Key Words: Hamiltonian systems, canonical integrators, symplectic integrators, Runge-Kutta-Nyström methods.
|
|
- Cory Adams
- 6 years ago
- Views:
Transcription
1 CANONICAL RUNGE-KUTTA-NYSTRÖM METHODS OF ORDERS 5 AND 6 DANIEL I. OKUNBOR AND ROBERT D. SKEEL DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 304 W. SPRINGFIELD AVE. URBANA, ILLINOIS Abstract. In ths paper, we construct canoncal explct 5-stage and 7-stage Runge-Kutta- Nyström methods of orders 5 and 6, respectvely, for Hamltonan dynamcal systems. Key Words: Hamltonan systems, canoncal ntegrators, symplectc ntegrators, Runge-Kutta-Nyström methods. AMS(MOS) Subject Classfcatons: 65L05 CR Subject Classfcatons: G..7. Introducton. There has been much recent nterest n dervng for Hamltonan systems () dq dt H(q, p) =, p dp dt p) = H(q,, q hgher order numercal ntegrators whch retan the canoncal (or symplectc) property of the flow of the orgnal system. Of partcular nterest have been explct Runge- Kutta-Nyström methods(rkn) for the specal separable Hamltonan (2) H(q, p) = 2 pt M p + V (q), where q and p are vectors representng, respectvely, the postons and momenta and where M s a dagonal matrx. The functon V (q) s assocated wth the potental energy and H the total energy. Ruth[9] was the frst to publsh results about canoncal numercal ntegrators. He showed that the 2nd-order -stage leapfrog/störmer/verlet method was canoncal and dscovered a 3-stage canoncal RKN method of order 3. Ruth s work was followed by consderable research n the area of constructng hgher order canoncal ntegrators[3, 6, 0, 2,, 2]. Forest and Ruth[3] derved an explct 3-stage canoncal ntegrator of order 4. Yoshda[2] was the frst to prove the exstence of canoncal ntegrators of arbtrarly hgh order. He showed how to construct a 3 k -stage method havng order 2k + 2 usng a composton of canoncal -stage method of order 2. He derved numercally 7- and 5-stage canoncal ntegrators, respectvely of orders 6 and 8 usng a Le group approach. Low stage number s desrable because of greater convenence (such as the generaton of more closely spaced output values). Stll much s unknown about the possbltes for hgher order canoncal methods nformaton that s useful n the search for practcal methods. In ths paper we derve Supported n part by the Natonal Scence Foundaton Grant DMS and Department of Energy Grant DE-FG02-9ER25099.
2 numercally 5th-order, 5-stage RKN methods and symmetrc 6th-order, 7-stage RKN methods n secton 3. A total of four 5th-order, 5-stage RKN methods are reported. Sxteen symmetrc 6th-order, 7-stage RKN methods were obtaned, three of these are equvalent (n the sense used n [6]) to the canoncal ntegrators constructed by Yoshda[2] for general separable Hamltonans. 2. Order and Canoncal Condtons. An s-stage Runge-Kutta-Nyström method for a system wth the Hamltonan (2) s gven by (3) s y = q n + c h q n + h 2 a j f(y j ), =, 2,..., s, j= s q n+ = q n + h q n + h 2 b f(y ), q n+ = q n + h = s B f(y ). = where q n = M p n and f(q) = M V (q). The method (3) s explct f a j = 0 for j. An explct s-stage RKN method wthout redundant stages s canoncal f [6, 7, ] (4) (5) b = B ( c ), s, a j = B j (c c j ), j <. If we assume that the condtons n (4) and (5) are satsfed, we have[4] the followng order condtons for RKN methods of order 5 t : B =, t 2 : B c = 2, t 3 : B c 2 = 3, t 4 : B B j (c c j ) = 6, j< t 5 : B c 3 = 4, t 6 : B B j c (c c j ) = 8, j< t 7 : B B j (c c j )c j = 24, t 8 : j< B c 4 = 5, t 9 : B B j c 2 (c c j ) = 0, t 0 : j< j< l< B B j B l (c c j )(c c l ) = 20, t : B B j c c j (c c j ) = 30, t 2 : j< 2 B B j c 2 j(c c j ) = 60, j<
3 t 3 : j< l<j B B j B l (c c j )(c j c l ) = 20. The condton t 7 s redundant (see Okunbor and Skeel [6]). We use a smlar approach as n [6] to show that t 2 and t 3 are also redundant: lhs of t 2 = B B j c 2 j(c c j ) j< = B B j c 2 (c c j ) j> = B B j c 2 (c c j ) B B j c 2 (c c j ) j< j = 0 ( B B j c 3 B B j c 2 ) j j = 0 ( ) = 60 = rhs of t 2, lhs of t 3 = B B j B l (c c j )(c j c l ) j< l<j = B B j B l (c c j )(c c l ) j> l< = B B j B l (c c j )(c c l ) B B j B l (c c j )(c c l ) j< l< j l< = 20 B j [ B B l (c c l )c B B l (c c l )c j ] j l< l< = 20 [ 8 B j c j B B l (c c l )] j = 20 ( ) = 20 = rhs of t 3. l l< The above results llustrate the proposton of Calvo and Sanz-Serna[] that states that f two Nyström trees that are equvalent (see defnton n the Appendx), then the Φ (see Appendx) that corresponds to one can be expressed n terms of Φ s of the other and trees of lower orders. In our case, the trees, f[z 2, f] and f[f[z 2 ]] n our specal notaton (see Appendx) that result respectvely, from t 9 and t 2 are equvalent. The same s true for trees, f[f[z] 2 ] and f[f[f]] that result respectvely, from t 0 and t Canoncal Runge-Kutta-Nyström methods th-Order 5-Stage Methods. In secton 2, we showed that t 7, t 2 and t 3 are redundant for a canoncal RKN method of 5th-order, leavng us wth 0 condtons nvolvng 0 parameters. These condtons were then solved for B and c. We resorted 3
4 Method B c Table 5th-order 5-stage Runge-Kutta-Nyström Methods to an teratve procedure because these condtons nvolve complcated expressons n B and c. We used the subroutnes HYBRD and HYBRJ of MINPACK obtaned from Netlb for determnng the soluton. HYBRD combnes Powell s method for optmzaton, QR factorzaton and the fnte dvded dfference method for computng the Jacoban matrx. HYBRJ s the same as HYBRD except that exact Jacoban matrx s used. The ntal guesses were obtaned randomly from a Gaussan dstrbuton wth mean 0 and standard devaton. About 0,000 dfferent ntal guesses were tred and only four methods were obtaned. These four methods, obtaned usng HYBRD were used as ntal solutons for the HYBRJ program to mprove the accuracy of method coeffcents. These four methods are shown n Table. The 2-norm of resduals n all 3 order condtons s 0 3 for method, 0 4 for methods 2, 0 5 for method 3 and 4. As s obvous method 3 s the adjont of method, and 4 the adjont of 2. The adjont of a method s obtaned by nterchangng h, q n and q n, respectvely, wth h, q n+ and q n+. The slght dfferences n coeffcents, ndcates the error n these values. We speculate that these are the only methods wth real coeffcents consderng the magntude of the number of ntal guesses tred. Very recently, we also found these methods n [8] th-Order 6-Stage Methods. To construct symmetrc methods of order 6, we start wth a 6-stage RKN method wth the followng condtons B = B 6, B 2 = B 5, B 3 = B 4, c = c 6, c 2 = c 5, c 3 = c 4. 4
5 Wth these condtons t, t 3, t 4 and t 7 can be wrtten as t : B 4 + B 5 + B 6 = 2, t 3 : B 4 c 4 ( c 4 ) + B 5 c 5 ( c 5 ) + B 6 c 6 ( c 6 ) = 2, t 4 : B 5 c 5 (B 4 + B 5 ) + B 6 c 6 ( B 6 ) + B 4 (B 4 c 4 + B 5 c 5 ) = 5 24, t 8 : B 4 c 2 4( c 4 ) 2 + B 5 c 2 5( c 5 ) 2 + B 6 c 2 6( c 6 ) 2 = 60. The condtons t 2, t 5 and t 6 are redundant gven t, t 3, t 4 and the symmetry condtons. For detals, see [5]. The condtons t 9, t 0 and t have complcated expressons even after they have been smplfed and they are omtted here. For a symmetrc 6-stage RKN method, t turns out that t, t 3, t 4, t 8 and two of t 9, t 0, t are enough to fnd B 4, B 5, B 6, c 4, c 5 and c 6. In all, there are 3 possble sets of equatons, namely t, t 3, t 4, t 8, t 9, t 0 ; t, t 3, t 4, t 8, t 9, t ; t, t 3, t 4, t 8, t 0, t. The three sets were solved by HYBRD and all solutons obtaned from each set never satsfed the mssng equaton after tryng 000 ntal guesses. We therefore state the followng conjecture. Conjecture. There s no symmetrc 6-stage RKN method of order th-Order 7-Stage Methods. The negatve result above motvated us to search for symmetrc 7-stage methods of order 6. The symmetry condtons n ths case are B = B 7, B 2 = B 6, B 3 = B 5, c = c 7, c 2 = c 6, c 3 = c 5, c 4 = 2. Wth these condtons t, t 3, t 4 and t 7 can now be wrtten as t : 2 B 4 + B 5 + B 6 + B 7 = 2, t 3 : 8 B 4 + B 5 c 5 ( c 5 ) + B 6 c 6 ( c 6 ) + B 7 c 7 ( c 7 ) = 2, t 4 : 8 B2 4 + B 5 c 5 (B 4 + B 5 ) + B 6 c 6 ( B 6 ) + B 7 c 7 ( B 7 ) 2B 6 B 7 c 6 = 5 24, 5
6 t 8 : 6 B 4 + 2B 5 c 2 5( c 5 ) 2 + 2B 6 c 2 6( c 6 ) 2 + 2B 7 c 2 7( c 7 ) 2 = 30. The unknowns n ths case are B 4, B 5, B 6, B 7, c 5, c 6 and c 7. Ths makes t lkely that the condtons t, t 3, t 4,t 8, t 9, t 0 and t can be solved unquely for those parameters. Agan, we used the routne HYBRD. After tryng 000 dfferent ntal guesses, we obtaned 6 dfferent methods as ndcated n Tables 2(a) and 2(b). The numbers n brackets represent the 2-norms of the resduals of all twenty-three order condtons. We dd not use HYBRJ to mprove the accuracy of method coeffcents as we dd n secton 3.. The 6th-order condtons are gven n the appendx for verfcaton purpose. These methods whch nclude the methods constructed by Yoshda are 7- stage all of order 6, counterexamples to what s suggested by Calvo and Sanz-Serna[]. Acknowledgement. We thank Skp Thompson for helpng us to obtan better accuracy for the method coeffcents wth the use of HYBRJ and an exact Jacoban matrx. REFERENCES [] M. P. Calvo and J. M. Sanz-Serna. Order condtons for canoncal Runge-Kutta-Nyström methods. BIT, 32:3 42, 992. [2] P. J. Channell and J. C. Scovel. Symplectc ntegraton of Hamltonan systems. Nonlnearty, 3:23 259, 990. [3] E. Forest and R. D. Ruth. Fourth-order symplectc ntegraton. Physca D, 43:05 7, 990. [4] E. Harer, S. P. Nørsett, and G. Wanner. Solvng Ordnary Dfferental Equatons I: Non-stff Systems. Sprnger-Verlag, Berln, 987. [5] D. Okunbor. Canoncal ntegraton methods for Hamltonan systems. Ph.D. Thess, n preparaton. [6] D. Okunbor and R. D. Skeel. Explct canoncal methods for Hamltonan systems. Math. Comp., 992. to appear. [7] D. Okunbor and R. D. Skeel. An explct Runge-Kutta-Nyström method s canoncal f and only f ts adjont s explct. J. SIAM. Numer. Anal., 29(2):52 527, 992. [8] M.-Z. Qn and W.-J. Zhu. Order condtons of two knds of canoncal dfference schemes. Manuscrpt, 992. [9] R. D. Ruth. A canoncal ntegraton technque. IEEE Trans. on Nucl. Sc., NS-30(4): , 983. [0] J. M. Sanz-Serna. The numercal ntegraton of Hamltonan systems. In Proc. of IMA Conference on Comput. ODEs. J. R. Cash and I. Gladwell, eds., Oxford Unv. Press. to appear. [] Y. B. Surs. Canoncal transformatons generated by methods of Runge-Kutta type for the numercal ntegraton of the system x = U. Zh. Vychsl. Mat. Mat. Fz., 29:202 2, x 989. (n Russan). Same as U.S.S.R. Comput. Maths. Phys., 29():38-44, 989. [2] H. Yoshda. Constructon of hgher order symplectc ntegrators. Physcs Letters A, 50: ,
7 Method (0 8 ) 2(0 ) B B 5 = B B 6 = B B 7 = B c c 5 = c c 6 = c c 7 = c Method 3(0 0 ) 4(0 3 ) B B 5 = B B 6 = B B 7 = B c 5 = c c 6 = c c 7 = c Method 5(0 4 ) 6(0 4 ) B B 5 = B B 6 = B B 7 = B c 5 = c c 6 = c c 7 = c Method 7(0 2 ) 8(0 2 ) B B 5 = B B 6 = B B 7 = B c 5 = c c 6 = c c 7 = c Table 2 (a) 6th-order 7-stage Runge-Kutta-Nyström methods Appendx Appendx A. Order Sx Condtons. Usng the notaton n [4], we have that an RKN method appled to a problem of the form y = f(y) s order p f and only f (6) b Φ (t) = (7) B Φ (t) = γ(t), 7, for Nyström trees wth ρ(t) p (ρ(t) + )γ(t) for Nyström trees wth ρ(t) p
8 Method 9(0 8 ) 0(0 ) B B 5 = B B 6 = B B 7 = B c 5 = c c 6 = c c 7 = c Method (0 4 ) 2(0 0 ) B B 5 = B B 6 = B B 7 = B c 5 = c c 6 = c c 7 = c Method 3(0 2 ) 4(0 6 ) B B 5 = B B 6 = B B 7 = B c 5 = c c 6 = c c 7 = c Method 5(0 3 ) 6(0 2 ) B B 5 = B B 6 = B B 7 = B c 5 = c c 6 = c c 7 = c Table 2 (b) 6th-order 7-stage Runge-Kutta-Nyström methods where ρ(t) s the order of the tree, γ(t) s the densty of the tree and Φ (t) corresponds to the elementary weght of the Nyström tree. If condtons (4) combne wth (7) then condtons (6) are superfluous. Therefore, we concentrate on the condtons (7). Frst, we let z = y and then use a specal notaton (smlar to what was used n our Mathematca program) to represent the trees or the elementary dfferentals. We gve here the correspondence between the elementary dfferentals n our notaton and the Nyström trees for a few of the elementary dfferentals. The fat node correspond to dervatve of z and the meagre node, to the dervatve of the y. The bottom node s the root of the tree. Two Nyström trees are equvalent n the sense defned n Calvo 8
9 and Sanz-Serna[], f they have equal number of fat nodes, equal number of meagre nodes and dentcal branches and dffers only n ther roots. The order 6 condtons are gven n Table 3, where α(t) s the weght of the elementary dfferental. FIGURE MISSING t ρ(t) α(t) γ(t) Φ (t) f[z 5 ] 6 6 c 5 f[f 2, z] j k a ja k c f[f, z 3 ] j a jc 3 f[z, f[f]] j k a ja jk c f[f, f[z]] j k a ja k c k f[z 2, f[z]] j a jc 2 c j f[f[f, z]] j k a ja jk c j f[z, f[z 2 ]] j a jc c 2 j f[f[z 3 ]] 6 20 j a jc 3 j f[f[f[z]]] j k a ja jk c k Table 3 Order Sx Condtons 9
Difference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationAppendix B. The Finite Difference Scheme
140 APPENDIXES Appendx B. The Fnte Dfference Scheme In ths appendx we present numercal technques whch are used to approxmate solutons of system 3.1 3.3. A comprehensve treatment of theoretcal and mplementaton
More informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
More informationEEE 241: Linear Systems
EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationChapter 12. Ordinary Differential Equation Boundary Value (BV) Problems
Chapter. Ordnar Dfferental Equaton Boundar Value (BV) Problems In ths chapter we wll learn how to solve ODE boundar value problem. BV ODE s usuall gven wth x beng the ndependent space varable. p( x) q(
More information4DVAR, according to the name, is a four-dimensional variational method.
4D-Varatonal Data Assmlaton (4D-Var) 4DVAR, accordng to the name, s a four-dmensonal varatonal method. 4D-Var s actually a drect generalzaton of 3D-Var to handle observatons that are dstrbuted n tme. The
More informationImplicit Integration Henyey Method
Implct Integraton Henyey Method In realstc stellar evoluton codes nstead of a drect ntegraton usng for example the Runge-Kutta method one employs an teratve mplct technque. Ths s because the structure
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationTrees and Order Conditions
Trees and Order Condtons Constructon of Runge-Kutta order condtons usng Butcher trees and seres. Paul Tranqull 1 1 Computatonal Scence Laboratory CSL) Department of Computer Scence Vrgna Tech. Trees and
More informationBallot Paths Avoiding Depth Zero Patterns
Ballot Paths Avodng Depth Zero Patterns Henrch Nederhausen and Shaun Sullvan Florda Atlantc Unversty, Boca Raton, Florda nederha@fauedu, ssull21@fauedu 1 Introducton In a paper by Sapounaks, Tasoulas,
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More informationBeyond Zudilin s Conjectured q-analog of Schmidt s problem
Beyond Zudln s Conectured q-analog of Schmdt s problem Thotsaporn Ae Thanatpanonda thotsaporn@gmalcom Mathematcs Subect Classfcaton: 11B65 33B99 Abstract Usng the methodology of (rgorous expermental mathematcs
More information12. The Hamilton-Jacobi Equation Michael Fowler
1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationOn Finite Rank Perturbation of Diagonalizable Operators
Functonal Analyss, Approxmaton and Computaton 6 (1) (2014), 49 53 Publshed by Faculty of Scences and Mathematcs, Unversty of Nš, Serba Avalable at: http://wwwpmfnacrs/faac On Fnte Rank Perturbaton of Dagonalzable
More informationUniqueness of Weak Solutions to the 3D Ginzburg- Landau Model for Superconductivity
Int. Journal of Math. Analyss, Vol. 6, 212, no. 22, 195-114 Unqueness of Weak Solutons to the 3D Gnzburg- Landau Model for Superconductvty Jshan Fan Department of Appled Mathematcs Nanjng Forestry Unversty
More informationAffine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationCHAPTER 14 GENERAL PERTURBATION THEORY
CHAPTER 4 GENERAL PERTURBATION THEORY 4 Introducton A partcle n orbt around a pont mass or a sphercally symmetrc mass dstrbuton s movng n a gravtatonal potental of the form GM / r In ths potental t moves
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More information10. Canonical Transformations Michael Fowler
10. Canoncal Transformatons Mchael Fowler Pont Transformatons It s clear that Lagrange s equatons are correct for any reasonable choce of parameters labelng the system confguraton. Let s call our frst
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More information2-π STRUCTURES ASSOCIATED TO THE LAGRANGIAN MECHANICAL SYSTEMS UDC 531.3: (045)=111. Victor Blãnuţã, Manuela Gîrţu
FACTA UNIVERSITATIS Seres: Mechancs Automatc Control and Robotcs Vol. 6 N o 1 007 pp. 89-95 -π STRUCTURES ASSOCIATED TO THE LAGRANGIAN MECHANICAL SYSTEMS UDC 531.3:53.511(045)=111 Vctor Blãnuţã Manuela
More informationReport on Image warping
Report on Image warpng Xuan Ne, Dec. 20, 2004 Ths document summarzed the algorthms of our mage warpng soluton for further study, and there s a detaled descrpton about the mplementaton of these algorthms.
More informationPoisson brackets and canonical transformations
rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order
More information2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification
E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More information6.3.4 Modified Euler s method of integration
6.3.4 Modfed Euler s method of ntegraton Before dscussng the applcaton of Euler s method for solvng the swng equatons, let us frst revew the basc Euler s method of numercal ntegraton. Let the general from
More informationOn a direct solver for linear least squares problems
ISSN 2066-6594 Ann. Acad. Rom. Sc. Ser. Math. Appl. Vol. 8, No. 2/2016 On a drect solver for lnear least squares problems Constantn Popa Abstract The Null Space (NS) algorthm s a drect solver for lnear
More informationCSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography
CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationDensity matrix. c α (t)φ α (q)
Densty matrx Note: ths s supplementary materal. I strongly recommend that you read t for your own nterest. I beleve t wll help wth understandng the quantum ensembles, but t s not necessary to know t n
More informationInexact Newton Methods for Inverse Eigenvalue Problems
Inexact Newton Methods for Inverse Egenvalue Problems Zheng-jan Ba Abstract In ths paper, we survey some of the latest development n usng nexact Newton-lke methods for solvng nverse egenvalue problems.
More informationOne-sided finite-difference approximations suitable for use with Richardson extrapolation
Journal of Computatonal Physcs 219 (2006) 13 20 Short note One-sded fnte-dfference approxmatons sutable for use wth Rchardson extrapolaton Kumar Rahul, S.N. Bhattacharyya * Department of Mechancal Engneerng,
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationSome Comments on Accelerating Convergence of Iterative Sequences Using Direct Inversion of the Iterative Subspace (DIIS)
Some Comments on Acceleratng Convergence of Iteratve Sequences Usng Drect Inverson of the Iteratve Subspace (DIIS) C. Davd Sherrll School of Chemstry and Bochemstry Georga Insttute of Technology May 1998
More informationFormal solvers of the RT equation
Formal solvers of the RT equaton Formal RT solvers Runge- Kutta (reference solver) Pskunov N.: 979, Master Thess Long characterstcs (Feautrer scheme) Cannon C.J.: 970, ApJ 6, 55 Short characterstcs (Hermtan
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
More informationChapter - 2. Distribution System Power Flow Analysis
Chapter - 2 Dstrbuton System Power Flow Analyss CHAPTER - 2 Radal Dstrbuton System Load Flow 2.1 Introducton Load flow s an mportant tool [66] for analyzng electrcal power system network performance. Load
More information1 GSW Iterative Techniques for y = Ax
1 for y = A I m gong to cheat here. here are a lot of teratve technques that can be used to solve the general case of a set of smultaneous equatons (wrtten n the matr form as y = A), but ths chapter sn
More informationOnline Appendix. t=1 (p t w)q t. Then the first order condition shows that
Artcle forthcomng to ; manuscrpt no (Please, provde the manuscrpt number!) 1 Onlne Appendx Appendx E: Proofs Proof of Proposton 1 Frst we derve the equlbrum when the manufacturer does not vertcally ntegrate
More informationLecture 2: Numerical Methods for Differentiations and Integrations
Numercal Smulaton of Space Plasmas (I [AP-4036] Lecture 2 by Lng-Hsao Lyu March, 2018 Lecture 2: Numercal Methods for Dfferentatons and Integratons As we have dscussed n Lecture 1 that numercal smulaton
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationResearch Article Green s Theorem for Sign Data
Internatonal Scholarly Research Network ISRN Appled Mathematcs Volume 2012, Artcle ID 539359, 10 pages do:10.5402/2012/539359 Research Artcle Green s Theorem for Sgn Data Lous M. Houston The Unversty of
More informationME 501A Seminar in Engineering Analysis Page 1
umercal Solutons of oundary-value Problems n Os ovember 7, 7 umercal Solutons of oundary- Value Problems n Os Larry aretto Mechancal ngneerng 5 Semnar n ngneerng nalyss ovember 7, 7 Outlne Revew stff equaton
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationEffects of Ignoring Correlations When Computing Sample Chi-Square. John W. Fowler February 26, 2012
Effects of Ignorng Correlatons When Computng Sample Ch-Square John W. Fowler February 6, 0 It can happen that ch-square must be computed for a sample whose elements are correlated to an unknown extent.
More informationSupplementary Notes for Chapter 9 Mixture Thermodynamics
Supplementary Notes for Chapter 9 Mxture Thermodynamcs Key ponts Nne major topcs of Chapter 9 are revewed below: 1. Notaton and operatonal equatons for mxtures 2. PVTN EOSs for mxtures 3. General effects
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
More informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More informationPHYS 705: Classical Mechanics. Hamilton-Jacobi Equation
1 PHYS 705: Classcal Mechancs Hamlton-Jacob Equaton Hamlton-Jacob Equaton There s also a very elegant relaton between the Hamltonan Formulaton of Mechancs and Quantum Mechancs. To do that, we need to derve
More informationDeriving the X-Z Identity from Auxiliary Space Method
Dervng the X-Z Identty from Auxlary Space Method Long Chen Department of Mathematcs, Unversty of Calforna at Irvne, Irvne, CA 92697 chenlong@math.uc.edu 1 Iteratve Methods In ths paper we dscuss teratve
More informationHigh resolution entropy stable scheme for shallow water equations
Internatonal Symposum on Computers & Informatcs (ISCI 05) Hgh resoluton entropy stable scheme for shallow water equatons Xaohan Cheng,a, Yufeng Ne,b, Department of Appled Mathematcs, Northwestern Polytechncal
More informationNON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS
IJRRAS 8 (3 September 011 www.arpapress.com/volumes/vol8issue3/ijrras_8_3_08.pdf NON-CENTRAL 7-POINT FORMULA IN THE METHOD OF LINES FOR PARABOLIC AND BURGERS' EQUATIONS H.O. Bakodah Dept. of Mathematc
More informationMarkov Chain Monte Carlo Lecture 6
where (x 1,..., x N ) X N, N s called the populaton sze, f(x) f (x) for at least one {1, 2,..., N}, and those dfferent from f(x) are called the tral dstrbutons n terms of mportance samplng. Dfferent ways
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationQuadratic invariants and multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs
Quadratc nvarants and mult-symplectcty of parttoned Runge-Kutta methods for Hamltonan PDEs Yajuan Sun Insttute of Computatonal Mathematcs and Scentfc/Engneerng Computng Academy of Mathematcs and System
More informationLecture 21: Numerical methods for pricing American type derivatives
Lecture 21: Numercal methods for prcng Amercan type dervatves Xaoguang Wang STAT 598W Aprl 10th, 2014 (STAT 598W) Lecture 21 1 / 26 Outlne 1 Fnte Dfference Method Explct Method Penalty Method (STAT 598W)
More informationWorkshop: Approximating energies and wave functions Quantum aspects of physical chemistry
Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:5-5: Copyrght 25 Dan Dll (dan@bu.edu) Department
More information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationComputation of Higher Order Moments from Two Multinomial Overdispersion Likelihood Models
Computaton of Hgher Order Moments from Two Multnomal Overdsperson Lkelhood Models BY J. T. NEWCOMER, N. K. NEERCHAL Department of Mathematcs and Statstcs, Unversty of Maryland, Baltmore County, Baltmore,
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More informationNUMERICAL DIFFERENTIATION
NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the
More informationGeorgia Tech PHYS 6124 Mathematical Methods of Physics I
Georga Tech PHYS 624 Mathematcal Methods of Physcs I Instructor: Predrag Cvtanovć Fall semester 202 Homework Set #7 due October 30 202 == show all your work for maxmum credt == put labels ttle legends
More informationAdditional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty
Addtonal Codes usng Fnte Dfference Method Benamn Moll 1 HJB Equaton for Consumpton-Savng Problem Wthout Uncertanty Before consderng the case wth stochastc ncome n http://www.prnceton.edu/~moll/ HACTproect/HACT_Numercal_Appendx.pdf,
More informationApplication of B-Spline to Numerical Solution of a System of Singularly Perturbed Problems
Mathematca Aeterna, Vol. 1, 011, no. 06, 405 415 Applcaton of B-Splne to Numercal Soluton of a System of Sngularly Perturbed Problems Yogesh Gupta Department of Mathematcs Unted College of Engneerng &
More informationHomework Notes Week 7
Homework Notes Week 7 Math 4 Sprng 4 #4 (a Complete the proof n example 5 that s an nner product (the Frobenus nner product on M n n (F In the example propertes (a and (d have already been verfed so we
More informationThe Minimum Universal Cost Flow in an Infeasible Flow Network
Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran
More informationGeneralized Linear Methods
Generalzed Lnear Methods 1 Introducton In the Ensemble Methods the general dea s that usng a combnaton of several weak learner one could make a better learner. More formally, assume that we have a set
More informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
More informationThe Exact Formulation of the Inverse of the Tridiagonal Matrix for Solving the 1D Poisson Equation with the Finite Difference Method
Journal of Electromagnetc Analyss and Applcatons, 04, 6, 0-08 Publshed Onlne September 04 n ScRes. http://www.scrp.org/journal/jemaa http://dx.do.org/0.46/jemaa.04.6000 The Exact Formulaton of the Inverse
More informationNumerical Solution of Ordinary Differential Equations
Numercal Methods (CENG 00) CHAPTER-VI Numercal Soluton of Ordnar Dfferental Equatons 6 Introducton Dfferental equatons are equatons composed of an unknown functon and ts dervatves The followng are examples
More informationIntegrals and Invariants of Euler-Lagrange Equations
Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours
UNIVERSITY OF TORONTO Faculty of Arts and Scence December 005 Examnatons STA47HF/STA005HF Duraton - hours AIDS ALLOWED: (to be suppled by the student) Non-programmable calculator One handwrtten 8.5'' x
More informationSOME RESULTS ON TRANSFORMATIONS GROUPS OF N-LINEAR CONNECTIONS IN THE 2-TANGENT BUNDLE
STUDIA UNIV. BABEŞ BOLYAI MATHEMATICA Volume LIII Number March 008 SOME RESULTS ON TRANSFORMATIONS GROUPS OF N-LINEAR CONNECTIONS IN THE -TANGENT BUNDLE GHEORGHE ATANASIU AND MONICA PURCARU Abstract. In
More informationLecture 3: Probability Distributions
Lecture 3: Probablty Dstrbutons Random Varables Let us begn by defnng a sample space as a set of outcomes from an experment. We denote ths by S. A random varable s a functon whch maps outcomes nto the
More informationVector Norms. Chapter 7 Iterative Techniques in Matrix Algebra. Cauchy-Bunyakovsky-Schwarz Inequality for Sums. Distances. Convergence.
Vector Norms Chapter 7 Iteratve Technques n Matrx Algebra Per-Olof Persson persson@berkeley.edu Department of Mathematcs Unversty of Calforna, Berkeley Math 128B Numercal Analyss Defnton A vector norm
More informationYong Joon Ryang. 1. Introduction Consider the multicommodity transportation problem with convex quadratic cost function. 1 2 (x x0 ) T Q(x x 0 )
Kangweon-Kyungk Math. Jour. 4 1996), No. 1, pp. 7 16 AN ITERATIVE ROW-ACTION METHOD FOR MULTICOMMODITY TRANSPORTATION PROBLEMS Yong Joon Ryang Abstract. The optmzaton problems wth quadratc constrants often
More informationThe binomial transforms of the generalized (s, t )-Jacobsthal matrix sequence
Int. J. Adv. Appl. Math. and Mech. 6(3 (2019 14 20 (ISSN: 2347-2529 Journal homepage: www.jaamm.com IJAAMM Internatonal Journal of Advances n Appled Mathematcs and Mechancs The bnomal transforms of the
More information= z 20 z n. (k 20) + 4 z k = 4
Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5
More informationSystem in Weibull Distribution
Internatonal Matheatcal Foru 4 9 no. 9 94-95 Relablty Equvalence Factors of a Seres-Parallel Syste n Webull Dstrbuton M. A. El-Dacese Matheatcs Departent Faculty of Scence Tanta Unversty Tanta Egypt eldacese@yahoo.co
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More information2.3 Nilpotent endomorphisms
s a block dagonal matrx, wth A Mat dm U (C) In fact, we can assume that B = B 1 B k, wth B an ordered bass of U, and that A = [f U ] B, where f U : U U s the restrcton of f to U 40 23 Nlpotent endomorphsms
More informationDUE: WEDS FEB 21ST 2018
HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION DUE: WEDS FEB 21ST 2018 1. Theory Beam bendng s a classcal engneerng analyss. The tradtonal soluton technque makes smplfyng assumptons such as a constant
More informationSolutions to exam in SF1811 Optimization, Jan 14, 2015
Solutons to exam n SF8 Optmzaton, Jan 4, 25 3 3 O------O -4 \ / \ / The network: \/ where all lnks go from left to rght. /\ / \ / \ 6 O------O -5 2 4.(a) Let x = ( x 3, x 4, x 23, x 24 ) T, where the varable
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationRemarks on the Properties of a Quasi-Fibonacci-like Polynomial Sequence
Remarks on the Propertes of a Quas-Fbonacc-lke Polynomal Sequence Brce Merwne LIU Brooklyn Ilan Wenschelbaum Wesleyan Unversty Abstract Consder the Quas-Fbonacc-lke Polynomal Sequence gven by F 0 = 1,
More informationSupplement: Proofs and Technical Details for The Solution Path of the Generalized Lasso
Supplement: Proofs and Techncal Detals for The Soluton Path of the Generalzed Lasso Ryan J. Tbshran Jonathan Taylor In ths document we gve supplementary detals to the paper The Soluton Path of the Generalzed
More informationBOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS
BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all
More informationLECTURE 9 CANONICAL CORRELATION ANALYSIS
LECURE 9 CANONICAL CORRELAION ANALYSIS Introducton he concept of canoncal correlaton arses when we want to quantfy the assocatons between two sets of varables. For example, suppose that the frst set of
More informationSome modelling aspects for the Matlab implementation of MMA
Some modellng aspects for the Matlab mplementaton of MMA Krster Svanberg krlle@math.kth.se Optmzaton and Systems Theory Department of Mathematcs KTH, SE 10044 Stockholm September 2004 1. Consdered optmzaton
More information