International Journal of Mathematical Archive-5(7), 2014, Available online through ISSN
|
|
- Patrick Greer
- 5 years ago
- Views:
Transcription
1 Iteratioal Joural of Mathematical Archive-5(7), 04, Available olie through wwwmaifo ISSN APPROXIMAION OF SMALLES EIGENVALUE AND IS CORRESPONDING EIGENVECOR BY POWER MEHOD Alaur Hussai Laskar ad Samira Behera* Departmet of Mathematics, Assam Uiversity, Silchar- 7880, Idia (Received O: ; Revised & Accepted O: ) ABSRAC Power method is ormally used to determie the largest eigevalue (i magitude) ad the correspodig eigevector of the system AX = λx I this study, we eamie power method for computig the smallest eigevalue ad its correspodig eigevector of real square matrices Our work is based o choosig of iitial vector i power method for acceleratio purpose Fially, we illustrate the method with eample ad results discussed Keywords: Domiat eigevalue, power method, Adjoit of a square matri, Iverse matri i terms of Adjoit matri INRODUCION We study the problem of calculatig the eigevalues ad eigevectors If oly a few eigevalues are to be calculated, the the umerical method will be differet tha if all eigevalues are required Eigevalues ad eigevectors play a importat part i the applicatios of liear algebra he aive method of fidig the eigevalues of a matri ivolves fidig the roots of the characteristic polyomial of the matri I idustrial sized matrices, however, this method is ot feasible, ad the eigevalues must be obtaied by other meas Fortuately, there eist several other techiques for fidig eigevalues ad eigevectors of a matri, some of which fall uder the realm of iterative methods hese methods work by repeatedly refiig approimatios to the eigevectors or eigevalues, ad ca be termiated wheever the approimatios reach a suitable degree of accuracy Iterative methods form the basis of much of moder day eigevalue computatio he geeral problem of fidig all eigevalues ad eigevectors of a o-symmetric matri A ca be quite ustable with respect to perturbatios i the coefficiets of A, ad this makes more difficult the desig of geeral methods ad computer programs he eigevalues of a symmetric matri A are quite stable with respect to perturbatios i A he eigevalues of a matri are usually calculated first, ad they are used i calculatig the eigevectors, if these are desired he mai eceptio to this rule is the power method described i this paper, which is useful i calculatig a sigle domiat eigevalue of a matri For obtaiig eigevalues ad eigevectors for low order matrices, ad 3 3 his ivolved firstly solvig the characteristic equatio det( A λi) = 0 for a give matri A his is a th order polyomial equatio ad, eve for as low as 3, solvig it is ot always straightforward For large eve obtaiig the characteristic equatio may be differet Cosequetly, i this paper we give a brief itroductio to alterative method, essetially umerical i ature, of obtaiig eigevalues ad perhaps eigevectors Algebraic procedures for determiig eigevalues ad eigevectors are impractical for most matrices of large order Istead, umerical methods that are efficiet ad stable whe programmed o high-speed computers have bee developed for this purpose Such methods are iterative, ad, i the ideal case, coverge to the eigevalues ad eigevectors of iterest I this paper, we outlie power method, ad summarize derivatios, procedures ad advatages he method to be eamied is the power method I sectio of this paper, we have discussed some basic cocepts regardig eigevalues ad eigevectors with eample required to uderstad the cocepts that are discussed I sectio 3, we have preseted power method with eample for approimatig smallest eigevalue ad its correspodig eigevector of the real square matri A Fially, i sectio 4, we summarized some cocludig remarks that are used i practice Correspodig author: Samira Behera* Departmet of Mathematics, Assam Uiversity, Silchar- 7880, Idia samirabehera998@gmailcom Iteratioal Joural of Mathematical Archive- 5(7), July 04 49
2 For the purposes of this paper, we restrict our attetio to real-valued, square matrices with a full set of real eigevalues PRELIMINARIES I this sectio, we recall some basic cocepts which would be used i the sequel Defiitio : he mior of a elemet of a determiat of order greater tha oe is the determiat of et lower order obtaied by deletig the row ad the colum of the give determiat i which the elemet occurs he mior of the elemet a i the determiat A is deoted by M Defiitio : he cofactor of a elemet of a determiat of order greater tha oe is the coefficiet of that elemet i the epasio of the determiat he cofactor of the elemet a i A is deoted by A he cofactor of a elemet a i A ca be determied i terms of its mior as A = ( ) i+ j M Defiitio 3: Correspodig to a square matri cofactor of a i A he which is deoted by AdjA A = a, we form a matri 04, IJMA All Rights Reserved 50 B= A, where A is the B (traspose of B ) is called the Adjoit Matri or Adjugate Matri of the Matri A Defiitio 4: A square matri A is ivertible if ad oly if A is o-sigular Defiitio 5: Let λ, λ,, λ be the eigevalues of a matri A λ is called the domiat eigevalue of A if λ > λ i, i =,, he eigevectors correspodig to λ are called domiat eigevectors of A Defiitio 6: Eigevectors correspodig to distict eigevalues are liearly idepedet However, two or more liearly idepedet eigevectors may correspod to the same eigevalue () () (3) A, A, A, be a sequece of matrices i m Defiitio 7: Let R We say that the sequece of matrices m coverges to a matri A R if the sequece A ( k ) i, j of real umbers coverges to A i, j for every pair i m, j, as k approaches ifiity hat is, a sequece of matrices coverges if the sequeces give by each etry of the matri all coverge 8 EIGENVALUES AND EIGENVECORS Cosider the equatio AX = λ X () Here, A is a matri, λ is a scalar ad X is a o-zero vector he solutio of () requires the solutio of λ he scalar λ (real or comple) is called the eigevalue or latet root or characteristic value of A X is called the correspodig eigevector or characteristic vector of the matri A he eigevalues of a matri are of great importace i physical problems hey occur i the aalysis of stability ad i the equatios of vibratios i structures or electrical circuits he stability of a aircraft is determied by the locatio of the eigevalues of a certai matri i the comple plae If A= a, the () ca be writte as a λ a a3 a a a λ a3 a X = 0 a a a3 a λ ()
3 Or[ A λi] X = 0 (3) his is a set of liear homogeeous equatios It will have a o-trivial solutio if ad oly if A λi = 0; that is, if ad oly if a λ a a3 a a a λ a3 a = 0 a a a a λ 3 he determiat is a polyomial of degree i λ he polyomial is called the characteristic polyomial of matri A It is usually deoted by P( λ ) he roots of this polyomial are the eigevalues (or the latet roots or the characteristic values) of the matri A If the values of λ are λ, λ,, λ, which may ot all be distict, the the eigevectors of the matri A are give by AX = λx, AX = λx,, AX = λx (5) Hece the determiatio of eigevalues of a matri A is othig but solvig a algebraic equatio of degree 9 WO IMPORAN PROPERIES OF EIGENVALUES AND EIGENVECORS Property : If X is a eigevector of A correspodig to the eigevalue λ ad A is ivertible, the X is a eigevector of correspodig to its eigevalue λ Property : If A is a o-sigular matri, the eigevalues of are the reciprocals of the eigevalues of A Proof: Let λ be a eigevalue of A ad X be a correspodig eigevector he AX = λ X X = A ( λx) = λ( A X) X = A X λ ( A is o-sigular λ 0) A X = X λ is a eigevalue of A ad X is a correspodig eigevector λ Coversely, suppose that k is a eigevalue of Sice A is o-sigular is o-sigular ad ( A ) = A, therefore it follows from the first part of this property that k is a eigevalue of A hus each eigevalue of is equal to the reciprocal of some eigevalue of A Hece the eigevalues of the reciprocals of the eigevalues of A Eample : Let us ow cosider the matri A = 5 04, IJMA All Rights Reserved 5 (4) are othig but to fid the eigevalues ad the correspodig eigevectors by direct method ie by algebraic procedures for verifyig the above two properties Solutio: he characteristic equatio is P( λ) = A λi = 0 λ = 0 5 λ
4 λ = ad λ = Which gives he correspodig eigevectors are obtaied thus: (i) For λ = Let the eigevector be X = he we have A = = 5 = ad + 5 = which gives the equatios = 4 which gives Hece the eigevector for eigevector is X = [ 4,] (ii) For λ = Let the eigevector be X = A = = 5 λ = is X [ 4, ] = Sice is arbitrary, we ca take = ad hece the he we have = ad + 5 =, which gives = 3 which gives the equatios Hece the eigevector for λ = is X [ 3, ] eigevector is X = [ 3,] hus, the eigevalues are 4, X = ad X = 3, respectively [ ] [ ] = Sice is arbitrary, we ca take = ad hece the λ = ad λ = ad the correspodig eigevectors are Now, to fid eigevalues ad eigevectors of the matri, we eed to fid ad for that we proceed as follows: Clearly, the matri A is o-sigular AdjA 5 = AdjA 5 A = = ( A ) A = 5 6 = 04, IJMA All Rights Reserved 5
5 he characteristic equatio of the matri P( λ) = λi = 0 5 λ 6 = 0 λ is which gives λ = ad λ = he correspodig eigevectors are obtaied thus: (i) For λ = Let the eigevector be X = he we have A = 5 6 = which gives the equatios = ad =, which gives = 4 Hece the eigevector for λ = is X = [ 4, ] Sice is arbitrary, we ca take = ad hece the eigevector is X = [ 4,] (ii) For λ = Let the eigevector be X = = A 5 6 = he we have = ad =, which gives = 3 which gives the equatios Hece the eigevector for λ = is X [ 3, ] eigevector is X = [ 3,] = Sice is arbitrary, we ca take = ad hece the From the above eample we have see that if X is a eigevector of A correspodig to the eigevalue λ ad A is ivertible, the X is a eigevector of eigevalues of correspodig to its eigevalue λ Also, we have see that the are the reciprocals of the eigevalues of A (obviously, the matri A is o-sigular ie A 0 04, IJMA All Rights Reserved 53
6 3 HE POWER MEHOD (IERAIVE MEHOD) his method is used for eigevalue problems where very few roots of the characteristic equatio are to be foud Let all the eigevalues be distict A arbitrary vector Y ca be epressed as Y = ax + ax + + ax = ax r r= r () o fid the umerically largest or domiat eigevalue ad its associate eigevector, we start with a arbitrary vector,0,0 It Y he vector is multiplied successively by the matri A A coveiet choice for ca also be take as [ 0,], [,], [ ] procedures require a iitial estimate of the quatity sought to be take Multiplyig the equatio () by A, we get () Y = AY = arax r = arλrx r r= r= () () Multiplyig by A agai ad lettig Y Y is [, 0] or [ ], or ay other vector of the correct size It must be oted that all iterative = AY, we get () = rλr r = rλr r = rλr r r= r= r= Y A a X a AX a X Proceedig like this, we get at the mth iteratio m Y = arλr Xr r= m m m = aλ X+ aλ X + + aλ X Suppose λ is the largest eigevalue he, m m m Y = λ [ ax + a( λ λ ) X + + a( λ λ ) X ] m he values ( i ) ( i ) m herefore, Y λ ax, a scalar multiple of λ λ ted to zero as m ad hece all the terms become egligible ecept the first term ( m ) m X, as m Also, Y + aλ + X for large m ( m+ ) Y ( m+ ) Y ad Y, we get λ for large m, the required Y herefore, takig the ratio of the magitudes of largest eigevalue It is clear that the rate of covergece depeds o the ratio of the moduli of the two largest eigevalues Whe this ratio is early uity, the covergece is very poor o avoid this, the followig procedure is adopted: (i) he arbitrary vector Y is selected such that the largest elemet of this vector is uity; ie the vector Y is put ito the ormalized form with the largest elemet uity (ii) he ormalized vector is multiplied by A (iii) he ew vector is ormalized by dividig each elemet by the largest elemet Let this largest elemet be l m (iv) he process is repeated util the values of l m ad l m + differ by some prescribed small value he value of l m gives the value of the largest eigevalue ad the vector Y is the eigevector correspodig to l m 3 SMALLES EIGENVALUE AND IS CORRESPONDING EIGENVECOR BY POWER MEHOD We have already stated that the eigevalues of, if A is o-sigular, are the reciprocals of the eigevalues of A herefore, the smallest eigevalue of A is the largest eigevalue of Hece we ca use the power method to determie the smallest eigevalue of A by workig with istead of A his procedure is illustrated i eample 04, IJMA All Rights Reserved 54
7 Eample : Let us ow cosider the same matri of eample ie eigevalue ad its correspodig eigevector by applyig power method to Solutio: Here, A = 5 istead of A 5 6 A = 5 We kow that from eample, = Let us fid the largest eigevalue of 35 7 Z = A ξ0 =, 05, 05 α = ξ = 5 46 Z = A ξ =, α 5, ξ 5 = = Z3 = A ξ =, α3 3, ξ3 3 = = Z4 = A ξ3 =, α 5 4 = 5, ξ4 = Z5 A ξ 4, α5 05, ξ = = = 5 = Z6 A ξ 5, α6 05, ξ = = = 6 = Z7 = A ξ6 =, 7 0, 7 0 α = ξ = Z8 = A ξ7 =, α8 = 005, ξ8 = to approimate the smallest by power method We begi with a iitial approimatio [ ] ξ 0 =, All these computatios show that α, α, coverges to, which is the largest eigevalue of ξ ξ ξ coverges to [ ] 0,,, ad X = 4, is the correspodig eigevector Sice the eigevalues of A are the, the smallest eigevalue of A is his is the same as the result we obtaied earlier (i reciprocals to those of eample by direct method ie by algebraic procedures) We have got the correspodig eigevector also the same as the oe obtaied earlier (i eample by direct method ie by algebraic procedures) 4 CONCLUSION I this paper, we have studied power method to approimate the smallest eigevalue ad its correspodig eigevector of real-valued square matrices Here, we used the ew iitial vector for the power method Maily, i this paper we have see that with eamples ad, if we apply the power method to, we will get the approimate largest eigevalue of ad its correspodig eigevector ad cosequetly we will get the approimate smallest eigevalue of A with the same eigevector as if X is a eigevector of A correspodig to the eigevalue λ ad A is ivertible, the X is a eigevector of correspodig to its eigevalue his approimate smallest λ eigevalue ad its correspodig eigevector appear to be approachig the eact smallest eigevalue ad its correspodig eigevector as we have obtaied earlier i eample by direct method ie by algebraic procedures 04, IJMA All Rights Reserved 55
8 REFERENCES [] GHGolub, CF Va Loa, Matri Computatios, Johs Hopkis Uiversity Press, 996 [] Richard Broso, Schaum s outlie of theory ad problems of matri operatios, McGRAW-HILL BOOK COMPANY [3] Devi Prasad, A Itroductio to Numerical Aalysis, hird Editio:006, Narosa Publishig House Pvt Ltd [4] C Moler, G Stewart, a Algorithm for Geeralized Matri Eigevalue Problems, SIAM J Numer Aal,Vol 0, No, 973 [5] Kedall E Atkiso, A Itroductio to Numerical Aalysis, Secod Editio: 988, Joh Wiley ad Sos Publishers [6] MK Jai, SRK Iyegar, RK Jai, Numerical Methods for Scietific ad Egieerig Computatio, Fourth Editio: 003, New Age Iteratioal (P) Limited, Publishers [7] GW Stewart, Itroductio to Matri Computatios, Academic Press, New York, 973 [8] GH Golub, CF Va Loa, Matri Computatios, Johs Hopkis Uiversity Press, Baltimore, 996 [9] JB Fraleigh, AR Beauregard, Liear Algebra, Addiso-Wesley Publishig Compay, 995 Source of support: Nil, Coflict of iterest: Noe Declared [Copy right 04 his is a Ope Access article distributed uder the terms of the Iteratioal Joural of Mathematical Archive (IJMA), which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited] 04, IJMA All Rights Reserved 56
Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationStochastic Matrices in a Finite Field
Stochastic Matrices i a Fiite Field Abstract: I this project we will explore the properties of stochastic matrices i both the real ad the fiite fields. We first explore what properties 2 2 stochastic matrices
More informationEigenvalues and Eigenvectors
5 Eigevalues ad Eigevectors 5.3 DIAGONALIZATION DIAGONALIZATION Example 1: Let. Fid a formula for A k, give that P 1 1 = 1 2 ad, where Solutio: The stadard formula for the iverse of a 2 2 matrix yields
More informationChapter 2 The Solution of Numerical Algebraic and Transcendental Equations
Chapter The Solutio of Numerical Algebraic ad Trascedetal Equatios Itroductio I this chapter we shall discuss some umerical methods for solvig algebraic ad trascedetal equatios. The equatio f( is said
More information1 Last time: similar and diagonalizable matrices
Last time: similar ad diagoalizable matrices Let be a positive iteger Suppose A is a matrix, v R, ad λ R Recall that v a eigevector for A with eigevalue λ if v ad Av λv, or equivaletly if v is a ozero
More informationInverse Matrix. A meaning that matrix B is an inverse of matrix A.
Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 50-004 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
More informationNotes on iteration and Newton s method. Iteration
Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f
More informationApply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.
Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α
More informationPAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
More informationBIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics
Mohd Yusuf Yasi / BIBECHANA 8 (2012) 31-36 : BMHSS, p. 31 BIBECHANA A Multidiscipliary Joural of Sciece, Techology ad Mathematics ISSN 2091-0762 (olie) Joural homepage: http://epjol.ifo/ide.php/bibechana
More informationSubject: Differential Equations & Mathematical Modeling-III
Power Series Solutios of Differetial Equatios about Sigular poits Subject: Differetial Equatios & Mathematical Modelig-III Lesso: Power series solutios of differetial equatios about Sigular poits Lesso
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationCommutativity in Permutation Groups
Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are
More informationCALCULATING FIBONACCI VECTORS
THE GENERALIZED BINET FORMULA FOR CALCULATING FIBONACCI VECTORS Stuart D Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithacaedu ad Dai Novak Departmet
More informationEstimation of Backward Perturbation Bounds For Linear Least Squares Problem
dvaced Sciece ad Techology Letters Vol.53 (ITS 4), pp.47-476 http://dx.doi.org/.457/astl.4.53.96 Estimatio of Bacward Perturbatio Bouds For Liear Least Squares Problem Xixiu Li School of Natural Scieces,
More information8. Applications To Linear Differential Equations
8. Applicatios To Liear Differetial Equatios 8.. Itroductio 8.. Review Of Results Cocerig Liear Differetial Equatios Of First Ad Secod Orders 8.3. Eercises 8.4. Liear Differetial Equatios Of Order N 8.5.
More informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More information-ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION
NEW NEWTON-TYPE METHOD WITH k -ORDER CONVERGENCE FOR FINDING SIMPLE ROOT OF A POLYNOMIAL EQUATION R. Thukral Padé Research Cetre, 39 Deaswood Hill, Leeds West Yorkshire, LS7 JS, ENGLAND ABSTRACT The objective
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts,
More informationThe Method of Least Squares. To understand least squares fitting of data.
The Method of Least Squares KEY WORDS Curve fittig, least square GOAL To uderstad least squares fittig of data To uderstad the least squares solutio of icosistet systems of liear equatios 1 Motivatio Curve
More informationComplex Analysis Spring 2001 Homework I Solution
Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle
More informationMatrix Algebra 2.2 THE INVERSE OF A MATRIX Pearson Education, Inc.
2 Matrix Algebra 2.2 THE INVERSE OF A MATRIX MATRIX OPERATIONS A matrix A is said to be ivertible if there is a matrix C such that CA = I ad AC = I where, the idetity matrix. I = I I this case, C is a
More informationMatrices and vectors
Oe Matrices ad vectors This book takes for grated that readers have some previous kowledge of the calculus of real fuctios of oe real variable It would be helpful to also have some kowledge of liear algebra
More informationChapter Unary Matrix Operations
Chapter 04.04 Uary atrix Operatios After readig this chapter, you should be able to:. kow what uary operatios meas, 2. fid the traspose of a square matrix ad it s relatioship to symmetric matrices,. fid
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationa for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a
Math S-b Lecture # Notes This wee is all about determiats We ll discuss how to defie them, how to calculate them, lear the allimportat property ow as multiliearity, ad show that a square matrix A is ivertible
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationTheorem: Let A n n. In this case that A does reduce to I, we search for A 1 as the solution matrix X to the matrix equation A X = I i.e.
Theorem: Let A be a square matrix The A has a iverse matrix if ad oly if its reduced row echelo form is the idetity I this case the algorithm illustrated o the previous page will always yield the iverse
More informationLemma Let f(x) K[x] be a separable polynomial of degree n. Then the Galois group is a subgroup of S n, the permutations of the roots.
15 Cubics, Quartics ad Polygos It is iterestig to chase through the argumets of 14 ad see how this affects solvig polyomial equatios i specific examples We make a global assumptio that the characteristic
More informationInternational Journal of Mathematical Archive-4(9), 2013, 1-5 Available online through ISSN
Iteratioal Joural o Matheatical Archive-4(9), 03, -5 Available olie through www.ija.io ISSN 9 5046 THE CUBIC RATE OF CONVERGENCE OF GENERALIZED EXTRAPOLATED NEWTON RAPHSON METHOD FOR SOLVING NONLINEAR
More informationSome Results on Certain Symmetric Circulant Matrices
Joural of Iformatics ad Mathematical Scieces Vol 7, No, pp 81 86, 015 ISSN 0975-5748 olie; 0974-875X prit Pulished y RGN Pulicatios http://wwwrgpulicatioscom Some Results o Certai Symmetric Circulat Matrices
More informationBounds for the Extreme Eigenvalues Using the Trace and Determinant
ISSN 746-7659, Eglad, UK Joural of Iformatio ad Computig Sciece Vol 4, No, 9, pp 49-55 Bouds for the Etreme Eigevalues Usig the Trace ad Determiat Qi Zhog, +, Tig-Zhu Huag School of pplied Mathematics,
More informationwavelet collocation method for solving integro-differential equation.
IOSR Joural of Egieerig (IOSRJEN) ISSN (e): 5-3, ISSN (p): 78-879 Vol. 5, Issue 3 (arch. 5), V3 PP -7 www.iosrje.org wavelet collocatio method for solvig itegro-differetial equatio. Asmaa Abdalelah Abdalrehma
More informationChimica Inorganica 3
himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule
More informationMAT 271 Project: Partial Fractions for certain rational functions
MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,
More informationOn Nonsingularity of Saddle Point Matrices. with Vectors of Ones
Iteratioal Joural of Algebra, Vol. 2, 2008, o. 4, 197-204 O Nosigularity of Saddle Poit Matrices with Vectors of Oes Tadeusz Ostrowski Istitute of Maagemet The State Vocatioal Uiversity -400 Gorzów, Polad
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationPOWER SERIES SOLUTION OF FIRST ORDER MATRIX DIFFERENTIAL EQUATIONS
Joural of Applied Mathematics ad Computatioal Mechaics 4 3(3) 3-8 POWER SERIES SOLUION OF FIRS ORDER MARIX DIFFERENIAL EQUAIONS Staisław Kukla Izabela Zamorska Istitute of Mathematics Czestochowa Uiversity
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oe-dimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationSubject: Differential Equations & Mathematical Modeling -III. Lesson: Power series solutions of Differential Equations. about ordinary points
Power series solutio of Differetial equatios about ordiary poits Subject: Differetial Equatios & Mathematical Modelig -III Lesso: Power series solutios of Differetial Equatios about ordiary poits Lesso
More informationMATH10212 Linear Algebra B Proof Problems
MATH22 Liear Algebra Proof Problems 5 Jue 26 Each problem requests a proof of a simple statemet Problems placed lower i the list may use the results of previous oes Matrices ermiats If a b R the matrix
More informationThe Jordan Normal Form: A General Approach to Solving Homogeneous Linear Systems. Mike Raugh. March 20, 2005
The Jorda Normal Form: A Geeral Approach to Solvig Homogeeous Liear Sstems Mike Raugh March 2, 25 What are we doig here? I this ote, we describe the Jorda ormal form of a matrix ad show how it ma be used
More informationU8L1: Sec Equations of Lines in R 2
MCVU U8L: Sec. 8.9. Equatios of Lies i R Review of Equatios of a Straight Lie (-D) Cosider the lie passig through A (-,) with slope, as show i the diagram below. I poit slope form, the equatio of the lie
More informationa for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a
Math E-2b Lecture #8 Notes This week is all about determiats. We ll discuss how to defie them, how to calculate them, lear the allimportat property kow as multiliearity, ad show that a square matrix A
More informationSolving a Nonlinear Equation Using a New Two-Step Derivative Free Iterative Methods
Applied ad Computatioal Mathematics 07; 6(6): 38-4 http://www.sciecepublishiggroup.com/j/acm doi: 0.648/j.acm.070606. ISSN: 38-5605 (Prit); ISSN: 38-563 (Olie) Solvig a Noliear Equatio Usig a New Two-Step
More informationChapter 4 : Laplace Transform
4. Itroductio Laplace trasform is a alterative to solve the differetial equatio by the complex frequecy domai ( s = σ + jω), istead of the usual time domai. The DE ca be easily trasformed ito a algebraic
More informationNewton Homotopy Solution for Nonlinear Equations Using Maple14. Abstract
Joural of Sciece ad Techology ISSN 9-860 Vol. No. December 0 Newto Homotopy Solutio for Noliear Equatios Usig Maple Nor Haim Abd. Rahma, Arsmah Ibrahim, Mohd Idris Jayes Faculty of Computer ad Mathematical
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More informationNumerical Solutions of Second Order Boundary Value Problems by Galerkin Residual Method on Using Legendre Polynomials
IOSR Joural of Mathematics (IOSR-JM) e-issn: 78-578, p-issn: 319-765X. Volume 11, Issue 6 Ver. IV (Nov. - Dec. 15), PP 1-11 www.iosrjourals.org Numerical Solutios of Secod Order Boudary Value Problems
More informationZ - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.
Z - Trasform The -trasform is a very importat tool i describig ad aalyig digital systems. It offers the techiques for digital filter desig ad frequecy aalysis of digital sigals. Defiitio of -trasform:
More informationNUMERICAL METHODS FOR SOLVING EQUATIONS
Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:
More informationIN many scientific and engineering applications, one often
INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, FEBRUARY 07 5 Secod Degree Refiemet Jacobi Iteratio Method for Solvig System of Liear Equatio Tesfaye Kebede Abstract Several
More informationMa 530 Introduction to Power Series
Ma 530 Itroductio to Power Series Please ote that there is material o power series at Visual Calculus. Some of this material was used as part of the presetatio of the topics that follow. What is a Power
More informationAlgorithm of Superposition of Boolean Functions Given with Truth Vectors
IJCSI Iteratioal Joural of Computer Sciece Issues, Vol 9, Issue 4, No, July ISSN (Olie: 694-84 wwwijcsiorg 9 Algorithm of Superpositio of Boolea Fuctios Give with Truth Vectors Aatoly Plotikov, Aleader
More informationLinear Differential Equations of Higher Order Basic Theory: Initial-Value Problems d y d y dy
Liear Differetial Equatios of Higher Order Basic Theory: Iitial-Value Problems d y d y dy Solve: a( ) + a ( )... a ( ) a0( ) y g( ) + + + = d d d ( ) Subject to: y( 0) = y0, y ( 0) = y,..., y ( 0) = y
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationResearch Article A New Second-Order Iteration Method for Solving Nonlinear Equations
Abstract ad Applied Aalysis Volume 2013, Article ID 487062, 4 pages http://dx.doi.org/10.1155/2013/487062 Research Article A New Secod-Order Iteratio Method for Solvig Noliear Equatios Shi Mi Kag, 1 Arif
More informationRecursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
More informationSOME METHODS FOR SOLVING FULLY FUZZY LINEAR SYSTEM OF EQUATIONS
SOME METHODS FOR SOLVING FULLY FUZZY LINEAR SYSTEM OF EQUATIONS Thesis submitted i partial fulfillmet of the requiremet for The award of the degree of Masters of Sciece i Mathematics ad Computig Submitted
More informationON THE HADAMARD PRODUCT OF BALANCING Q n B AND BALANCING Q n
TWMS J App Eg Math V5, N, 015, pp 01-07 ON THE HADAMARD PRODUCT OF ALANCING Q AND ALANCING Q MATRIX MATRIX PRASANTA KUMAR RAY 1, SUJATA SWAIN, Abstract I this paper, the matrix Q Q which is the Hadamard
More informationMost text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t
Itroductio to Differetial Equatios Defiitios ad Termiolog Differetial Equatio: A equatio cotaiig the derivatives of oe or more depedet variables, with respect to oe or more idepedet variables, is said
More informationA Hadamard-type lower bound for symmetric diagonally dominant positive matrices
A Hadamard-type lower boud for symmetric diagoally domiat positive matrices Christopher J. Hillar, Adre Wibisoo Uiversity of Califoria, Berkeley Jauary 7, 205 Abstract We prove a ew lower-boud form of
More informationApplication of Jordan Canonical Form
CHAPTER 6 Applicatio of Jorda Caoical Form Notatios R is the set of real umbers C is the set of complex umbers Q is the set of ratioal umbers Z is the set of itegers N is the set of o-egative itegers Z
More informationIterative Techniques for Solving Ax b -(3.8). Assume that the system has a unique solution. Let x be the solution. Then x A 1 b.
Iterative Techiques for Solvig Ax b -(8) Cosider solvig liear systems of them form: Ax b where A a ij, x x i, b b i Assume that the system has a uique solutio Let x be the solutio The x A b Jacobi ad Gauss-Seidel
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationProperties and Tests of Zeros of Polynomial Functions
Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by
More informationResearch Article Approximate Riesz Algebra-Valued Derivations
Abstract ad Applied Aalysis Volume 2012, Article ID 240258, 5 pages doi:10.1155/2012/240258 Research Article Approximate Riesz Algebra-Valued Derivatios Faruk Polat Departmet of Mathematics, Faculty of
More information, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx)
Cosider the differetial equatio y '' k y 0 has particular solutios y1 si( kx) ad y cos( kx) I geeral, ay liear combiatio of y1 ad y, cy 1 1 cy where c1, c is also a solutio to the equatio above The reaso
More informationA Block Cipher Using Linear Congruences
Joural of Computer Sciece 3 (7): 556-560, 2007 ISSN 1549-3636 2007 Sciece Publicatios A Block Cipher Usig Liear Cogrueces 1 V.U.K. Sastry ad 2 V. Jaaki 1 Academic Affairs, Sreeidhi Istitute of Sciece &
More information17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15
17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig
More informationPolynomials with Rational Roots that Differ by a Non-zero Constant. Generalities
Polyomials with Ratioal Roots that Differ by a No-zero Costat Philip Gibbs The problem of fidig two polyomials P(x) ad Q(x) of a give degree i a sigle variable x that have all ratioal roots ad differ by
More informationSession 5. (1) Principal component analysis and Karhunen-Loève transformation
200 Autum semester Patter Iformatio Processig Topic 2 Image compressio by orthogoal trasformatio Sessio 5 () Pricipal compoet aalysis ad Karhue-Loève trasformatio Topic 2 of this course explais the image
More informationA NEW CLASS OF 2-STEP RATIONAL MULTISTEP METHODS
Jural Karya Asli Loreka Ahli Matematik Vol. No. (010) page 6-9. Jural Karya Asli Loreka Ahli Matematik A NEW CLASS OF -STEP RATIONAL MULTISTEP METHODS 1 Nazeeruddi Yaacob Teh Yua Yig Norma Alias 1 Departmet
More information1+x 1 + α+x. x = 2(α x2 ) 1+x
Math 2030 Homework 6 Solutios # [Problem 5] For coveiece we let α lim sup a ad β lim sup b. Without loss of geerality let us assume that α β. If α the by assumptio β < so i this case α + β. By Theorem
More informationBasic Iterative Methods. Basic Iterative Methods
Abel s heorem: he roots of a polyomial with degree greater tha or equal to 5 ad arbitrary coefficiets caot be foud with a fiite umber of operatios usig additio, subtractio, multiplicatio, divisio, ad extractio
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More information1 1 2 = show that: over variables x and y. [2 marks] Write down necessary conditions involving first and second-order partial derivatives for ( x0, y
Questio (a) A square matrix A= A is called positive defiite if the quadratic form waw > 0 for every o-zero vector w [Note: Here (.) deotes the traspose of a matrix or a vector]. Let 0 A = 0 = show that:
More informationON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS
Joural of Algebra, Number Theory: Advaces ad Applicatios Volume, Number, 00, Pages 7-89 ON SOME DIOPHANTINE EQUATIONS RELATED TO SQUARE TRIANGULAR AND BALANCING NUMBERS OLCAY KARAATLI ad REFİK KESKİN Departmet
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationChapter 2: Numerical Methods
Chapter : Numerical Methods. Some Numerical Methods for st Order ODEs I this sectio, a summar of essetial features of umerical methods related to solutios of ordiar differetial equatios is give. I geeral,
More informationECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations
ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows
More informationNumerical Solution of Non-linear Equations
Numerical Solutio of Noliear Equatios. INTRODUCTION The most commo reallife problems are oliear ad are ot ameable to be hadled by aalytical methods to obtai solutios of a variety of mathematical problems.
More informationSection 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations
Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?
More informationSection 1 of Unit 03 (Pure Mathematics 3) Algebra
Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course
More informationTHE EIGENVALUE PROBLEM
November 5, 2013 APPENDIX D HE EIGENVALUE PROLEM Eigevalues ad Eigevectors are Properties of the Equatios that Simulate the ehavior of a Real Structure D.1 INRODUCION he classical mathematical eigevalue
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationAssignment 2 Solutions SOLUTION. ϕ 1 Â = 3 ϕ 1 4i ϕ 2. The other case can be dealt with in a similar way. { ϕ 2 Â} χ = { 4i ϕ 1 3 ϕ 2 } χ.
PHYSICS 34 QUANTUM PHYSICS II (25) Assigmet 2 Solutios 1. With respect to a pair of orthoormal vectors ϕ 1 ad ϕ 2 that spa the Hilbert space H of a certai system, the operator  is defied by its actio
More informationNumerical Conformal Mapping via a Fredholm Integral Equation using Fourier Method ABSTRACT INTRODUCTION
alaysia Joural of athematical Scieces 3(1): 83-93 (9) umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod 1 Ali Hassa ohamed urid ad Teh Yua Yig 1, Departmet of athematics, Faculty
More informationAH Checklist (Unit 3) AH Checklist (Unit 3) Matrices
AH Checklist (Uit 3) AH Checklist (Uit 3) Matrices Skill Achieved? Kow that a matrix is a rectagular array of umbers (aka etries or elemets) i paretheses, each etry beig i a particular row ad colum Kow
More informationResearch Article Powers of Complex Persymmetric Antitridiagonal Matrices with Constant Antidiagonals
Hidawi Publishig orporatio ISRN omputatioal Mathematics, Article ID 4570, 5 pages http://dx.doi.org/0.55/04/4570 Research Article Powers of omplex Persymmetric Atitridiagoal Matrices with ostat Atidiagoals
More informationTHE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.
THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of
More informationChapter 9 - CD companion 1. A Generic Implementation; The Common-Merge Amplifier. 1 τ is. ω ch. τ io
Chapter 9 - CD compaio CHAPTER NINE CD-9.2 CD-9.2. Stages With Voltage ad Curret Gai A Geeric Implemetatio; The Commo-Merge Amplifier The advaced method preseted i the text for approximatig cutoff frequecies
More informationTopics in Eigen-analysis
Topics i Eige-aalysis Li Zajiag 28 July 2014 Cotets 1 Termiology... 2 2 Some Basic Properties ad Results... 2 3 Eige-properties of Hermitia Matrices... 5 3.1 Basic Theorems... 5 3.2 Quadratic Forms & Noegative
More informationA Note on the Symmetric Powers of the Standard Representation of S n
A Note o the Symmetric Powers of the Stadard Represetatio of S David Savitt 1 Departmet of Mathematics, Harvard Uiversity Cambridge, MA 0138, USA dsavitt@mathharvardedu Richard P Staley Departmet of Mathematics,
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More information