IN many scientific and engineering applications, one often
|
|
- Jesse Fitzgerald
- 5 years ago
- Views:
Transcription
1 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 iterative techiques for the solutio of liear system of equatios have bee proposed i differet literature i the pasti this paper, we preset a Secod degree of refiemet Jacobi Iteratio method for solvig system of liear equatio, Ax = b ad we cosider few umerical examples ad spectral radius to show that the effective of the Secod degree of refiemet Jacobi Iteratio Method (SDRJ) i compariso with other methods of First degree Jacobi (FDJ), First degree Refiemet Jacobi (FDRJ) ad Secod degree Jacobi (SDJ) method Idex Terms Jacobi iteratio, secod degree refiemet, system of liear equatios I INTRODUCTION IN may scietific ad egieerig applicatios, oe ofte comes across with a problem of fidig the solutio of a system of liear equatios writte as the followig equatio i matrix form: Ax = b () where A is a osigular matrix of size, x ad b are -dimesioal vectors Splittig the matrix A [] as : A = D L U () where D is a diagoal matrix ad L ad U are strictly lower ad upper triagular part of A respectively A geeral first degree liear statioary iterative method for the solutio of the system of equatio () may be defied i the form: x (+) = Hx () +C (3) where x (+) ad x () are the approximatio for x at the ( + ) th ad th iteratios respectively, H is called the iterative matrix depedig o matrix A ad C is a colum vector The iteratio system x (+) = Hx () +C is coverge if ad oly if the spectral radius of H are less tha uity, ie σ(h) < The first degree iterative method of Jacobi (FDJ) method for the solutio of () is defied as: x (+) = D (L +U)x () + D b (4) ad the first degree refiemet Jacobi (FDRJ) method ca be obtaied i the form of : where x (+) = (D (L +U)) x () + (I + D (L +U))D b (5) X (+) = H RJ X () +C RJ, (6) H RJ = [D (L +U)],C RJ = [I + D (L +U)]D b (7) Mauscript received December, 06; accepted February, 07 The author is with the Departmet of Mathematics, College of Sciece, Bahir Dar Uiversity, Bahir Dar, Ethiopia tk_ke@yahoocom The liear statioary ay secod degree method is give by [] x (+) = x () + a(x () x ( ) ) + (x (+) x () ) (8) Here x (+) appearig i the right had side as give i equatio (3) is completely cosistet for ay costat a ad such that 0 X (+) = X () + a(x () X ( ) ) + (HX () +C X () ) X (+) = X () + ax () ax ( ) + HX () + C X () X (+) = X () + ax () X () + HX () + C ax ( ) X (+) = ( + a b)x () + HX () ax ( ) + C X (+) = [( + a )I + H]X () ax ( ) + C where X (+) = GX () + H X ( ) + K (9) G = ( + a )I + H (0) H = ai () K = C () The liear statioary ay secod degree method is give by [] ca be writte i umber (6) or umber (7) with (8), (9) ad (0) coditios O the other way equatio () ca be solved usig the secod degree Jacobi(SDJ) statioary iterative method usig x (+) = D (L +U)x () ax ( ) + k x (+) = [D (L +U)x () + k ] ax ( ) (3) For optimal values of a ad Where k = D b If A is a row strictly diagoal domiat (SDD) matrix, the the Jacobi method coverges for ay arbitrary choice of the iitial approximatio [3] I this paper, we costruct a ew method of solvig a liear system of the form Ax = b that arise i ay egieerig ad applied sciece The outlie of this paper is as follows: we itroduce secod degree refiemet Jacobi (SDRJ) iterative method i accordace this we will see the relatioship betwee spectral radius of first degree Jacobi (FDJ), first degree refiemet Jacobi (FDRJ), Secod degree Jacobi (SDJ) methods ad Secod degree refiemet Jacobi iteratio (SDRJ) methods are give Based o the methods ad results, few umerical examples are cosidered to show that the efficiecy of the ew method i compariso with the existig FDJ, FDRJ ad SDJ methods Fially discussio ad coclusio made at Sectio V
2 6 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, FEBRUARY 07 II SECOND DEGREE REFINEMENT JACOBI (SDRJ) ITERATIVE METHOD Theorem : If matrix A is o sigular PD ad SDD matrix with A = D L U, the the Secod degree of Refiemet Jacobi iterative method is: [x (+) = [D (L +U)] x () aix ( ) + (I + D (L +U))D b] for ay iitial guess ad the optimal values for a ad Give: A is o sigular PD ad SDD matrix ad A = D-L- U Required: the secod degree of refiemet Jacobi iterative method is: x (+) = [D (L +U)+k ]x () a x ( ) Proof: ow cosider equatio (5)ad (6), so oe ca get : x (+) = x () +a(x () x ( ) )+ (H RJ x () +C RJ x () ) (4) This also ca be writte as follows after some computatio: X (+) = G RJ x () + F RJ x ( ) + K RJ (5) where G RJ = ( + a )I + H RJ,F RJ = ai ad K RJ = C RJ By usig Golub ad Varga [] ( ) ( )( ) ( ) x () 0 I x ( ) 0 x (+) = F RJ G RJ x () + (6) K RJ ( x () x (+) ) ( x ( ) = Ĝ x () ( 0 I where Ĝ = F RJ G RJ ) ( 0 + K RJ ) ), (7) The above equatio coverges to the exact solutio if σ(ĝ) <, ie the spectra radius of Ĝ is less tha oe I order to solve the spectra radius of Ĝ, first we have to solve the eigevalues λ of Ĝ σ(ĝ) < iff all roots λ RJ of det(λ RJ I λ RJ G RJ F RJ ) = 0 (8) ie det(λ RJ I λ RJ G RJ F RJ ) = 0 det(λ RJ I λ RJ [( + a )I + H RJ ] + ai) = 0 det(λ RJ I λ RJ ( + a )I λ RJ H RJ + ai) = 0 det( λ RJ [ λ RJ I + ( + a ) I + H RJ a λ RJ I] = 0 ( λ RJ ) det(h RJ + ( + a ) I λ RJ + a I) = 0 λ RJ det(h RJ + ( + a ) I λ RJ + a I) = 0,sice( λ RJ ) 0 λ RJ (9) Thus, the eigevalues λ RJ of Ĝ are related to the eigevalues of H RJ with H J is + ( + a ) = (a + λ RJ ) (0) λ RJ As the image of the circle, Let the eigevalue λ RJ = ve iθ = v(cosθ +isiθ) is the ellipse, the substitutig this i equatio (8), we obtai + ( + a ) + + a + + a = (veiθ ) + a ve iθ = veiθ + a ve iθ = = ( + a ) v(cosθ + isiθ) + vcosθ + + i vsiθ a(cosθ isiθ) v + acosθ v i asiθ v = (v+ a v )cosθ + a +i (v a )siθ () v ie Re = (v + a v )cosθ (+a ) Im = (v a v )siθ cosθ = Re + (+a ) (v + a v ) () siθ = We kow that cos θ + si θ = [ Re + (+a ) (v + a v ) ] + [ IM (v + a v ) (3) Im (v a v )] = (4) cetre = c(h,k) = ( ( + a ),0) (5) cetre = c(h,k) = ( ( + a ),0) (6) Legth of semi-major axis = a / = (v + a v ) (7) Legth of semi-mior axis = b / = (v a v ) (8) Foci = F = (h c,0) = ( + a a,0) = (α,0) (9) Foci = F = (h + c,0) = ( + + a,0) = (β,0) v = (h a,0) = ( + a (v + a v ),0) v = (h + a,0) = ( + a + (v + a v ),0) v 3 = (h,k + b ) = ( + a, (v a v )) v 4 = (h,k b ) = ( + a, (v a v )) (30)
3 KEBEDE et al: SECOND DEGREE REFINEMENT JACOBI ITERATION METHOD FOR SOLVING SYSTEM OF LINEAR EQUATION 7 Before we prove the theorem let us prove the followig Lemmas Lemma : If is real, the α β <, for ay foci α ad β which are real Proof: We kow that is a real umber We require that α β <, for ay foci α ad β which are real The proof is as follows: = (v + a v )cosθ + a, sice is a real umber I this equatio θ varies (v+ a v ) + a µ m (v+ a v )cosθ + a, sice cosθ α µ J β < (3) Because α ad β from equatio (6) ad (7) ad to be coverget ρ(h RJ ) < so all the eigevalues must be less tha Lemma 3: If the eigevalues of H RJ < are real ad lie i the iterval Proof: α µ J β <, the the optimal choices of a ad must satisfy the followig coditios: b) α + β c) a) v = a (3) = + a (33) = v (34) d) v = (α + β) ( + v ) (35) Give: the eigevalues of H RJ < are real ad lie i the iterval α µ J β < Required: proof of a) util d) Proof : a) we kow µ m is real, the (v a v )siθ = 0, we have siθ = 0 or (v a v ) = 0, so we get v = a or siθ = 0, from the secod equatio we have θ = π, = 0,, Therefore V = a b) From the (6) ad (7) ad from Lemma (a),we get: α = v + a ad β = v + a α + β = + a (mid poit formula) c) We kow from (b) above we have α = v +a β = v, the oe ca get +a d) From Lemma (b), we have α + β α + β (α + β) = v = + a Divide equatio (3) by (33),we get β α (α+β) = ( + a ) ad = + ( + a ) = + a (36) = v +a (α + β) = v + a v = (α + β) ( + v ) Lemma 4: If is the spectral radius of H RJ, the = (α + β) (37) Proof: Give: is the spectral radius of H RJ Required: = β α (α+β) Proof: we kow that = (v + a v )cosθ +a By defiitio of derivative of fuctios i calculus d dθ = d dθ [ (v+ a v )cosθ + a ] = (v+ a v )siθ To calculate the maximum ad miimum value, the above equatio equates to zero (v + a v )siθ = 0 siθ = 0 θ = 0,π,π Whe θ = 0, the = (v + a v ) +a Whe θ = π, the = (v + a v ) +a Whe θ = π, the = (v + a v ) +a From the above the maximum value occurs at θ = 0 ad π max i= () = (v + a v ) + a The miimum value occurs at θ = π = β mi i= = (v + a v ) + a = α = max = max (v + a i= i= v )cosθ + a = (v + a v ) + a,sice cosθ ad
4 8 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, FEBRUARY 07 = mi i= ( ) = mi (v + a v )cosθ + a = (v + a v ) + a = v + a = v + a + = (α + β) + = β by equatio (3) ad (33) Therefore = β = mi i= ( ) = (v + a v ) + a = v + a + = α Therefore = α From the previous two results, we obtai = ad = (α + β) The divide the previous two equatios, we get = β α (α+β) Now let us determie the values of a ad First, let us fid a from Lemma 3d equatio (3) v = by lemma 3d, we have (α + β) ( + v ) v = ( + v ) v v + = 0 This is the equatio of quadratic whose graph is a parabola ad the miimum value occurs at p=( µ, RJ µ ) sice µ RJ RJ > 0 Oe ca solve by quadratic formula of the above equatio: v = ± 4 4 = ± v = + ad v = The smallest value is v = Let + v = ω a = ω + v = + µ RJ µ RJ a =, ( + ) sice a = v Secodly, let us fid = = 4v β α 4v = ( + v ) by usig equatio (7) 4 = ( + µ RJ )( (α + β)) Lemma 5: If matrix A is positive defiite matrix ad if H RJ is Jacobi iterative matrix, the β = α = = σ(h RJ ) Proof: Give: matrix A is positive defiite matrix ad if H RJ is Jacobi iterative matrix Required: β = α = = σh RJ Proof: I order to prove this Lemma, we have to use Lemma 3 = max i= = (v + a v ) + a = (v + a v ) + a β = α = Now we ca fid the optimal value of a ad µ RJ = α = β ie a = b (+ µ RJ ) =, sice β = α + µ RJ α +β = 0 Now let us fid secod degree of Refiemet Jacobi (SDRJ) method: sice β = α From the secod degree + a + a = α + β = α α ( + a ) = 0 x (+) = G RJ x () + F RJ x ( ) + k RJ x (+) = [( + a)i + H RJ ]x () + ( ai)x ( ) + C RJ where x (+) = H RJ x () ax ( ) + CRJ RJ x (+) = [D (L +U)x () + k ] ax ( ), a =, = ( + µ ) + µ µ III RELATIONSHIP BETWEEN SPECTRAL RADIUS As we have see above the spectral radius of First degree Jacobi method(fdj) is µ First degree Refiemet Jacobi method(fdj) is = µ Secod degree Jacobi method(sdj) is a = µ + µ Secod degree Refiemet Jacobi method(sdgj) is a = µ (+ µ ) µ µ ) That is oe ca see (+ + µ µ sice + µ > 0 ad also µ sice 0 < µ < µ
5 KEBEDE et al: SECOND DEGREE REFINEMENT JACOBI ITERATION METHOD FOR SOLVING SYSTEM OF LINEAR EQUATION 9 IV NUMERICAL EXAMPLES a Solve the followig SDD matrix usig FDJ,FDGJ,SDJ ad SDRJ iterative methods 4x x x 3 = 3 x + 6x + x 3 = 9 x + x + 7x 3 = 6 Solutio: all results are based o the give data ad we get the spectral radius as follows Method FDJ FDRJ SDJ SDRJ Spectral radius b Solve the followig PD matrix usig FDJ, FDGJ, SDJ ad SDRJ iterative methods 3x x x 3 = x + 3x + x 3 = 3 x + x + 4x 3 = 7 Solutio: all results are based o the give data we get the spectral radius as follows Method FDJ FDRJ SDJ SDRJ Spectral radius The detailed experimetal results are writte i the appedix Table I shows that FDRJ method coverges faster tha FDJ method for SDD matrix Table II shows that SDRJ method coverges faster tha the SDJ method for SDD matrix Table III shows that FDRJ method coverges faster tha FDJ method for PD matrix Table IV shows that SDRJ method coverges faster tha the SDJ method for PD matrix REFERENCES [] V K Vatti ad T K Eeyew, A refiemet of gauss-seidel method for solvig of liear system of equatios, It J Cotemp Math Scieces, vol 6, o 3, pp 7, 0 [] D Youg ad D Kicaid, Liear statioary secod-degree methods for the solutio of large liear systems, Uiv of Texas, Ceter for Numerical Aalysis, Austi, TX (Uited States), Tech Rep, 990 [3] F Naeimi Dafchahi, A ew refiemet of jacobi method for solutio of liear system equatios Ax=b, It J Cotemp Math Scieces,, vol 3, o 7, pp 89 87, 008 APPENDIX TABLE I ALL NUMERICAL RESULT OF SDD MATRIX FOR FDJ AND FDRJ OF EXAMPLE First degree Jacobi (FDJ) First degree refiemet Jacobi (FDRJ) () x () x () x () x V CONCLUSIONS As we have see i this report for SDD ad PD matrix, we ca otice that FDJ, FDRJ ad SDJ are reasoable to approximate the exact solutio of system of liear equatios at a certai give coditio But they are relatively slow to coverge to the exact solutio However, the Secod degree of refiemet Jacobi iterative method for solvig system of liear equatios are reasoable ad efficiet way of approximatig the exact solutio of system of liear equatios Numerical results of spectral radius show that, SDRJ methods coverge with a small umber of iteratio steps for solvig systems of liear equatios I geeral, the results of umerical examples cosidered clearly demostrate the accuracy of the methods developed i this paper It is cojectured that the rate of covergece of the method that developed i this paper ca be further ehaced by usig extrapolatig techiques TABLE II ALL NUMERICAL RESULT OF SDD MATRIX FOR SDJ AND SDRJ OF EXAMPLE Secod degree Jacobi (SDJ) Secod degree refiemet Jacobi (SDRJ) () x () x () x () x VI ACKNOWLEDGEMENTS I would like to express my sicere appreciatio to Professor Vatti, Bassava Kumar, Departmet of Egieerig mathematics, College of Egieerig, Adhra Uiversity Visakhapatam , Idia My Studet Gashaye Dessalew ad Yilkal Abiyu ad also my fried Bekalu Tarekeg, Abraham Kassa ad Teshager Assefa for their commets ad suggestio for preparatio of the work
6 0 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, FEBRUARY 07 TABLE III ALL NUMERICAL RESULT OF POSITIVE DEFINITE (PD) MATRIX FOR FDJ AND FDRJ OF EXAMPLE First degree Jacobi (FDJ) First degree refiemet Jacobi (FDRJ) () x () x () x () x TABLE IV ALL NUMERICAL RESULT OF POSITIVE DEFINITE (PD) MATRIX FOR SDJ AND SDRJ OF EXAMPLE Secod degree Jacobi (SDJ) Secod degree refiemet Jacobi (SDFJ) () x () x () x () x
Iterative 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 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 informationSolution of Differential Equation from the Transform Technique
Iteratioal Joural of Computatioal Sciece ad Mathematics ISSN 0974-3189 Volume 3, Number 1 (2011), pp 121-125 Iteratioal Research Publicatio House http://wwwirphousecom Solutio of Differetial Equatio from
More informationA collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation
Iteratioal Joural of Mathematics Research. ISSN 0976-5840 Volume 9 Number 1 (017) pp. 45-51 Iteratioal Research Publicatio House http://www.irphouse.com A collocatio method for sigular itegral equatios
More informationNEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE
UPB Sci Bull, Series A, Vol 79, Iss, 207 ISSN 22-7027 NEW FAST CONVERGENT SEQUENCES OF EULER-MASCHERONI TYPE Gabriel Bercu We itroduce two ew sequeces of Euler-Mascheroi type which have fast covergece
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This
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 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 informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
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 informationMATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and
MATH01 Real Aalysis (2008 Fall) Tutorial Note #7 Sequece ad Series of fuctio 1: Poitwise Covergece ad Uiform Covergece Part I: Poitwise Covergece Defiitio of poitwise covergece: A sequece of fuctios f
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
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 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 informationMath 113, Calculus II Winter 2007 Final Exam Solutions
Math, Calculus II Witer 7 Fial Exam Solutios (5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute x x + dx The check your aswer usig the Evaluatio Theorem Solutio: I this
More information( ) (( ) ) ANSWERS TO EXERCISES IN APPENDIX B. Section B.1 VECTORS AND SETS. Exercise B.1-1: Convex sets. are convex, , hence. and. (a) Let.
Joh Riley 8 Jue 03 ANSWERS TO EXERCISES IN APPENDIX B Sectio B VECTORS AND SETS Exercise B-: Covex sets (a) Let 0 x, x X, X, hece 0 x, x X ad 0 x, x X Sice X ad X are covex, x X ad x X The x X X, which
More informationMATH 10550, EXAM 3 SOLUTIONS
MATH 155, EXAM 3 SOLUTIONS 1. I fidig a approximate solutio to the equatio x 3 +x 4 = usig Newto s method with iitial approximatio x 1 = 1, what is x? Solutio. Recall that x +1 = x f(x ) f (x ). Hece,
More informationLecture 8: Solving the Heat, Laplace and Wave equations using finite difference methods
Itroductory lecture otes o Partial Differetial Equatios - c Athoy Peirce. Not to be copied, used, or revised without explicit writte permissio from the copyright ower. 1 Lecture 8: Solvig the Heat, Laplace
More informationA note on the modified Hermitian and skew-hermitian splitting methods for non-hermitian positive definite linear systems
A ote o the modified Hermitia ad skew-hermitia splittig methods for o-hermitia positive defiite liear systems Shi-Liag Wu Tig-Zhu Huag School of Applied Mathematics Uiversity of Electroic Sciece ad Techology
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 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 informationSome New Iterative Methods for Solving Nonlinear Equations
World Applied Scieces Joural 0 (6): 870-874, 01 ISSN 1818-495 IDOSI Publicatios, 01 DOI: 10.589/idosi.wasj.01.0.06.830 Some New Iterative Methods for Solvig Noliear Equatios Muhammad Aslam Noor, Khalida
More informationHigher-order iterative methods by using Householder's method for solving certain nonlinear equations
Math Sci Lett, No, 7- ( 7 Mathematical Sciece Letters A Iteratioal Joural http://dxdoiorg/785/msl/5 Higher-order iterative methods by usig Householder's method for solvig certai oliear equatios Waseem
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 informationMIDTERM 3 CALCULUS 2. Monday, December 3, :15 PM to 6:45 PM. Name PRACTICE EXAM SOLUTIONS
MIDTERM 3 CALCULUS MATH 300 FALL 08 Moday, December 3, 08 5:5 PM to 6:45 PM Name PRACTICE EXAM S Please aswer all of the questios, ad show your work. You must explai your aswers to get credit. You will
More informationMath 312 Lecture Notes One Dimensional Maps
Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity 21-23 February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example,
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 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 informationf(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim
Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =
More informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More informationMath 257: Finite difference methods
Math 257: Fiite differece methods 1 Fiite Differeces Remember the defiitio of a derivative f f(x + ) f(x) (x) = lim 0 Also recall Taylor s formula: (1) f(x + ) = f(x) + f (x) + 2 f (x) + 3 f (3) (x) +...
More informationThe Phi Power Series
The Phi Power Series I did this work i about 0 years while poderig the relatioship betwee the golde mea ad the Madelbrot set. I have fially decided to make it available from my blog at http://semresearch.wordpress.com/.
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 informationTwo-step Extrapolated Newton s Method with High Efficiency Index
Jour of Adv Research i Damical & Cotrol Systems Vol. 9 No. 017 Two-step Etrapolated Newto s Method with High Efficiecy Ide V.B. Kumar Vatti Dept. of Egieerig Mathematics Adhra Uiversity Visakhapatam Idia.
More informationMath 113 Exam 4 Practice
Math Exam 4 Practice Exam 4 will cover.-.. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for
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 informationComparison 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 informationModified Decomposition Method by Adomian and. Rach for Solving Nonlinear Volterra Integro- Differential Equations
Noliear Aalysis ad Differetial Equatios, Vol. 5, 27, o. 4, 57-7 HIKARI Ltd, www.m-hikari.com https://doi.org/.2988/ade.27.62 Modified Decompositio Method by Adomia ad Rach for Solvig Noliear Volterra Itegro-
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 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 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 informationMATH 31B: MIDTERM 2 REVIEW
MATH 3B: MIDTERM REVIEW JOE HUGHES. Evaluate x (x ) (x 3).. Partial Fractios Solutio: The umerator has degree less tha the deomiator, so we ca use partial fractios. Write x (x ) (x 3) = A x + A (x ) +
More informationNumerical Solution of the Two Point Boundary Value Problems By Using Wavelet Bases of Hermite Cubic Spline Wavelets
Australia Joural of Basic ad Applied Scieces, 5(): 98-5, ISSN 99-878 Numerical Solutio of the Two Poit Boudary Value Problems By Usig Wavelet Bases of Hermite Cubic Splie Wavelets Mehdi Yousefi, Hesam-Aldie
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas 1 to compute
Math, Calculus II Fial Eam Solutios. 5 poits) Use the limit defiitio of the defiite itegral ad the sum formulas to compute 4 d. The check your aswer usig the Evaluatio Theorem. ) ) Solutio: I this itegral,
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 informationINEQUALITIES BJORN POONEN
INEQUALITIES BJORN POONEN 1 The AM-GM iequality The most basic arithmetic mea-geometric mea (AM-GM) iequality states simply that if x ad y are oegative real umbers, the (x + y)/2 xy, with equality if ad
More informationExplicit Group Methods in the Solution of the 2-D Convection-Diffusion Equations
Proceedigs of the World Cogress o Egieerig 00 Vol III WCE 00 Jue 0 - July 00 Lodo U.K. Explicit Group Methods i the Solutio of the -D Covectio-Diffusio Equatios a Kah Bee orhashidah Hj. M. Ali ad Choi-Hog
More informationFor use only in [the name of your school] 2014 FP2 Note. FP2 Notes (Edexcel)
For use oly i [the ame of your school] 04 FP Note FP Notes (Edexcel) Copyright wwwpgmathscouk - For AS, A otes ad IGCSE / GCSE worksheets For use oly i [the ame of your school] 04 FP Note BLANK PAGE Copyright
More informationMETHOD OF FUNDAMENTAL SOLUTIONS FOR HELMHOLTZ EIGENVALUE PROBLEMS IN ELLIPTICAL DOMAINS
Please cite this article as: Staisław Kula, Method of fudametal solutios for Helmholtz eigevalue problems i elliptical domais, Scietific Research of the Istitute of Mathematics ad Computer Sciece, 009,
More information3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials
Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered
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 informationSynopsis of Euler s paper. E Memoire sur la plus grande equation des planetes. (Memoir on the Maximum value of an Equation of the Planets)
1 Syopsis of Euler s paper E105 -- Memoire sur la plus grade equatio des plaetes (Memoir o the Maximum value of a Equatio of the Plaets) Compiled by Thomas J Osler ad Jase Adrew Scaramazza Mathematics
More informationMon Feb matrix inverses. Announcements: Warm-up Exercise:
Math 225-4 Week 6 otes We will ot ecessarily fiish the material from a give day's otes o that day We may also add or subtract some material as the week progresses, but these otes represet a i-depth outlie
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 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 informationBenaissa Bernoussi Université Abdelmalek Essaadi, ENSAT de Tanger, B.P. 416, Tanger, Morocco
EXTENDING THE BERNOULLI-EULER METHOD FOR FINDING ZEROS OF HOLOMORPHIC FUNCTIONS Beaissa Beroussi Uiversité Abdelmalek Essaadi, ENSAT de Tager, B.P. 416, Tager, Morocco e-mail: Beaissa@fstt.ac.ma Mustapha
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 informationMATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.
MATH 1080: Calculus of Oe Variable II Fall 2017 Textbook: Sigle Variable Calculus: Early Trascedetals, 7e, by James Stewart Uit 3 Skill Set Importat: Studets should expect test questios that require a
More informationIt is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.
Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable
More informationSOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.
SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad
More informationCALCULUS BASIC SUMMER REVIEW
CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=
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 informationPhysics 116A Solutions to Homework Set #1 Winter Boas, problem Use equation 1.8 to find a fraction describing
Physics 6A Solutios to Homework Set # Witer 0. Boas, problem. 8 Use equatio.8 to fid a fractio describig 0.694444444... Start with the formula S = a, ad otice that we ca remove ay umber of r fiite decimals
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 informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
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 informationSection 13.3 Area and the Definite Integral
Sectio 3.3 Area ad the Defiite Itegral We ca easily fid areas of certai geometric figures usig well-kow formulas: However, it is t easy to fid the area of a regio with curved sides: METHOD: To evaluate
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationSection A assesses the Units Numerical Analysis 1 and 2 Section B assesses the Unit Mathematics for Applied Mathematics
X0/70 NATIONAL QUALIFICATIONS 005 MONDAY, MAY.00 PM 4.00 PM APPLIED MATHEMATICS ADVANCED HIGHER Numerical Aalysis Read carefully. Calculators may be used i this paper.. Cadidates should aswer all questios.
More informationLecture #20. n ( x p i )1/p = max
COMPSCI 632: Approximatio Algorithms November 8, 2017 Lecturer: Debmalya Paigrahi Lecture #20 Scribe: Yua Deg 1 Overview Today, we cotiue to discuss about metric embeddigs techique. Specifically, we apply
More informationSolutions to quizzes Math Spring 2007
to quizzes Math 4- Sprig 7 Name: Sectio:. Quiz a) x + x dx b) l x dx a) x + dx x x / + x / dx (/3)x 3/ + x / + c. b) Set u l x, dv dx. The du /x ad v x. By Itegratio by Parts, x(/x)dx x l x x + c. l x
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More informationCALCULUS IN A NUTSHELL INTRODUCTION:
CALCULUS IN A NUTSHELL INTRODUCTION: Studets are usually itroduced to basic calculus i twelfth grade i high school or the first year of college. The course is typically stretched out over oe year ad ivolves
More informationEllipsoid Method for Linear Programming made simple
Ellipsoid Method for Liear Programmig made simple Sajeev Saxea Dept. of Computer Sciece ad Egieerig, Idia Istitute of Techology, Kapur, INDIA-08 06 December 3, 07 Abstract I this paper, ellipsoid method
More informationLinear Regression Demystified
Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to
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 informationMH1101 AY1617 Sem 2. Question 1. NOT TESTED THIS TIME
MH AY67 Sem Questio. NOT TESTED THIS TIME ( marks Let R be the regio bouded by the curve y 4x x 3 ad the x axis i the first quadrat (see figure below. Usig the cylidrical shell method, fid the volume of
More informationPeriod Function of a Lienard Equation
Joural of Mathematical Scieces (4) -5 Betty Joes & Sisters Publishig Period Fuctio of a Lieard Equatio Khalil T Al-Dosary Departmet of Mathematics, Uiversity of Sharjah, Sharjah 77, Uited Arab Emirates
More informationComputation of Error Bounds for P-matrix Linear Complementarity Problems
Mathematical Programmig mauscript No. (will be iserted by the editor) Xiaoju Che Shuhuag Xiag Computatio of Error Bouds for P-matrix Liear Complemetarity Problems Received: date / Accepted: date Abstract
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More information1. C only. 3. none of them. 4. B only. 5. B and C. 6. all of them. 7. A and C. 8. A and B correct
M408D (54690/54695/54700), Midterm # Solutios Note: Solutios to the multile-choice questios for each sectio are listed below. Due to radomizatio betwee sectios, exlaatios to a versio of each of the multile-choice
More informationA GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca
Idia J Pure Appl Math 45): 75-89 February 204 c Idia Natioal Sciece Academy A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS Mircea Merca Departmet of Mathematics Uiversity
More informationThe log-behavior of n p(n) and n p(n)/n
Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity
More informationAdditional Notes on Power Series
Additioal Notes o Power Series Mauela Girotti MATH 37-0 Advaced Calculus of oe variable Cotets Quick recall 2 Abel s Theorem 2 3 Differetiatio ad Itegratio of Power series 4 Quick recall We recall here
More informationTaylor polynomial solution of difference equation with constant coefficients via time scales calculus
TMSCI 3, o 3, 129-135 (2015) 129 ew Treds i Mathematical Scieces http://wwwtmscicom Taylor polyomial solutio of differece equatio with costat coefficiets via time scales calculus Veysel Fuat Hatipoglu
More informationAP Calculus BC Review Applications of Derivatives (Chapter 4) and f,
AP alculus B Review Applicatios of Derivatives (hapter ) Thigs to Kow ad Be Able to Do Defiitios of the followig i terms of derivatives, ad how to fid them: critical poit, global miima/maima, local (relative)
More informationComplex Number Theory without Imaginary Number (i)
Ope Access Library Joural Complex Number Theory without Imagiary Number (i Deepak Bhalchadra Gode Directorate of Cesus Operatios, Mumbai, Idia Email: deepakm_4@rediffmail.com Received 6 July 04; revised
More informationSection 5.5. Infinite Series: The Ratio Test
Differece Equatios to Differetial Equatios Sectio 5.5 Ifiite Series: The Ratio Test I the last sectio we saw that we could demostrate the covergece of a series a, where a 0 for all, by showig that a approaches
More informationCS537. Numerical Analysis and Computing
CS57 Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 Jauary 9 9 What is the Root May physical system ca be
More informationDifferentiable Convex Functions
Differetiable Covex Fuctios The followig picture motivates Theorem 11. f ( x) f ( x) f '( x)( x x) ˆx x 1 Theorem 11 : Let f : R R be differetiable. The, f is covex o the covex set C R if, ad oly if for
More informationCS321. Numerical Analysis and Computing
CS Numerical Aalysis ad Computig Lecture Locatig Roots o Equatios Proessor Ju Zhag Departmet o Computer Sciece Uiversity o Ketucky Leigto KY 456-6 September 8 5 What is the Root May physical system ca
More informationPLEASE MARK YOUR ANSWERS WITH AN X, not a circle! 1. (a) (b) (c) (d) (e) 3. (a) (b) (c) (d) (e) 5. (a) (b) (c) (d) (e) 7. (a) (b) (c) (d) (e)
Math 0560, Exam 3 November 6, 07 The Hoor Code is i effect for this examiatio. All work is to be your ow. No calculators. The exam lasts for hour ad 5 mi. Be sure that your ame is o every page i case pages
More informationSCORE. Exam 2. MA 114 Exam 2 Fall 2016
MA 4 Exam Fall 06 Exam Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use
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 informationIntroduction to Optimization Techniques. How to Solve Equations
Itroductio to Optimizatio Techiques How to Solve Equatios Iterative Methods of Optimizatio Iterative methods of optimizatio Solutio of the oliear equatios resultig form a optimizatio problem is usually
More informationAnalytic Continuation
Aalytic Cotiuatio The stadard example of this is give by Example Let h (z) = 1 + z + z 2 + z 3 +... kow to coverge oly for z < 1. I fact h (z) = 1/ (1 z) for such z. Yet H (z) = 1/ (1 z) is defied for
More informationSolutions to Homework 7
Solutios to Homework 7 Due Wedesday, August 4, 004. Chapter 4.1) 3, 4, 9, 0, 7, 30. Chapter 4.) 4, 9, 10, 11, 1. Chapter 4.1. Solutio to problem 3. The sum has the form a 1 a + a 3 with a k = 1/k. Sice
More informationMath 116 Practice for Exam 3
Math 6 Practice for Eam 3 Geerated April 4, 26 Name: SOLUTIONS Istructor: Sectio Number:. This eam has questios. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur
More informationDECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS. Park Road, Islamabad, Pakistan
Mathematical ad Computatioal Applicatios, Vol. 9, No. 3, pp. 30-40, 04 DECOMPOSITION METHOD FOR SOLVING A SYSTEM OF THIRD-ORDER BOUNDARY VALUE PROBLEMS Muhammad Aslam Noor, Khalida Iayat Noor ad Asif Waheed
More information