (k) ɛ k, 0 < k Proof The exstence and unueness of K k follows mmedately from Lemma The central neualty can be establshed usng the well orderng prncple The detals can be found n a report by Mkkelsen and Kågström [] We now move to prove the estmate gven by Theorem It s straghtforward to verfy that eualty s acheved for matrces A gven by euaton () where B = O k, A = I k, C = ɛi k, and O k s the k by k zero matrx, I k s the k by k dentty matrx and ɛ < In order to prove the general neualty t suffces to consder the nteracton between two neghborng parttons Ths follows mmedately from the propertes of the nfnty norm Let G k be a compact representaton of k block rows drawn from the orgnal matrx A, e G k = B k A k C k B A C B A C B k A k C k and let H k be a compact representaton of the correspondng rows of the assocated SPIKE matrx Then G k H k and H k has the form ()
9 8 Carl Chrstan Kjelgaard Mkkelsen U k I V k U I V H k = () U I V U k I V k Our task s to show that the auxlary matrx Z k gven by»»» Z Z I V U 0 Z k = = Z Z U I 0 V () satsfes Z k ɛ k, k =,,, We contnue to reduce H k usng row operatons We repartton H k n order to focus our attenton on the two central block rows, e U k I V k U I V G k U I V U k I V k Then we predvde wth the central by block matrx and obtan U k I V k Z I Z G k Z I Z U k I V k and t s clear that there exsts a matrx K k such that G k K k and
10 The explct SPIKE algorthm: Iteratve soluton of the reduced system 9 and the matrx Z k satsfes U I V U I V U I V K k = U I V U I V U I V ()» U V Z k = () U V At ths pont we have reduced the problem of provng Theorem to a straghtforward applcaton of Theorem Concluson The explct SPIKE algorthm by Dongarra and Sameh [] extends naturally to systems whch are block trdagonal and strctly dagonally domnant by rows Moreover, the analyss of the method s smplfed by focusng on the number of dagonal blocks rather than the bandwdth The parallel bottleneck remans the soluton of the reduced system Rx r = g r whch s strctly dagonally domnant and block trdagonal The sgnfcance of the off dagonal blocks can be measured usng the auxlary matrx B gven by B = T (T R) = I T R, where T denotes the man block dagonal of R If B s suffcently small, then we can gnore the off dagonal blocks and approxmate x r wth the soluton of the truncated reduced system T x tr = g r In general, we can solve the reduced system teratvely usng the man block dagonal T as a precondtoner and the convergence rate s controlled by the sze of B Our man contrbuton s Theorem whch establshes a tght upper bound on B References SC Chen, DJ Kuck and A Sameh, Practcal parallel band trangular system solvers, ACM Trans Math Software,, (98), pp 0 JJ Dongarra and A Sameh, On some parallel banded system solvers, Parallel Comput, vol (98), pp  DH Lawre and A Sameh, The computaton and communcaton complexty of a parallel banded system solver, ACM Trans Math, Software, vol 0 (98), pp 89
11 0 Carl Chrstan Kjelgaard Mkkelsen M Manguoglu, A Sameh and O Schenk, PSPIKE: A Parallel Hybrd Sparse Lnear System Solver, In Proceedngs of EUROPAR009, LNCS 0 (009), pp C C K Mkkelsen and M Manguoglu, Analyss of the truncated SPIKE algorthm, SIMAX, vol 0, no (008), pp 009 C C K Mkkelsen and Bo Kågström, Analyss of ncomplete cyclc reducton for narrow banded and strctly dagonally domnant lnear systems Tech Rep UMINF 0, Department of Computng Scence, Umeå Unversty (0) Submtted to PPAM0 A Sameh and DJ Kuck, On stable parallel lnear systems solvers, J ACM, vol (98), pp E Polzz and A Sameh, A parallel hybrd banded system solver: The SPIKE algorthm, Parallel Comput, vol (00), pp 9 9 E Polzz and A Sameh, SPIKE: A parallel envronment for solvng banded lnear systems, Comput Fluds, vol (00), pp 0
v a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationThe eigenvalue derivatives of linear damped systems
Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by YeongJeu Sun Department of Electrcal Engneerng IShou Unversty Kaohsung, Tawan 840, R.O.C emal: yjsun@su.edu.tw
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationChapter 3 Group Theory p. 1  Remark: This is only a brief summary of most important results of groups theory with respect
Chapter 3 Group Theory p.  3. Compact Course: Groups Theory emark: Ths s only a bref summary of most mportant results of groups theory wth respect to the applcatons dscussed n the followng chapters. For
More informationSPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:
SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and
More informationgreatest common divisor
4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no
More informationSolution of Algebraic and Transcendental Equations
CHAPTER Soluton of Algerac and Transcendental Equatons. INTRODUCTION One of the most common prolem encountered n engneerng analyss s that gven a functon f (, fnd the values of for whch f ( = 0. The soluton
More information9.1 The Cumulative Sum Control Chart
Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s
More informationA Computer Technique for Solving LP Problems with Bounded Variables
Dhaka Unv. J. Sc. 60(2): 163168, 2012 (July) A Computer Technque for Solvng LP Problems wth Bounded Varables S. M. Atqur Rahman Chowdhury * and Sanwar Uddn Ahmad Department of Mathematcs; Unversty of
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationHYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION
HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION Abdul Ghapor Hussn Centre for Foundaton Studes n Scence Unversty of Malaya 563 KUALA LUMPUR Emal: ghapor@umedumy Abstract Ths paper
More informationA linear recurrence sequence of composite numbers
LMS J Comput Math 15 (2012) 360 373 C 2012 Author do:101112/s1461157012001143 A lnear recurrence sequence of composte numbers Jonas Šurys Abstract We prove that for each postve nteger k n the range 2 k
More informationIMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS 1
Nov Sad J. Math. Vol. 36, No. 2, 2006, 009 IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUAREROOT INTERVAL METHOD FOR MULTIPLE ZEROS Modrag S. Petkovć 2, Dušan M. Mloševć 3 Abstract. A new theorem concerned
More information1 Example 1: Axisaligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationNonparametric Estimation of Asymmetric First Price Auctions: A Simplified Approach
Nonparametrc Estmaton of Asymmetrc Frst Prce Auctons: A Smplfed Approach Bn Zhang, Kemal Guler Intellgent Enterprse Technologes Laboratory HP Laboratores Palo Alto HPL200286(R.) November 23, 2004 frst
More informationPLANAR GRAPHS. Plane graph (or embedded graph) A graph that is drawn on the plane without edge crossing, is called a Plane graph
PLANAR GRAPHS Basc defntons Isomorphc graphs Two graphs G(V,E) and G2(V2,E2) are somorphc f there s a onetoone correspondence F of ther vertces such that the followng holds:  u,v V, uv E, => F(u)F(v)
More informationLecture 3. 1 Largest singular value The Behavior of Algorithms in Practice 2/14/2
18.409 The Behavor of Algorthms n Practce 2/14/2 Lecturer: Dan Spelman Lecture 3 Scrbe: Arvnd Sankar 1 Largest sngular value In order to bound the condton number, we need an upper bound on the largest
More informationNew bounds in BalogSzemerédiGowers theorem
New bounds n BalogSzemerédGowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A
More informationThe Greedy Method. Introduction. 0/1 Knapsack Problem
The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton
More informationAryabhata s Root Extraction Methods. Abhishek Parakh Louisiana State University Aug 31 st 2006
Aryabhata s Root Extracton Methods Abhshek Parakh Lousana State Unversty Aug 1 st 1 Introducton Ths artcle presents an analyss of the root extracton algorthms of Aryabhata gven n hs book Āryabhatīya [1,
More informationMoment of a force about a point and about an axis
3. STATICS O RIGID BODIES In the precedng chapter t was assumed that each of the bodes consdered could be treated as a sngle partcle. Such a vew, however, s not always possble, and a body, n general, should
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy Scurve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy Scurve Regresson ChengWu Chen, Morrs H. L. Wang and TngYa Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationLogical Development Of Vogel s Approximation Method (LDVAM): An Approach To Find Basic Feasible Solution Of Transportation Problem
INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN 77866 Logcal Development Of Vogel s Approxmaton Method (LD An Approach To Fnd Basc Feasble Soluton Of Transportaton
More informationSolution : (a) FALSE. Let C be a binary oneerror correcting code of length 9. Then it follows from the Sphere packing bound that.
MATH 29T Exam : Part I Solutons. TRUE/FALSE? Prove your answer! (a) (5 pts) There exsts a bnary oneerror correctng code of length 9 wth 52 codewords. (b) (5 pts) There exsts a ternary oneerror correctng
More informationUsing AlphaBeta Associative Memories to Learn and Recall RGB Images
Usng AlphaBeta Assocatve Memores to Learn and Recall RGB Images Cornelo YáñezMárquez, María Elena CruzMeza, Flavo Arturo SánchezGarfas, and Itzamá LópezYáñez Centro de Investgacón en Computacón, Insttuto
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationLoop Parallelization
  Loop Parallelzaton C52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I,J]+B[I,J] ED FOR ED FOR analyze
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationWhen Network Effect Meets Congestion Effect: Leveraging Social Services for Wireless Services
When Network Effect Meets Congeston Effect: Leveragng Socal Servces for Wreless Servces aowen Gong School of Electrcal, Computer and Energy Engeerng Arzona State Unversty Tempe, AZ 8587, USA xgong9@asuedu
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationFault tolerance in cloud technologies presented as a service
Internatonal Scentfc Conference Computer Scence 2015 Pavel Dzhunev, PhD student Fault tolerance n cloud technologes presented as a servce INTRODUCTION Improvements n technques for vrtualzaton and performance
More informationJoint Scheduling of Processing and Shuffle Phases in MapReduce Systems
Jont Schedulng of Processng and Shuffle Phases n MapReduce Systems Fangfe Chen, Mural Kodalam, T. V. Lakshman Department of Computer Scence and Engneerng, The Penn State Unversty Bell Laboratores, AlcatelLucent
More informationEE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN
EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson  3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson  6 Hrs.) Voltage
More informationn + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)
MATH 16T Exam 1 : Part I (InClass) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total
More informationA Prefix Code Matching Parallel LoadBalancing Method for SolutionAdaptive Unstructured Finite Element Graphs on Distributed Memory Multicomputers
Ž. The Journal of Supercomputng, 15, 25 49 2000 2000 Kluwer Academc Publshers. Manufactured n The Netherlands. A Prefx Code Matchng Parallel LoadBalancng Method for SolutonAdaptve Unstructured Fnte Element
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationMultivariate EWMA Control Chart
Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant
More informationYves Genin, Yurii Nesterov, Paul Van Dooren. CESAME, Universite Catholique de Louvain. B^atiment Euler, Avenue G. Lema^tre 46
Submtted to ECC 99 as a regular paper n Lnear Systems Postve transfer functons and convex optmzaton 1 Yves Genn, Yur Nesterov, Paul Van Dooren CESAME, Unverste Catholque de Louvan B^atment Euler, Avenue
More information38123 Povo Trento (Italy), Via Sommarive 14 GENETICALLYDESIGNED ARBITRARY LENGTH ALMOST DIFFERENCE SETS
UNIVERSITY OF TRENTO DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL INFORMAZIONE 38123 Povo Trento (Italy), Va Sommarve 14 http://www.ds.untn.t GENETICALLYDESIGNED ARBITRARY LENGTH ALMOST DIFFERENCE SETS G.
More informationHW #2 Solutions: M552 Spring 2006
HW #2 Solutons: M552 Sprng 2006 1. (3.1Trefethen & Bau) Prove that f W s an arbtrary nonsngular matrx, the functon W defned by x W = Wx s a vector norm. ANS: We need to show () x W 0, x W = 0 x = 0 Gven
More informationA Fast Incremental Spectral Clustering for Large Data Sets
2011 12th Internatonal Conference on Parallel and Dstrbuted Computng, Applcatons and Technologes A Fast Incremental Spectral Clusterng for Large Data Sets Tengteng Kong 1,YeTan 1, Hong Shen 1,2 1 School
More informationProduction. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem.
Producer Theory Producton ASSUMPTION 2.1 Propertes of the Producton Set The producton set Y satsfes the followng propertes 1. Y s nonempty If Y s empty, we have nothng to talk about 2. Y s closed A set
More informationJournal of Computational and Applied Mathematics. Breakdownfree version of ILU factorization for nonsymmetric positive definite matrices
Journal of Computatonal and Appled Mathematcs 230 (2009) 699 705 Contents lsts avalable at ScenceDrect Journal of Computatonal and Appled Mathematcs ournal homepage: wwwelsevercom/locate/cam Breakdownfree
More informationThe Analysis of Outliers in Statistical Data
THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate
More informationOn fourth order simultaneously zerofinding method for multiple roots of complex polynomial equations 1
General Mathematcs Vol. 6, No. 3 (2008), 9 3 On fourth order smultaneously zerofndng method for multple roots of complex polynomal euatons Nazr Ahmad Mr and Khald Ayub Abstract In ths paper, we present
More informationThe Performance Analysis Of A M/M/2/2+1 Retrial Queue With Unreliable Server
Journal of Statstcal Scence and Applcaton, October 5, Vol. 3, No. 9, 6374 do:.765/384/5.9.3 D DAV I D PUBLISHING The Performance Analyss Of A M/M//+ Retral Queue Wth Unrelable Server R. Kalyanaraman
More informationA Note on the Decomposition of a Random Sample Size
A Note on the Decomposton of a Random Sample Sze Klaus Th. Hess Insttut für Mathematsche Stochastk Technsche Unverstät Dresden Abstract Ths note addresses some results of Hess 2000) on the decomposton
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More informationNonlinear data mapping by neural networks
Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal
More informationRing structure of splines on triangulations
www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAMReport 201448 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon
More informationU.C. Berkeley CS270: Algorithms Lecture 4 Professor Vazirani and Professor Rao Jan 27,2011 Lecturer: Umesh Vazirani Last revised February 10, 2012
U.C. Berkeley CS270: Algorthms Lecture 4 Professor Vazran and Professor Rao Jan 27,2011 Lecturer: Umesh Vazran Last revsed February 10, 2012 Lecture 4 1 The multplcatve weghts update method The multplcatve
More informationMANY machine learning and pattern recognition applications
1 Trace Rato Problem Revsted Yangqng Ja, Fepng Ne, and Changshu Zhang Abstract Dmensonalty reducton s an mportant ssue n many machne learnng and pattern recognton applcatons, and the trace rato problem
More informationGraph Theory and Cayley s Formula
Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll
More information6. EIGENVALUES AND EIGENVECTORS 3 = 3 2
EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a nonzero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :
More informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2  Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of noncoplanar vectors Scalar product
More informationThreedimensional Gantt Chart Based Resourceconstrained Multiple Projects Scheduling and Critical Chain Identification
Threedmensonal Gantt Chart Based Resourceconstraned Multple Proects Schedulng and Crtcal Chan Identfcaton J. Q. Wang,, S. F. Zhang,, J. Chen,, S. Wang,, Y. F. Zhang, Insttute of System Integrated & ngneerng
More informationAlternate Approximation of Concave Cost Functions for
Alternate Approxmaton of Concave Cost Functons for Process Desgn and Supply Chan Optmzaton Problems Dego C. Cafaro * and Ignaco E. Grossmann INTEC (UNL CONICET), Güemes 3450, 3000 Santa Fe, ARGENTINA Department
More informationL10: Linear discriminants analysis
L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss
More informationData Broadcast on a MultiSystem Heterogeneous Overlayed Wireless Network *
JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 24, 819840 (2008) Data Broadcast on a MultSystem Heterogeneous Overlayed Wreless Network * Department of Computer Scence Natonal Chao Tung Unversty Hsnchu,
More informationA Secure PasswordAuthenticated Key Agreement Using Smart Cards
A Secure PasswordAuthentcated Key Agreement Usng Smart Cards Ka Chan 1, WenChung Kuo 2 and JnChou Cheng 3 1 Department of Computer and Informaton Scence, R.O.C. Mltary Academy, Kaohsung 83059, Tawan,
More informationJournal of Computational and Applied Mathematics
Journal of Computatonal and Appled Mathematcs 37 (03) 6 35 Contents lsts avalable at ScVerse ScenceDrect Journal of Computatonal and Appled Mathematcs journal homepage: wwwelsevercom/locate/cam The nverse
More informationPoint cloud to point cloud rigid transformations. Minimizing Rigid Registration Errors
Pont cloud to pont cloud rgd transformatons Russell Taylor 600.445 1 600.445 Fall 000014 Copyrght R. H. Taylor Mnmzng Rgd Regstraton Errors Typcally, gven a set of ponts {a } n one coordnate system and
More informationConversion between the vector and raster data structures using Fuzzy Geographical Entities
Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationForecasting the Direction and Strength of Stock Market Movement
Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract  Stock market s one of the most complcated systems
More informationRiskbased Fatigue Estimate of Deep Water Risers  Course Project for EM388F: Fracture Mechanics, Spring 2008
Rskbased Fatgue Estmate of Deep Water Rsers  Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn
More informationAN OPTIMAL ALGORITHM FOR CONFLICTFREE COLORING FOR TREE OF RINGS
AN OPTIMAL ALGORITHM FOR CONFLICTFREE COLORING FOR TREE OF RINGS Enollah Pra The Busness Tranng Center of TabrzIran pra_ep2006@yahoo.com ABSTRACT An optmal algorthm s presented about ConflctFree Colorng
More informationA FASTER EXTERNAL SORTING ALGORITHM USING NO ADDITIONAL DISK SPACE
47 A FASTER EXTERAL SORTIG ALGORITHM USIG O ADDITIOAL DISK SPACE Md. Rafqul Islam +, Mohd. oor Md. Sap ++, Md. Sumon Sarker +, Sk. Razbul Islam + + Computer Scence and Engneerng Dscplne, Khulna Unversty,
More informationMulticlass MultiServer Thresholdbased Systems: a. Study of Noninstantaneous Server Activation
Multclass MultServer Thresholdbased Systems: a Study of Nonnstantaneous Server Actvaton 1 ChengFu Chou, Leana Golubchk, and John C. S. Lu Abstract In ths paper, we consder performance evaluaton of
More informationThe example below solves a system in the unknowns α and β:
The Fnd Functon The functon Fnd returns a soluton to a system of equatons gven by a solve block. You can use Fnd to solve a lnear system, as wth lsolve, or to solve nonlnear systems. The example below
More informationAn Algorithm for DataDriven Bandwidth Selection
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 25, NO. 2, FEBRUARY 2003 An Algorthm for DataDrven Bandwdth Selecton Dorn Comancu, Member, IEEE Abstract The analyss of a feature space
More informationA Constant Factor Approximation for the Single Sink Edge Installation Problem
A Constant Factor Approxmaton for the Sngle Snk Edge Installaton Problem Sudpto Guha Adam Meyerson Kamesh Munagala Abstract We present the frst constant approxmaton to the sngle snk buyatbulk network
More informationComplex Number Representation in RCBNS Form for Arithmetic Operations and Conversion of the Result into Standard Binary Form
Complex Number epresentaton n CBNS Form for Arthmetc Operatons and Converson of the esult nto Standard Bnary Form Hatm Zan and. G. Deshmukh Florda Insttute of Technology rgd@ee.ft.edu ABSTACT Ths paper
More informationA Programming Model for the Cloud Platform
Internatonal Journal of Advanced Scence and Technology A Programmng Model for the Cloud Platform Xaodong Lu School of Computer Engneerng and Scence Shangha Unversty, Shangha 200072, Chna luxaodongxht@qq.com
More informationSecure Network Coding Over the Integers
Secure Network Codng Over the Integers Rosaro Gennaro Jonathan Katz Hugo Krawczyk Tal Rabn Abstract Network codng has receved sgnfcant attenton n the networkng communty for ts potental to ncrease throughput
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationParticle Swarm Optimization for Scheduling to Minimize Tardiness Penalty and Power Cost
Partcle Swarm Optmzaton for Schedulng to Mnmze Tardness Penalty and Power Cost KueTang Fang and Bertrand M.T. Ln Department of Informaton and Fnance Management Insttute of Informaton Management Natonal
More informationThe Application of Fractional Brownian Motion in Option Pricing
Vol. 0, No. (05), pp. 738 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qngxn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com
More informationSketching Sampled Data Streams
Sketchng Sampled Data Streams Florn Rusu, Aln Dobra CISE Department Unversty of Florda Ganesvlle, FL, USA frusu@cse.ufl.edu adobra@cse.ufl.edu Abstract Samplng s used as a unversal method to reduce the
More informationII. PROBABILITY OF AN EVENT
II. PROBABILITY OF AN EVENT As ndcated above, probablty s a quantfcaton, or a mathematcal model, of a random experment. Ths quantfcaton s a measure of the lkelhood that a gven event wll occur when the
More informationGRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 NORM
GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 NORM BARRIOT JeanPerre, SARRAILH Mchel BGI/CNES 18.av.E.Beln 31401 TOULOUSE Cedex 4 (France) Emal: jeanperre.barrot@cnes.fr 1/Introducton The
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More informationEstimating projects duration in uncertain environments: Monte Carlo simulations strike back
Estmatng proects duraton n uncertan envronments: Monte Carlo smulatons strke back Stefana Tatton 1, Massmlano M. Schrald ( Tor Vergata Unversty of Rome, Dept. of Enterprse Engneerng, Va del Poltecnco,
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationSection C2: BJT Structure and Operational Modes
Secton 2: JT Structure and Operatonal Modes Recall that the semconductor dode s smply a pn juncton. Dependng on how the juncton s based, current may easly flow between the dode termnals (forward bas, v
More informationOn the Solution of Indefinite Systems Arising in Nonlinear Optimization
On the Soluton of Indefnte Systems Arsng n Nonlnear Optmzaton Slva Bonettn, Valera Ruggero and Federca Tnt Dpartmento d Matematca, Unverstà d Ferrara Abstract We consder the applcaton of the precondtoned
More informationEnabling P2P Oneview Multiparty Video Conferencing
Enablng P2P Onevew Multparty Vdeo Conferencng Yongxang Zhao, Yong Lu, Changja Chen, and JanYn Zhang Abstract MultParty Vdeo Conferencng (MPVC) facltates realtme group nteracton between users. Whle P2P
More informationAsynchronous Neighbor Discovery on Dutycycled Mobile Devices: Integer and NonInteger Schedules
Asynchronous Neghbor Dscovery on Dutycycled Moble Devces: Integer and NonInteger Schedules Sxa Chen Central Connectcut State Unversty schen@ccsu.edu Yanyuan Qn Unversty of Connectcut yanyuan.qn@uconn.edu
More informationNetwork Maximal Correlation Soheil Feizi, Ali Makhdoumi, Ken Duffy, Manolis Kellis, and Muriel Medard
Computer Scence and Artfcal Intellgence Laboratory Techncal Report MITCSAILTR205028 September 2, 205 Network Maxmal Correlaton Sohel Fez, Al Makhdoum, Ken Duffy, Manols Kells, and Murel Medard massachusetts
More informationPrediction of Wind Energy with Limited Observed Data
Predcton of Wnd Energy wth Lmted Observed Data Shgeto HIRI, khro HOND Nagasak R&D Center, MITSISHI HEVY INDSTRIES, LTD, Nagasak, 8539 JPN Masaak SHIT Nagasak Shpyard & Machnery Works, MITSISHI HEVY INDSTRIES,
More informationLinear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
More informationAn O(N log N) fast direct solver for partial Hierarchically SemiSeparable matrices With application to radial basis function interpolation
An O() fast drect solver for partal Herarchcally SemSeparable matrces Wth applcaton to radal bass functon nterpolaton Svaram Ambkasaran Erc Darve Receved: date / Accepted: date Ths artcle descrbes a fast
More informationOn File Delay Minimization for Content Uploading to Media Cloud via Collaborative Wireless Network
On Fle Delay Mnmzaton for Content Uploadng to Meda Cloud va Collaboratve Wreless Network Ge Zhang and Yonggang Wen School of Computer Engneerng Nanyang Technologcal Unversty Sngapore Emal: {zh0001ge, ygwen}@ntu.edu.sg
More informationPowerofTwo Policies for Single Warehouse MultiRetailer Inventory Systems with Order Frequency Discounts
Powerofwo Polces for Sngle Warehouse MultRetaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)
More informationA Fault Tree Analysis Strategy Using Binary Decision Diagrams.
A Fault Tree Analyss Strategy Usng Bnary Decson Dagrams. Karen A. Reay and John D. Andrews Loughborough Unversty, Loughborough, Lecestershre, LE 3TU. Abstract The use of Bnary Decson Dagrams (BDDs) n fault
More informationComment on Rotten Kids, Purity, and Perfection
Comment Comment on Rotten Kds, Purty, and Perfecton PerreAndré Chappor Unversty of Chcago Iván Wernng Unversty of Chcago and Unversdad Torcuato d Tella After readng Cornes and Slva (999), one gets the
More information