The Computation of Matrix Function in Particular the Matrix Expotential
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Regular Linear Systems on Cp1 and Their Monodromy Groups V.P
Astérisque V. P. KOSTOV Regular linear systems onCP1 and their monodromy groups Astérisque, tome 222 (1994), p. 259-283 <http://www.numdam.org/item?id=AST_1994__222__259_0> © Société mathématique de France, 1994, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ REGULAR LINEAR SYSTEMS ON CP1 AND THEIR MONODROMY GROUPS V.P. KOSTOV 1. INTRODUCTION 1.1 A meromorphic linear system of differential equations on CP1 can be pre sented in the form X = A(t)X (1) where A{t) is a meromorphic on CP1 n x n matrix function, " • " = d/dt. Denote its poles ai,... ,ap+i, p > 1. We consider the dependent variable X to be also n x n-matrix. Definition. System (1) is called fuchsian if all the poles of the matrix- function A(t) axe of first order. Definition. System (1) is called regular at the pole a,j if in its neighbour hood the solutions of the system are of moderate growth rate, i.e. IW-a^l^Odt-a^), Ni E R, j = 1,...., p + 1 Here || • || denotes an arbitrary norm in gl(n, C) and we consider a restriction of the solution to a sector with vertex at ctj and of a sufficiently small radius, i.e. -
The Rational and Jordan Forms Linear Algebra Notes
The Rational and Jordan Forms Linear Algebra Notes Satya Mandal November 5, 2005 1 Cyclic Subspaces In a given context, a "cyclic thing" is an one generated "thing". For example, a cyclic groups is a one generated group. Likewise, a module M over a ring R is said to be a cyclic module if M is one generated or M = Rm for some m 2 M: We do not use the expression "cyclic vector spaces" because one generated vector spaces are zero or one dimensional vector spaces. 1.1 (De¯nition and Facts) Suppose V is a vector space over a ¯eld F; with ¯nite dim V = n: Fix a linear operator T 2 L(V; V ): 1. Write R = F[T ] = ff(T ) : f(X) 2 F[X]g L(V; V )g: Then R = F[T ]g is a commutative ring. (We did considered this ring in last chapter in the proof of Caley-Hamilton Theorem.) 2. Now V acquires R¡module structure with scalar multiplication as fol- lows: Define f(T )v = f(T )(v) 2 V 8 f(T ) 2 F[T ]; v 2 V: 3. For an element v 2 V de¯ne Z(v; T ) = F[T ]v = ff(T )v : f(T ) 2 Rg: 1 Note that Z(v; T ) is the cyclic R¡submodule generated by v: (I like the notation F[T ]v, the textbook uses the notation Z(v; T ).) We say, Z(v; T ) is the T ¡cyclic subspace generated by v: 4. If V = Z(v; T ) = F[T ]v; we say that that V is a T ¡cyclic space, and v is called the T ¡cyclic generator of V: (Here, I di®er a little from the textbook.) 5. -
Polynomial Flows in the Plane
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ADVANCES IN MATHEMATICS 55, 173-208 (1985) Polynomial Flows in the Plane HYMAN BASS * Department of Mathematics, Columbia University. New York, New York 10027 AND GARY MEISTERS Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588 Contents. 1. Introduction. I. Polynomial flows are global, of bounded degree. 2. Vector fields and local flows. 3. Change of coordinates; the group GA,(K). 4. Polynomial flows; statement of the main results. 5. Continuous families of polynomials. 6. Locally polynomial flows are global, of bounded degree. II. One parameter subgroups of GA,(K). 7. Introduction. 8. Amalgamated free products. 9. GA,(K) as amalgamated free product. 10. One parameter subgroups of GA,(K). 11. One parameter subgroups of BA?(K). 12. One parameter subgroups of BA,(K). 1. Introduction Let f: Rn + R be a Cl-vector field, and consider the (autonomous) system of differential equations with initial condition x(0) = x0. (lb) The solution, x = cp(t, x,), depends on t and x0. For which f as above does the flow (p depend polynomially on the initial condition x,? This question was discussed in [M2], and in [Ml], Section 6. We present here a definitive solution of this problem for n = 2, over both R and C. (See Theorems (4.1) and (4.3) below.) The main tool is the theorem of Jung [J] and van der Kulk [vdK] * This material is based upon work partially supported by the National Science Foun- dation under Grant NSF MCS 82-02633. -
Linear Systems and Control: a First Course (Course Notes for AAE 564)
Linear Systems and Control: A First Course (Course notes for AAE 564) Martin Corless School of Aeronautics & Astronautics Purdue University West Lafayette, Indiana [email protected] August 25, 2008 Contents 1 Introduction 1 1.1 Ingredients..................................... 4 1.2 Somenotation................................... 4 1.3 MATLAB ..................................... 4 2 State space representation of dynamical systems 5 2.1 Linearexamples.................................. 5 2.1.1 Afirstexample .............................. 5 2.1.2 Theunattachedmass........................... 6 2.1.3 Spring-mass-damper ........................... 6 2.1.4 Asimplestructure ............................ 7 2.2 Nonlinearexamples............................... 8 2.2.1 Afirstnonlinearsystem ......................... 8 2.2.2 Planarpendulum ............................. 9 2.2.3 Attitudedynamicsofarigidbody. .. 9 2.2.4 Bodyincentralforcemotion. 10 2.2.5 Ballisticsindrag ............................. 11 2.2.6 Doublependulumoncart . .. .. 12 2.2.7 Two-linkroboticmanipulator . .. 13 2.3 Discrete-timeexamples . ... 14 2.3.1 Thediscreteunattachedmass . 14 2.4 Generalrepresentation . ... 15 2.4.1 Continuous-time ............................. 15 2.4.2 Discrete-time ............................... 18 2.4.3 Exercises.................................. 20 2.5 Vectors....................................... 22 2.5.1 Vector spaces and IRn .......................... 22 2.5.2 IR2 andpictures.............................. 24 2.5.3 Derivatives ............................... -
MAT247 Algebra II Assignment 5 Solutions
MAT247 Algebra II Assignment 5 Solutions 1. Find the Jordan canonical form and a Jordan basis for the map or matrix given in each part below. (a) Let V be the real vector space spanned by the polynomials xiyj (in two variables) with i + j ≤ 3. Let T : V V be the map Dx + Dy, where Dx and Dy respectively denote 2 differentiation with respect to x and y. (Thus, Dx(xy) = y and Dy(xy ) = 2xy.) 0 1 1 1 1 1 ! B0 1 2 1 C (b) A = B C over Q @0 0 1 -1A 0 0 0 2 Solution: (a) The operator T is nilpotent so its characterictic polynomial splits and its only eigenvalue is zero and K0 = V. We have Im(T) = spanf1; x; y; x2; xy; y2g; Im(T 2) = spanf1; x; yg Im(T 3) = spanf1g T 4 = 0: Thus the longest cycle for eigenvalue zero has length 4. Moreover, since the number of cycles of length at least r is given by dim(Im(T r-1)) - dim(Im(T r)), the number of cycles of lengths at least 4, 3, 2, 1 is respectively 1, 2, 3, and 4. Thus the number of cycles of lengths 4,3,2,1 is respectively 1,1,1,1. Denoting the r × r Jordan block with λ on the diagonal by Jλ,r, the Jordan canonical form is thus 0 1 J0;4 B J0;3 C J = B C : @ J0;2 A J0;1 We now proceed to find a corresponding Jordan basis, i.e. -
Quantum Information
Quantum Information J. A. Jones Michaelmas Term 2010 Contents 1 Dirac Notation 3 1.1 Hilbert Space . 3 1.2 Dirac notation . 4 1.3 Operators . 5 1.4 Vectors and matrices . 6 1.5 Eigenvalues and eigenvectors . 8 1.6 Hermitian operators . 9 1.7 Commutators . 10 1.8 Unitary operators . 11 1.9 Operator exponentials . 11 1.10 Physical systems . 12 1.11 Time-dependent Hamiltonians . 13 1.12 Global phases . 13 2 Quantum bits and quantum gates 15 2.1 The Bloch sphere . 16 2.2 Density matrices . 16 2.3 Propagators and Pauli matrices . 18 2.4 Quantum logic gates . 18 2.5 Gate notation . 21 2.6 Quantum networks . 21 2.7 Initialization and measurement . 23 2.8 Experimental methods . 24 3 An atom in a laser field 25 3.1 Time-dependent systems . 25 3.2 Sudden jumps . 26 3.3 Oscillating fields . 27 3.4 Time-dependent perturbation theory . 29 3.5 Rabi flopping and Fermi's Golden Rule . 30 3.6 Raman transitions . 32 3.7 Rabi flopping as a quantum gate . 32 3.8 Ramsey fringes . 33 3.9 Measurement and initialisation . 34 1 CONTENTS 2 4 Spins in magnetic fields 35 4.1 The nuclear spin Hamiltonian . 35 4.2 The rotating frame . 36 4.3 On-resonance excitation . 38 4.4 Excitation phases . 38 4.5 Off-resonance excitation . 39 4.6 Practicalities . 40 4.7 The vector model . 40 4.8 Spin echoes . 41 4.9 Measurement and initialisation . 42 5 Photon techniques 43 5.1 Spatial encoding . -
New Algorithms in the Frobenius Matrix Algebra for Polynomial Root-Finding
City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports CUNY Academic Works 2014 TR-2014006: New Algorithms in the Frobenius Matrix Algebra for Polynomial Root-Finding Victor Y. Pan Ai-Long Zheng How does access to this work benefit ou?y Let us know! More information about this work at: https://academicworks.cuny.edu/gc_cs_tr/397 Discover additional works at: https://academicworks.cuny.edu This work is made publicly available by the City University of New York (CUNY). Contact: [email protected] New Algoirthms in the Frobenius Matrix Algebra for Polynomial Root-finding ∗ Victor Y. Pan[1,2],[a] and Ai-Long Zheng[2],[b] Supported by NSF Grant CCF-1116736 and PSC CUNY Award 64512–0042 [1] Department of Mathematics and Computer Science Lehman College of the City University of New York Bronx, NY 10468 USA [2] Ph.D. Programs in Mathematics and Computer Science The Graduate Center of the City University of New York New York, NY 10036 USA [a] [email protected] http://comet.lehman.cuny.edu/vpan/ [b] [email protected] Abstract In 1996 Cardinal applied fast algorithms in Frobenius matrix algebra to complex root-finding for univariate polynomials, but he resorted to some numerically unsafe techniques of symbolic manipulation with polynomials at the final stages of his algorithms. We extend his work to complete the computations by operating with matrices at the final stage as well and also to adjust them to real polynomial root-finding. Our analysis and experiments show efficiency of the resulting algorithms. 2000 Math. -
On Multivariate Interpolation
On Multivariate Interpolation Peter J. Olver† School of Mathematics University of Minnesota Minneapolis, MN 55455 U.S.A. [email protected] http://www.math.umn.edu/∼olver Abstract. A new approach to interpolation theory for functions of several variables is proposed. We develop a multivariate divided difference calculus based on the theory of non-commutative quasi-determinants. In addition, intriguing explicit formulae that connect the classical finite difference interpolation coefficients for univariate curves with multivariate interpolation coefficients for higher dimensional submanifolds are established. † Supported in part by NSF Grant DMS 11–08894. April 6, 2016 1 1. Introduction. Interpolation theory for functions of a single variable has a long and distinguished his- tory, dating back to Newton’s fundamental interpolation formula and the classical calculus of finite differences, [7, 47, 58, 64]. Standard numerical approximations to derivatives and many numerical integration methods for differential equations are based on the finite dif- ference calculus. However, historically, no comparable calculus was developed for functions of more than one variable. If one looks up multivariate interpolation in the classical books, one is essentially restricted to rectangular, or, slightly more generally, separable grids, over which the formulae are a simple adaptation of the univariate divided difference calculus. See [19] for historical details. Starting with G. Birkhoff, [2] (who was, coincidentally, my thesis advisor), recent years have seen a renewed level of interest in multivariate interpolation among both pure and applied researchers; see [18] for a fairly recent survey containing an extensive bibli- ography. De Boor and Ron, [8, 12, 13], and Sauer and Xu, [61, 10, 65], have systemati- cally studied the polynomial case. -
MATH 237 Differential Equations and Computer Methods
Queen’s University Mathematics and Engineering and Mathematics and Statistics MATH 237 Differential Equations and Computer Methods Supplemental Course Notes Serdar Y¨uksel November 19, 2010 This document is a collection of supplemental lecture notes used for Math 237: Differential Equations and Computer Methods. Serdar Y¨uksel Contents 1 Introduction to Differential Equations 7 1.1 Introduction: ................................... ....... 7 1.2 Classification of Differential Equations . ............... 7 1.2.1 OrdinaryDifferentialEquations. .......... 8 1.2.2 PartialDifferentialEquations . .......... 8 1.2.3 Homogeneous Differential Equations . .......... 8 1.2.4 N-thorderDifferentialEquations . ......... 8 1.2.5 LinearDifferentialEquations . ......... 8 1.3 Solutions of Differential equations . .............. 9 1.4 DirectionFields................................. ........ 10 1.5 Fundamental Questions on First-Order Differential Equations............... 10 2 First-Order Ordinary Differential Equations 11 2.1 ExactDifferentialEquations. ........... 11 2.2 MethodofIntegratingFactors. ........... 12 2.3 SeparableDifferentialEquations . ............ 13 2.4 Differential Equations with Homogenous Coefficients . ................ 13 2.5 First-Order Linear Differential Equations . .............. 14 2.6 Applications.................................... ....... 14 3 Higher-Order Ordinary Linear Differential Equations 15 3.1 Higher-OrderDifferentialEquations . ............ 15 3.1.1 LinearIndependence . ...... 16 3.1.2 WronskianofasetofSolutions . ........ 16 3.1.3 Non-HomogeneousProblem -
The Exponential of a Matrix
5-28-2012 The Exponential of a Matrix The solution to the exponential growth equation dx kt = kx is given by x = c e . dt 0 It is natural to ask whether you can solve a constant coefficient linear system ′ ~x = A~x in a similar way. If a solution to the system is to have the same form as the growth equation solution, it should look like At ~x = e ~x0. The first thing I need to do is to make sense of the matrix exponential eAt. The Taylor series for ez is ∞ n z z e = . n! n=0 X It converges absolutely for all z. It A is an n × n matrix with real entries, define ∞ n n At t A e = . n! n=0 X The powers An make sense, since A is a square matrix. It is possible to show that this series converges for all t and every matrix A. Differentiating the series term-by-term, ∞ ∞ ∞ ∞ n−1 n n−1 n n−1 n−1 m m d At t A t A t A t A At e = n = = A = A = Ae . dt n! (n − 1)! (n − 1)! m! n=0 n=1 n=1 m=0 X X X X At ′ This shows that e solves the differential equation ~x = A~x. The initial condition vector ~x(0) = ~x0 yields the particular solution At ~x = e ~x0. This works, because e0·A = I (by setting t = 0 in the power series). Another familiar property of ordinary exponentials holds for the matrix exponential: If A and B com- mute (that is, AB = BA), then A B A B e e = e + . -
Scientific Computing: an Introductory Survey
Eigenvalue Problems Existence, Uniqueness, and Conditioning Computing Eigenvalues and Eigenvectors Scientific Computing: An Introductory Survey Chapter 4 – Eigenvalue Problems Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted for noncommercial, educational use only. Michael T. Heath Scientific Computing 1 / 87 Eigenvalue Problems Existence, Uniqueness, and Conditioning Computing Eigenvalues and Eigenvectors Outline 1 Eigenvalue Problems 2 Existence, Uniqueness, and Conditioning 3 Computing Eigenvalues and Eigenvectors Michael T. Heath Scientific Computing 2 / 87 Eigenvalue Problems Eigenvalue Problems Existence, Uniqueness, and Conditioning Eigenvalues and Eigenvectors Computing Eigenvalues and Eigenvectors Geometric Interpretation Eigenvalue Problems Eigenvalue problems occur in many areas of science and engineering, such as structural analysis Eigenvalues are also important in analyzing numerical methods Theory and algorithms apply to complex matrices as well as real matrices With complex matrices, we use conjugate transpose, AH , instead of usual transpose, AT Michael T. Heath Scientific Computing 3 / 87 Eigenvalue Problems Eigenvalue Problems Existence, Uniqueness, and Conditioning Eigenvalues and Eigenvectors Computing Eigenvalues and Eigenvectors Geometric Interpretation Eigenvalues and Eigenvectors Standard eigenvalue problem : Given n × n matrix A, find scalar λ and nonzero vector x such that A x = λ x λ is eigenvalue, and x is corresponding eigenvector -
Stabilization, Estimation and Control of Linear Dynamical Systems with Positivity and Symmetry Constraints
Stabilization, Estimation and Control of Linear Dynamical Systems with Positivity and Symmetry Constraints A Dissertation Presented by Amirreza Oghbaee to The Department of Electrical and Computer Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering Northeastern University Boston, Massachusetts April 2018 To my parents for their endless love and support i Contents List of Figures vi Acknowledgments vii Abstract of the Dissertation viii 1 Introduction 1 2 Matrices with Special Structures 4 2.1 Nonnegative (Positive) and Metzler Matrices . 4 2.1.1 Nonnegative Matrices and Eigenvalue Characterization . 6 2.1.2 Metzler Matrices . 8 2.1.3 Z-Matrices . 10 2.1.4 M-Matrices . 10 2.1.5 Totally Nonnegative (Positive) Matrices and Strictly Metzler Matrices . 12 2.2 Symmetric Matrices . 14 2.2.1 Properties of Symmetric Matrices . 14 2.2.2 Symmetrizer and Symmetrization . 15 2.2.3 Quadratic Form and Eigenvalues Characterization of Symmetric Matrices . 19 2.3 Nonnegative and Metzler Symmetric Matrices . 22 3 Positive and Symmetric Systems 27 3.1 Positive Systems . 27 3.1.1 Externally Positive Systems . 27 3.1.2 Internally Positive Systems . 29 3.1.3 Asymptotic Stability . 33 3.1.4 Bounded-Input Bounded-Output (BIBO) Stability . 34 3.1.5 Asymptotic Stability using Lyapunov Equation . 37 3.1.6 Robust Stability of Perturbed Systems . 38 3.1.7 Stability Radius . 40 3.2 Symmetric Systems . 43 3.3 Positive Symmetric Systems . 47 ii 4 Positive Stabilization of Dynamic Systems 50 4.1 Metzlerian Stabilization . 50 4.2 Maximizing the stability radius by state feedback .