Starting algorithms for partitioned RungeKutta methods: the pair Lobatto IIIAIIIB


 Jodie Summers
 2 years ago
 Views:
Transcription
1 Monografías de la Real Academia de Ciencias de Zaragoza. : 49 8, 4). Starting algorithms for partitioned RungeKutta methods: the pair Lobatto IIIAIIIB I. Higueras and T. Roldán Departamento de Matemática e Informática. Universidad Pública de Navarra. 3 Pamplona, Spain. Abstract Partitioned RungeKutta PRK) methods are suitable for the numerical integration of separable Hamiltonian systems. For implicit PRK methods, the computational effort may be dominated by the cost of solving the nonlinear systems. For these nonlinear systems, good starting values minimize both the risk of a failed iteration and the effort required for convergence. In this paper we consider the Lobatto IIIAIIIB pair. Applying the theory developed by the authors for PRK methods, we obtain optimum predictors for this pair. Numerical experiments show the efficiency of these predictors. Key words and expressions: Starting algorithms, stage value predictors, partitioned RungeKutta methods, Hamiltonian systems. MSC: L, L, L8, H. Introduction We consider partitioned differential equations of the form y = fy, z) yx ) = y, z = gy, z) zx ) = z, ) where f : IR l IR m IR l and g : IR l IR m IR m are sufficiently smooth functions. An example of this type of systems are Hamiltonian systems p = H q, ) q = 49 H p, 3)
2 where the Hamiltonian function H has the special structure Hp, q) = Tp) + V q). In this case, the Hamiltonian system )3) is of the form p q = fq) = gp) with f = q V, g = p T. For the numerical integration of ) we consider partitioned RungeKutta PRK) methods. The idea of the PRK methods is to take two different RungeKutta RK) methods, A, b, Â,ˆb), and to treat the variables y with the first method, A, b), and the z variables with the second one, Â,ˆb). In this way, the numerical solution from t n, y n, z n ) to t n+, y n+, z n+ ) with the sstage PRK method A, b, Â,ˆb) is given by y n+ = y n + h b t I l )fy n+, Z n+ ), z n+ = z n + h ˆb t I m )gy n+, Z n+ ), where the internal stages vectors Y n+, Z n+ are obtained from the system Y n+ = e y n + h A I l )fy n+, Z n+ ), 4) Z n+ = e z n + h Â I m)gy n+, Z n+ ). ) As usual, we have denoted by the Kronecker product and by e =,..., ) t. For implicit PRK methods, the computational effort may be dominated by the cost of solving the nonlinear systems 4)). These nonlinear systems are solved with an iterative method that requires starting values Y ) n+, Z ) n+, and it is well known that these values must be as accurate as possible, because in other case, the number of iterations in each step may be too high or even worse, the convergence may fail. A common way to proceed is to take as starting values the last computed numerical solution, i.e., Y ) n+ = e y n, Z ) n+ = e z n. We will refer to them as the trivial predictors. However it is also possible to involve the last computed internal stages, Y n, Z n, and the approximation y n, and consider starting algorithms of the form Y ) n+ = b y n + B I l )Y n, ) Z ) n+ = c z n + C I m )Z n, 7) where b, c IR s, B and C are s s matrices which have to be determined. Several kinds of starting algorithms for RK methods, also called stage value predictors, have been studied by different authors [], [], [4], [7], [4], [3], [], [], [], []). The
3 starting algorithms considered for example in [], [] or [7] are of the form )7), whereas the ones considered for example in [], [4] or [] also involve function evaluations of the internal stages. Although in this paper we only deal with Hamiltonian systems without restrictions ODEs), we are also interested in Hamiltonian systems with restrictions DAEs). For ODE systems we can use any of the stage value predictors mentioned in the previous paragraph, but this is not the case for DAEs. Predictors involving function evaluations cannot be used for this type of systems [], [8]). This is the reason why we use predictors of the form )7). These predictors have already been studied for PRK methods in [9]. In this paper we consider the PRK methods Lobatto IIIAIIIB, that have been proved to be suitable ones for Hamiltonian systems [], [8]) and construct good starting algorithms for them. This is done in Section. The numerical experiments done in Section 3, show the efficiency of these starting algorithms. Starting Algorithms for Lobatto IIIAIIIB Methods We consider the PRK methods A, b, Â,ˆb), where A, b) is the Lobatto IIIA method and Â,ˆb) is the Lobatto IIB method. We will refer to this PRK method as the Lobatto IIIAIIB method. Observe that these methods share the same nodes, that is to say c = ĉ. Furthermore, the sstage Lobatto IIIA method satisfies Cs), and thus Ac q = c q /q, q =,..., s; whereas the sstage Lobatto IIIB method satisfies Cs ), and therefore Âĉ q = ĉ q /q, q =,..., s Figure : The pair Lobatto IIIAIIIB for s = Figure : The pair Lobatto IIIAIIIB for s = 4
4 In [9] the general order conditions for predictors of the form )7) have been obtained. Fortunately, these conditions are reduced considerably if the method PRK satisfies some simplifying assumptions, as it occurs with Lobatto IIIAIIIB methods. For s 3, the sstage Lobatto IIIA and IIIB methods satisfy at least C3) and C) respectively. In this case, the general order conditions in [9] for the variable y are reduced to the following ones Consistency: b + B e = e, Order : B c = e + r c, Order : B A c = eb t c + rae + rc), 8) Order 3: B Ac = eb t c + rae + rc), B AÂc = ebt Âc + raeb t c + r AÂe + rc), whereas the order conditions for the variable z are reduced to Consistency: c + C e = e, Order : C c = e + r c, Order : C Â c = eˆb t c + râe + rc), 9) Order 3: C Âc = eˆb t c + râe + rc), C Â c = eˆb t Âc + râeˆb t c + r Â e + rc). The parameter r = h n+ /h n, where h n+ denotes the current stepsize, h n+ = t n+ t n, and h n denotes the last stepsize, h n = t n t n. For s = 3, if we take into account the number of parameters available and the number of order conditions, we get that the maximum order achieved is for both variables, y and z. It is still possible to impose one of the two conditions of order 3, but not both of them. For s 4, the sstage Lobatto IIIA and IIIB methods satisfy at least C4) and C) respectively. Both of them satisfy b t c = ˆb t c = /. In this case, in 8) and 9) the two conditions of order 3 are equivalent. Hence for s = 4, it is possible to achieve order 3 for both variables, y and z. In order to simplify the implementation, it is interesting to initialize the variables y and z with the same coefficients. This situation corresponds to the case b = c and B = C in )7).
5 In that case, for s 3, up to order, the joint order conditions 8) and 9) are reduced to the following ones Consistency: b + B e = e, Order : B c = e + r c, Order : B A c = eb t c + rae + rc), B Â c = eˆb t c + râe + rc). Now it is easy to conclude that there exists a unique joint predictor of order for the case s = 3. This predictor is given by the following coefficients r r b = r3 + r), B = r + 3r) r + r) + 3r + r ). ) r + r) r + 3r) 4r + r) + 3r + r For s 4, up to order 3, the joint order conditions 8) and 9) are reduced to the following ones Consistency: b + B e = e, Order : B c = e + r c, Order : B A c = eb t c + rae + rc), Order 3: B Ac = eb t c + rae + rc), B Âc = eˆb t c + râe + rc). We immediately obtain that for s = 4 there exists a unique joint predictor of order 3. This predictor is given by the following coefficients r 3 ) + + r ) ) r r 3 b = + ) r ) ), ) r r 3 r 3 r 9 r 3 + r 3 b b b 3 b 4 B =, ) b 3 b 3 b 33 b 34 b 4 b 4 b 43 b 44 3
6 where b = + ) r 3 + ) r ) r 3, ) ) + 3 r 9 + r b = ) r 3, b 3 = r + 3 ) r + ) r 3, b 4 = ) r + 3 ) r + ) r 3, b 3 = + + ) r ) r b 3 = r ) r + + ) r 3, ) ) 3 r 9 r b 33 = + b 34 = ) r ) r ) r 3, + ) r 3, + ) r 3, b 4 = + r + r + 4r 3, b 4 = + ) r b 43 = + ) r b 44 = + r + r + r ) r + 3 ) r + r 3, r 3, For Lobatto IIIA, the first stage of the variable y is explicit. As Y n, = y n and Y n,s = y n, in ) and )) we get Y ) = y n, and therefore, as expected, the first stage of the variable y is not initialized. Observe that this is not the case for Lobatto IIIB method. In this case the first stage is implicit and the coefficients in ) and )) are necessary to obtain a predictor. 3 Numerical Experiments In this section we test the predictor constructed in the previous section for the 3stage pair Lobatto IIIAIIIB. We have selected an academic problem and the well known restricted threebody problem. In both cases we have proceeded in the same way. The stopping criteria for the Newton iterations is Y k) Y k) TOL. 4
7 The problems have been integrated for different values of T OL and for different stepsizes. In Tables  we show the average number of iterations per step needed when the trivial predictor is used, and the ones when the optimum predictor given by ) is used. For each value of TOL and h, the values on the left correspond to the trivial predictor, whereas the values on the right correspond to the optimum predictor given by ). 3. Problem The problem is given by y = 4z + t) + t y) = 3) z = y t z+t) z) = The exact solution is yt) = sint + t, zt) = cost t. We have integrated this problem for t [, ]. h TOL.E3.E.E7.E./../.9 3./..E3./../../..E3./../../..E3.898/../../.9 Table : Iterations per step. Problem It can be seen that except in one case, the number of iterations is lower with the predictor constructed in this paper. The exception is for h =.E3 and TOL =.E7 where the number of iterations coincides. We remark that in this case, the numerical solution obtained with the optimum predictor is more accurate. 3. The restricted threebody problem We have considered the restricted threebody problem from [3] x = v x x) = x y = v y y) = y z = v z z) = z v x = v y + x [ ] x+µ µ x µ + µ r 3 v r 3 x ) = v x v y = v x + y [ ] µ + µ r 3 r y 3 vy ) = v y v z = [ ] µ + µ r 3 r z 3 vz ) = v z 4)
8 where µ = µ, and r = x + µ ) + y + z r = x µ ) + y + z. We have integrated this problem for t [, ] with different values of the parameter µ and different initial conditions. Case I We take µ =.8 and the initial values.4,,,,, ). The numerical results are shown in Table. h TOL.E3.E.E7.E./.84.4/.3 3.9/.43.E3.8/.3.3/.8.874/.87.E3./..3/.49./..E3.93/../..77/.938 Table : Iterations per step: µ =.8 Case II We take µ =.9 and the initial values.4,,,,.99,.). The numerical results are shown in Table 3. h TOL.E3.E.E7.E./..94/.4.4/.74.E3./.3.49/.3.9/.3.E3.4/../..9/..E3.9/../.3.4/.37 Table 3: Iterations per step: µ =.9 Case III We take µ =.9994 and the initial values.74,,,,.43, ). The numerical results are shown in Table 4. h TOL.E.E7.E9.E./../../..E3./../../..E3./../../..E3./../../. Table 4: Iterations per step: µ =.9994
9 In Tables 4 we again see that the number of iterations needed when we use the predictors constructed in this paper is lower than the ones needed when the trivial predictor is used. We remark that the reduction in the number of iterations when solving the nonlinear systems implies a reduction in the total CPU time needed for the numerical integration, and hence it may be essential when long time integrations are done. Acknowledgments This research has been supported by the Ministerio de Ciencia y Tecnología, Project BFM88. References [] Calvo, M., Laburta, P. and Montijano, J. I., 3: Starting algorithms for Gauss RungeKutta methods for Hamiltonian systems, Computers and Mathematics with Applications 4, 4 4. [] Calvo, M., Laburta, P. and Montijano, J. I., 3: Twostep high order starting values for implicit RungeKutta methods, Advances in Computational Mathematics 9, 4 4. [3] Calvo, M. P., : High order starting iterates for implicit RungeKutta methods. An improvement for variablestep symplectic integrators, IMA Journal of Numerical Analysis, 3. [4] González Pinto, S., Montijano, J. I. and Rodríguez, S. P., : On the starting algorithms for fully implicit RungeKutta methods, BIT 4, [] González Pinto, S., Montijano, J. I. and Rodríguez, S. P., 3: Stabilized starting algorithms for collocation RungeKutta methods, Computers and Mathematics with Applications 4, [] González Pinto, S., Montijano, J. I. and Rodríguez, S. P., 3: Variable order starting algorithms for implicit RungeKutta methods on stiff problems, Applied Numerical Mathematics 44, [7] Higueras, I. and Roldán, T., : Starting algorithms for some DIRK methods, Numerical Algorithms 3, [8] Higueras, I. and Roldán, T., 3: IRK methods for index and 3 DAEs: Starting algorithms, BIT 43,
10 [9] Higueras, I. and Roldán, T., 3: Stage value predictors for additive Runge Kutta methods, submitted for publication. [] Jay, L., 99: Symplectic partitioned RungeKutta methods for constrained Hamiltonian systems, SIAM Journal on Numerical Analysis 33, [] Jay, L., 998: Structure preservation for constrained dynamics with super partitioned additive RungeKutta methods, SIAM Journal on Scientific Computing, [] Laburta, M., 997: Starting algorithms for IRK methods, Journal of Computational and Applied Mathematics 83, [3] Murray, C. D. and Dermott S. F., 999: Solar System Dynamics, Cambridge University Press, Cambridge. [4] Olsson, H. and Söderlind, G., 999: Stage value predictors and efficient Newton iterations in implicit RungeKutta methods, SIAM Journal on Scientific Computing, 8. [] Roldán, T. and Higueras, I., 999: IRK methods for DAE: Starting algorithms, Journal of Computational and Applied Mathematics, [] Sand, J., 99, Methods for starting iterations schemes for implicit RungeKutta formulae, Institute of Mathematical Applied Conference Series, New Series vol. 39, Oxford University Press, New York,. [7] Sanz Serna, J. and Calvo, M., 994: Numerical Hamiltonian Problems, Chapman & Hall, London. [8] Sun, G., 993: Symplectic partitioned Runge Kutta methods, Journal of Computational Mathematics,
AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS
AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEYINTERSCIENCE A John Wiley & Sons, Inc.,
More informationExtrasolar massive planets with small semimajor axes?
Monografías de la Real Academia de Ciencias de Zaragoza. 25: 115 120, (2004). Extrasolar massive planets with small semimajor axes? S. Fernández, D. Giuliodori and M. A. Nicotra Observatorio Astronómico.
More informationGeneral Framework for an Iterative Solution of Ax b. Jacobi s Method
2.6 Iterative Solutions of Linear Systems 143 2.6 Iterative Solutions of Linear Systems Consistent linear systems in real life are solved in one of two ways: by direct calculation (using a matrix factorization,
More informationOn the Efficient Implementation of Implicit RungeKutta Methods
MATHEMATICS OF COMPUTATION, VOLUME 33, NUMBER 146 APRIL 1979, PAGES 557561 On the Efficient Implementation of Implicit RungeKutta Methods By J. M. Varah Abstract. Extending some recent ideas of Butcher,
More information(Quasi)Newton methods
(Quasi)Newton methods 1 Introduction 1.1 Newton method Newton method is a method to find the zeros of a differentiable nonlinear function g, x such that g(x) = 0, where g : R n R n. Given a starting
More informationClassification and reduction of polynomial Hamiltonians with three degrees of freedom
Monografías de la Real Academia de Ciencias de Zaragoza. 22: 101 109, (2003). Classification and reduction of polynomial Hamiltonians with three degrees of freedom S. Gutiérrez, J. Palacián and P. Yanguas
More informationUniversity of Lille I PC first year list of exercises n 7. Review
University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients
More informationEnergy conserving time integration methods for the incompressible NavierStokes equations
Energy conserving time integration methods for the incompressible NavierStokes equations B. Sanderse WSC Spring Meeting, June 6, 2011, Delft June 2011 ECNM11064 Energy conserving time integration
More informationGeneral Theory of Differential Equations Sections 2.8, 3.13.2, 4.1
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.13.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions
More informationFamilies of symmetric periodic orbits in the three body problem and the figure eight
Monografías de la Real Academia de Ciencias de Zaragoza. 25: 229 24, (24). Families of symmetric periodic orbits in the three body problem and the figure eight F. J. MuñozAlmaraz, J. Galán and E. Freire
More informationSymmetric planar non collinear relative equilibria for the Lennard Jones potential 3 body problem with two equal masses
Monografías de la Real Academia de Ciencias de Zaragoza. 25: 93 114, (2004). Symmetric planar non collinear relative equilibria for the Lennard Jones potential 3 body problem with two equal masses M. Corbera,
More informationLEVERRIERFADEEV ALGORITHM AND CLASSICAL ORTHOGONAL POLYNOMIALS
This is a reprint from Revista de la Academia Colombiana de Ciencias Vol 8 (106 (004, 3947 LEVERRIERFADEEV ALGORITHM AND CLASSICAL ORTHOGONAL POLYNOMIALS by J Hernández, F Marcellán & C Rodríguez LEVERRIERFADEEV
More informationModern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh
Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem
More informationImplementation of the Bulirsch Stöer extrapolation method
Implementation of the Bulirsch Stöer extrapolation method Sujit Kirpekar Department of Mechanical Engineering kirpekar@newton.berkeley.edu December, 003 Introduction This paper outlines the pratical implementation
More informationNumerical Analysis An Introduction
Walter Gautschi Numerical Analysis An Introduction 1997 Birkhauser Boston Basel Berlin CONTENTS PREFACE xi CHAPTER 0. PROLOGUE 1 0.1. Overview 1 0.2. Numerical analysis software 3 0.3. Textbooks and monographs
More informationNumerically integrating equations of motion
Numerically integrating equations of motion 1 Introduction to numerical ODE integration algorithms Many models of physical processes involve differential equations: the rate at which some thing varies
More informationNumerical methods for American options
Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment
More informationImplicit RungeKutta Methods for Orbit Propagation
Implicit RungeKutta Methods for Orbit Propagation Jeffrey M. Aristoff and Aubrey B. Poore Numerica Corporation, 4850 Hahns Peak Drive, Suite 200, Loveland, Colorado, 80538, USA Accurate and efficient
More informationJoint models for classification and comparison of mortality in different countries.
Joint models for classification and comparison of mortality in different countries. Viani D. Biatat 1 and Iain D. Currie 1 1 Department of Actuarial Mathematics and Statistics, and the Maxwell Institute
More informationA simple application of the implicit function theorem
Boletín de la Asociación Matemática Venezolana, Vol. XIX, No. 1 (2012) 71 DIVULGACIÓN MATEMÁTICA A simple application of the implicit function theorem Germán LozadaCruz Abstract. In this note we show
More informationThe basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23
(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, highdimensional
More informationSIXTY STUDY QUESTIONS TO THE COURSE NUMERISK BEHANDLING AV DIFFERENTIALEKVATIONER I
Lennart Edsberg, Nada, KTH Autumn 2008 SIXTY STUDY QUESTIONS TO THE COURSE NUMERISK BEHANDLING AV DIFFERENTIALEKVATIONER I Parameter values and functions occurring in the questions belowwill be exchanged
More information10.3 POWER METHOD FOR APPROXIMATING EIGENVALUES
58 CHAPTER NUMERICAL METHODS. POWER METHOD FOR APPROXIMATING EIGENVALUES In Chapter 7 you saw that the eigenvalues of an n n matrix A are obtained by solving its characteristic equation n c nn c nn...
More informationCentral configuration in the planar n + 1 body problem with generalized forces.
Monografías de la Real Academia de Ciencias de Zaragoza. 28: 1 8, (2006). Central configuration in the planar n + 1 body problem with generalized forces. M. Arribas, A. Elipe Grupo de Mecánica Espacial.
More informationCS 29473 Software Engineering for Scientific Computing. http://www.cs.berkeley.edu/~colella/cs294fall2013. Lecture 16: Particle Methods; Homework #4
CS 29473 Software Engineering for Scientific Computing http://www.cs.berkeley.edu/~colella/cs294fall2013 Lecture 16: Particle Methods; Homework #4 Discretizing TimeDependent Problems From here on in,
More informationHomework 2 Solutions
Homework Solutions Igor Yanovsky Math 5B TA Section 5.3, Problem b: Use Taylor s method of order two to approximate the solution for the following initialvalue problem: y = + t y, t 3, y =, with h = 0.5.
More informationA new continuous dependence result for impulsive retarded functional differential equations
CADERNOS DE MATEMÁTICA 11, 37 47 May (2010) ARTIGO NÚMERO SMA#324 A new continuous dependence result for impulsive retarded functional differential equations M. Federson * Instituto de Ciências Matemáticas
More informationIs there chaos in Copenhagen problem?
Monografías de la Real Academia de Ciencias de Zaragoza 30, 43 50, (2006). Is there chaos in Copenhagen problem? Roberto Barrio, Fernando Blesa and Sergio Serrano GME, Universidad de Zaragoza Abstract
More informationNUMERICAL ANALYSIS (Syllabus for the academic years and onwards)
ACHARYA NAGARJUNA UNIVERSITY CURRICULUM  B.A / B.Sc MATHEMATICS  PAPER  IV (ELECTIVE  ) NUMERICAL ANALYSIS (Syllabus for the academic years  and onwards) Paper IV (Elective )  Curriculum 9 Hours
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationArticle from: ARCH 2014.1 Proceedings
Article from: ARCH 20141 Proceedings July 31August 3, 2013 1 Comparison of the Standard Rating Methods and the New General Rating Formula Muhamed Borogovac Boston Mutual Life Insurance Company*; muhamed_borogovac@bostonmutualcom
More informationG.A. Pavliotis. Department of Mathematics. Imperial College London
EE1 MATHEMATICS NUMERICAL METHODS G.A. Pavliotis Department of Mathematics Imperial College London 1. Numerical solution of nonlinear equations (iterative processes). 2. Numerical evaluation of integrals.
More informationA Simple Observation Concerning Contraction Mappings
Revista Colombiana de Matemáticas Volumen 46202)2, páginas 229233 A Simple Observation Concerning Contraction Mappings Una simple observación acerca de las contracciones German LozadaCruz a Universidade
More information3.2 Sources, Sinks, Saddles, and Spirals
3.2. Sources, Sinks, Saddles, and Spirals 6 3.2 Sources, Sinks, Saddles, and Spirals The pictures in this section show solutions to Ay 00 C By 0 C Cy D 0. These are linear equations with constant coefficients
More informationThe Right Scalene Triangle Problem
The Right Scalene Triangle Problem This paper solves the problem of the right scalene triangle through a general sequential solution. A simplified solution is also presented. by R. A. Frino November 24
More informationReference: Introduction to Partial Differential Equations by G. Folland, 1995, Chap. 3.
5 Potential Theory Reference: Introduction to Partial Differential Equations by G. Folland, 995, Chap. 3. 5. Problems of Interest. In what follows, we consider Ω an open, bounded subset of R n with C 2
More informationStability of the LMS Adaptive Filter by Means of a State Equation
Stability of the LMS Adaptive Filter by Means of a State Equation Vítor H. Nascimento and Ali H. Sayed Electrical Engineering Department University of California Los Angeles, CA 90095 Abstract This work
More informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationORDINARY DIFFERENTIAL EQUATIONS
ORDINARY DIFFERENTIAL EQUATIONS GABRIEL NAGY Mathematics Department, Michigan State University, East Lansing, MI, 48824. SEPTEMBER 4, 25 Summary. This is an introduction to ordinary differential equations.
More informationMATH1231 Algebra, 2015 Chapter 7: Linear maps
MATH1231 Algebra, 2015 Chapter 7: Linear maps A/Prof. Daniel Chan School of Mathematics and Statistics University of New South Wales danielc@unsw.edu.au Daniel Chan (UNSW) MATH1231 Algebra 1 / 43 Chapter
More informationESSENTIAL COMPUTATIONAL FLUID DYNAMICS
ESSENTIAL COMPUTATIONAL FLUID DYNAMICS Oleg Zikanov WILEY JOHN WILEY & SONS, INC. CONTENTS PREFACE xv 1 What Is CFD? 1 1.1. Introduction / 1 1.2. Brief History of CFD / 4 1.3. Outline of the Book / 6 References
More informationWe shall turn our attention to solving linear systems of equations. Ax = b
59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system
More informationDEFORMATION OF DIRAC STRUCTURES ALONG ISOTROPIC SUBBUNDLES. and MARCO ZAMBON
Vol. 65 (2010) REPORTS ON MATHEMATICAL PHYSICS No. 2 DEFORMATION OF DIRAC STRUCTURES ALONG ISOTROPIC SUBBUNDLES IVÁN CALVO Laboratorio Nacional de Fusión, Asociación EURATOMCIEMAT, E28040 Madrid, Spain
More informationINTERPOLATION. Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y).
INTERPOLATION Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y). As an example, consider defining and x 0 =0, x 1 = π 4, x
More informationRoots of Polynomials
Roots of Polynomials (Com S 477/577 Notes) YanBin Jia Sep 24, 2015 A direct corollary of the fundamental theorem of algebra is that p(x) can be factorized over the complex domain into a product a n (x
More informationMaximum Likelihood Estimation of an ARMA(p,q) Model
Maximum Likelihood Estimation of an ARMA(p,q) Model Constantino Hevia The World Bank. DECRG. October 8 This note describes the Matlab function arma_mle.m that computes the maximum likelihood estimates
More informationPractical Guide to the Simplex Method of Linear Programming
Practical Guide to the Simplex Method of Linear Programming Marcel Oliver Revised: April, 0 The basic steps of the simplex algorithm Step : Write the linear programming problem in standard form Linear
More informationH.Calculating Normal Vectors
Appendix H H.Calculating Normal Vectors This appendix describes how to calculate normal vectors for surfaces. You need to define normals to use the OpenGL lighting facility, which is described in Chapter
More informationAdvanced CFD Methods 1
Advanced CFD Methods 1 Prof. Patrick Jenny, FS 2014 Date: 15.08.14, Time: 13:00, Student: Federico Danieli Summary The exam took place in Prof. Jenny s office, with his assistant taking notes on the answers.
More information10.2 ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS. The Jacobi Method
578 CHAPTER 1 NUMERICAL METHODS 1. ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS As a numerical technique, Gaussian elimination is rather unusual because it is direct. That is, a solution is obtained after
More informationNumerical Analysis Introduction. Student Audience. Prerequisites. Technology.
Numerical Analysis Douglas Faires, Youngstown State University, (Chair, 20122013) Elizabeth Yanik, Emporia State University, (Chair, 20132015) Graeme Fairweather, Executive Editor, Mathematical Reviews,
More informationON GALOIS REALIZATIONS OF THE 2COVERABLE SYMMETRIC AND ALTERNATING GROUPS
ON GALOIS REALIZATIONS OF THE 2COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for
More informationDATA ANALYSIS II. Matrix Algorithms
DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where
More information24. The Branch and Bound Method
24. The Branch and Bound Method It has serious practical consequences if it is known that a combinatorial problem is NPcomplete. Then one can conclude according to the present state of science that no
More informationLecture 7: Finding Lyapunov Functions 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1
More informationMath 209 Solutions to Assignment 7. x + 2y. 1 x + 2y i + 2. f x = cos(y/z)), f y = x z sin(y/z), f z = xy z 2 sin(y/z).
Math 29 Solutions to Assignment 7. Find the gradient vector field of the following functions: a fx, y lnx + 2y; b fx, y, z x cosy/z. Solution. a f x x + 2y, f 2 y x + 2y. Thus, the gradient vector field
More informationFACTORING PEAK POLYNOMIALS
FACTORING PEAK POLYNOMIALS SARA BILLEY, MATTHEW FAHRBACH, AND ALAN TALMAGE Abstract. Let S n be the symmetric group of permutations π = π 1 π 2 π n of {1, 2,..., n}. An index i of π is a peak if π i 1
More information9. Particular Solutions of Nonhomogeneous second order equations Undetermined Coefficients
September 29, 201 91 9. Particular Solutions of Nonhomogeneous second order equations Undetermined Coefficients We have seen that in order to find the general solution to the second order differential
More informationDOCUMENTS DE TREBALL DE LA FACULTAT D ECONOMIA I EMPRESA. Col.lecció d Economia
DOCUMENTS DE TREBALL DE LA FACULTAT D ECONOMIA I EMPRESA Col.lecció d Economia E13/89 On the nucleolus of x assignment games F. Javier Martínez de Albéniz Carles Rafels Neus Ybern F. Javier Martínez de
More informationUNCERTAINTY TREATMENT IN STOCK MARKET MODELING
UNCERTAINTY TREATMENT IN STOCK MARKET MODELING Stanislaw Raczynski Universidad Panamericana Augusto Rodin 498, 03910 Mexico City, Mexico email: stanracz@netservice.com.mx Abstract A model of stock market
More informationLecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 10 Boundary Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction
More informationGEC320 COURSE COMPACT. Four hours per week for 15 weeks (60 hours)
GEC320 COURSE COMPACT Course Course code: GEC 320 Course title: Course status: Course Duration Numerical Methods (2 units) Compulsory Four hours per week for 15 weeks (60 hours) Lecturer Data Name: Engr.
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign
More informationComputing divisors and common multiples of quasilinear ordinary differential equations
Computing divisors and common multiples of quasilinear ordinary differential equations Dima Grigoriev CNRS, Mathématiques, Université de Lille Villeneuve d Ascq, 59655, France Dmitry.Grigoryev@math.univlille1.fr
More information5. Factoring by the QF method
5. Factoring by the QF method 5.0 Preliminaries 5.1 The QF view of factorability 5.2 Illustration of the QF view of factorability 5.3 The QF approach to factorization 5.4 Alternative factorization by the
More informationPerformance of collocation software for singular BVPs
Performance of collocation software for singular BVPs Winfried Auzinger Othmar Koch Dirk Praetorius Gernot Pulverer Ewa Weinmüller Technical Report ANUM Preprint No. / Institute for Analysis and Scientific
More informationLecture Notes on Polynomials
Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex
More informationRow Ideals and Fibers of Morphisms
Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion
More informationU.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
More informationFigure 2.1: Center of mass of four points.
Chapter 2 Bézier curves are named after their inventor, Dr. Pierre Bézier. Bézier was an engineer with the Renault car company and set out in the early 196 s to develop a curve formulation which would
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationMINIMAL BOOKS OF RATIONALES
MINIMAL BOOKS OF RATIONALES José Apesteguía Miguel A. Ballester D.T.2005/01 MINIMAL BOOKS OF RATIONALES JOSE APESTEGUIA AND MIGUEL A. BALLESTER Abstract. Kalai, Rubinstein, and Spiegler (2002) propose
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationCorrections to the First Printing
Corrections to the First Printing Chapter 2 (i) Page 48, Paragraph 1: cells/µ l should be cells/µl without the space. (ii) Page 48, Paragraph 2: Uninfected cells T i should not have the asterisk. Chapter
More information1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form
1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and
More informationFOR ACCELERATION OF ITERATIVE ALGORITHMS
DISTRIBUTED MINIMAL RESIDUAL (DMR) METHOD Q/ 2% FOR ACCELERATION OF ITERATIVE ALGORITHMS Seungsoo Lee and George S. Dulikravich Pennsylvania State University University Park, Pennsylvania, U.S.A. ABSTRACT
More informationOptimization of Supply Chain Networks
Optimization of Supply Chain Networks M. Herty TU Kaiserslautern September 2006 (2006) 1 / 41 Contents 1 Supply Chain Modeling 2 Networks 3 Optimization Continuous optimal control problem Discrete optimal
More informationFACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z
FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization
More informationLINEAR SYSTEMS. Consider the following example of a linear system:
LINEAR SYSTEMS Consider the following example of a linear system: Its unique solution is x +2x 2 +3x 3 = 5 x + x 3 = 3 3x + x 2 +3x 3 = 3 x =, x 2 =0, x 3 = 2 In general we want to solve n equations in
More informationHeriotWatt University. M.Sc. in Actuarial Science. Life Insurance Mathematics I. Tutorial 5
1 HeriotWatt University M.Sc. in Actuarial Science Life Insurance Mathematics I Tutorial 5 1. Consider the illnessdeath model in Figure 1. A life age takes out a policy with a term of n years that pays
More informationMain concepts: Linear systems, nonlinear systems, BCH theorem, order conditions
Chapter 13 Splitting Methods Main concepts: Linear sstems, nonlinear sstems, BCH theorem, order conditions In this section, we present a natural approach to method building which is based on splitting
More informationThe Matrix Elements of a 3 3 Orthogonal Matrix Revisited
Physics 116A Winter 2011 The Matrix Elements of a 3 3 Orthogonal Matrix Revisited 1. Introduction In a class handout entitled, ThreeDimensional Proper and Improper Rotation Matrices, I provided a derivation
More informationAdaptive Online Gradient Descent
Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationNumerical simulation of the manoeuvrability of an underwater vehicle
XXI Congreso de Ecuaciones Diferenciales y Aplicaciones XI Congreso de Matemática Aplicada Ciudad Real, 2125 septiembre 29 (pp. 1 8) Numerical simulation of the manoeuvrability of an underwater vehicle
More informationA Learning Based Method for SuperResolution of Low Resolution Images
A Learning Based Method for SuperResolution of Low Resolution Images Emre Ugur June 1, 2004 emre.ugur@ceng.metu.edu.tr Abstract The main objective of this project is the study of a learning based method
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION Introduction In the previous chapter, we explored a class of regression models having particularly simple analytical
More informationABSTRACT FOR THE 1ST INTERNATIONAL WORKSHOP ON HIGH ORDER CFD METHODS
1 ABSTRACT FOR THE 1ST INTERNATIONAL WORKSHOP ON HIGH ORDER CFD METHODS Sreenivas Varadan a, Kentaro Hara b, Eric Johnsen a, Bram Van Leer b a. Department of Mechanical Engineering, University of Michigan,
More informationNonlinear Model Predictive Control of Hammerstein and Wiener Models Using Genetic Algorithms
Nonlinear Model Predictive Control of Hammerstein and Wiener Models Using Genetic Algorithms AlDuwaish H. and Naeem, Wasif Electrical Engineering Department/King Fahd University of Petroleum and Minerals
More informationprinceton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora
princeton univ. F 13 cos 521: Advanced Algorithm Design Lecture 6: Provable Approximation via Linear Programming Lecturer: Sanjeev Arora Scribe: One of the running themes in this course is the notion of
More informationWorking Paper Series
RGEA Universidade de Vigo http://webs.uvigo.es/rgea Working Paper Series A Market Game Approach to Differential Information Economies Guadalupe Fugarolas, Carlos HervésBeloso, Emma Moreno García and
More informationGeneralized Inverse Computation Based on an Orthogonal Decomposition Methodology.
International Conference on Mathematical and Statistical Modeling in Honor of Enrique Castillo. June 2830, 2006 Generalized Inverse Computation Based on an Orthogonal Decomposition Methodology. Patricia
More information3 Polynomial Interpolation
3 Polynomial Interpolation Read sections 7., 7., 7.3. 7.3.3 (up to p. 39), 7.3.5. Review questions 7. 7.4, 7.8 7.0, 7. 7.4, 7.7, 7.8. All methods for solving ordinary differential equations which we considered
More informationBINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract
BINOMIAL OPTIONS PRICING MODEL Mark Ioffe Abstract Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple
More information(57) (58) (59) (60) (61)
Module 4 : Solving Linear Algebraic Equations Section 5 : Iterative Solution Techniques 5 Iterative Solution Techniques By this approach, we start with some initial guess solution, say for solution and
More informationLecture 3: Linear methods for classification
Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,
More informationValuation of American Options
Valuation of American Options Among the seminal contributions to the mathematics of finance is the paper F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political
More informationError Control and Adaptivity for Reduced Basis Approximations of Parametrized Evolution Equations. Mario Ohlberger
Error Control and Adaptivity for Reduced Basis Approximations of Parametrized Evolution Equations Mario Ohlberger In cooperation with: M. Dihlmann, M. Drohmann, B. Haasdonk, G. Rozza Workshop on A posteriori
More informationInstituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC Coimbra
Instituto de Engenharia de Sistemas e Computadores de Coimbra Institute of Systems Engineering and Computers INESC Coimbra João Clímaco and Marta Pascoal A new method to detere unsupported nondoated solutions
More informationThe equivalence of logistic regression and maximum entropy models
The equivalence of logistic regression and maximum entropy models John Mount September 23, 20 Abstract As our colleague so aptly demonstrated ( http://www.winvector.com/blog/20/09/thesimplerderivationoflogisticregression/
More information