Bulletin of the Transilvania University of Braşov Vol 10(59), No Series III: Mathematics, Informatics, Physics,
|
|
- Sharon Waters
- 5 years ago
- Views:
Transcription
1 Bulletin of the Trnsilvni University of Brşov Vol (59), No Series III: Mthemtics, Informtics, Physics, ON THE CONVERGENCE OF SOME MODIFIED NEWTON METHODS THROUGH COMPUTER ALGEBRA Ernest SCHEIBER Abstrct For some modified Newton methods to solve non-liner eqution the convergence is estblished nd the convergence order is computed using Computer Algebr Softwre. 22 ACM Subject Clssifiction: G..5, K..3. Key words: roots of nonliner equtions, modified Newton method, computer uses in eduction. Introduction We present the possibility to estblish the convergence nd compute the convergence order of method to solve non-liner eqution using Computer Algebr Softwre (CAS). The pplied procedure is bsed on well known convergence result, i.e. [?]. Severl modified Newton methods re known. Some of them re derived from different qudrture formuls [], [2], [7], [5]. We shll pply the convergence result to these methods, but the computtions re mde using CAS. The sme rgumenttion for convergence ws used in [4] nd [3], too. It implies tht the convergence occurs only when the initil pproximtion is properly chosen nd tht the convergence order is 3. In [9] we used the sme pproch for some methods to simultneously compute ll the roots of polynomil. We give unitry simplified presenttion of the convergence results for severl modified Newton methods with the usge of Mthemtic CAS [2]. The note is orgnized s follows. In Section 2 we recll the convergence result tht will be used. In Section 3 the convergence conditions re verified for some modified Newton methods using Mthemtic. e-mil: scheiber@unitbv.ro
2 28 Ernest Scheiber 2 A convergence frmework Let Ω C n be n open convex subset, T : Ω C n, T (z) = (T (z),..., T n (z)) T n m times differentible opertor such tht T (m) (z) is continuous nd the sequence (z (k) ) k N defined by z (k+) = T (z (k) ), z (k) = (z (k), z(k) 2,..., z(k) n ) T () z (k+) i = T i (z (k) ), i {, 2,..., n}, k N. In C n we shll use the mx norm z = mx{ z, z 2,..., z n }. We remind result enbling to estblish the convergence of such methods nd lower bound of their convergence order [?]. The min ingredient of the convergence theorem is the following well known result, but for completeness we shll give the proof of the result tht we shll use. Theorem. [] Let X, Y be normed spces, D n open convex subset of X nd T : D Y n m times Frèchet differentible opertor. Then, for ny x, y D T (y) T (x) j= Using this result, we hve Theorem 2. Let α Ω. If. T (α) = α, j! T (j) (x) (y x)... (y x) }{{} j times 2. T (α) = T (α) =... = T () (α) = y x m sup T (m) (ζ). (2) ζ [x,y] then there exists r > such tht for ny z () C n, z () α < r, the sequence z (k+) = T (z (k) ), k N, () converges to α. Proof. Let r > be such tht V = {z C n : z α r } Ω nd C = mx z V T (m) (z). There exists < r r such tht C r m < r ( ) C r <. We denote V = {z C n : z α r}. If z V, then (2) nd the present hypothesis implies T (z) α = T (z) T (α) z α m sup ζ [α,z] j= j! T (j) (α) (z α)... (z α) }{{} j times T (m) (ζ) C r m < r,
3 Modified Newton methods through computer lgebr 29 thus T (z) V. For z = z (k) from the bove reltions we obtin z (k+) α = T (z (k) ) α C z(k) α m. (3) Using recursively the inequlity (3), we find = z (k) α C z(k ) α m C ( ) +m C z (k 2) α m2... ( ) m k C r m k = ( (C ( ) m C z(k 2) α m = ( ) +m+...+m k C z () α mk ) r ) m k r, for k. x Let lim k x k = x. If lim k+ x k x k x = ρ, with < ρ <, then r is the r convergence order of the sequence (x k ) k N. From the inequlity (3) it results tht the convergence order of the sequence (z (k) ) k N is t lest m. 3 Modified Newton methods Let there be differentible function F : Ω R R nd the non-liner eqution F (x) =, (4) such tht F () =, F (). The itertion formul of modified Newton methods is Denoting x n+ = x n F (x n) F (x n ) G(x n), n N. (5) T (x) = x F (x) G(x), in order to prove the third order convergence of the method (5) the following reltions must be verified T () = T () =, T () = T (3) (). (6) The convergence occurs when the initil pproximtion x is properly chosen. The Hlley s method [3] is defined by x n+ = x n 2F (x n )F (x n ) 2F 2 (x n ) F (x n )F (x n ), G(x) = 2F 2 (x) 2F 2 (x) F (x)f (x).
4 2 Ernest Scheiber If T (x) = x 2F (x)f (x 2F 2 (x) F (x)f (x) then the computtion in Mthemtic 2 is performed by the code below: F [] = T [x ]:=x (2F [x]f [x])/(2f [x] 2 F [x]f [x]) T [x]/.x D[T [x], x]/.x D[T [x], {x, 2}]/.x Simplify[D[T [x], {x, 3}]/.x ] 3F [] 2 2F []F (3) [] 2F [] 2 The sme scheme will be pplied to the following methods. In [] Weerkoon nd Fernndo hd introduced the third order convergence method x n+ = x n For 2F (x n ) ( ), G(x) = F (x n ) + F x n F (xn) F (x n) T 2(x) = x 2F (x) ( ) + F x F (x) we found F [] = T2[x ]:=x 2F [x]/(f [x] + F [x F [x]/f [x]]) T2[x]/.x D[T2[x], x]/.x D[T2[x], {x, 2}]/.x Simplify[D[N[x], {x, 3}]/.x ] 3T [] 2 +T []T (3) [] 2T [] 2 Frontini nd Sormni [2] considered the method 2 ( ). (7) + F x F (x) F (x n ) x n+ = x n ( ), G(x) = F x n F (xn) 2F (x n) ( F x F (x) 2 ). (8) 2 The settings nd given commnds re printed with bold chrcters.
5 Modified Newton methods through computer lgebr 2 So, for the Mthemtic code is F [] = T3[x ]:=x F [x]/f [x F [x]/(2f [x])] T3[x]/.x D[T3[x], x]/.x D[T3[x], {x, 2}]/.x D[T3[x], {x, 3}]/.x 3F [] 2 2F [] 2 F (3) [] 4F [] F (x) T 3(x) = x ( ). F x F (x) 2 In [4], [] the following method is defined x n+ = x n F (x n) 2 F (x n ) + ( ), (9) F x n F (xn) G(x) = 2 F (x n) + ( ). F x F (x) The corresponding Mthemtic code for this method is F [] = T4[x ]:=x F [x]/2(/f [x] + /F [x F [x]/f [x]]) T4[x]/.x D[T4[x], x]/.x D[T4[x], {x, 2}]/.x D[T4[x], {x, 3}]/.x ( ) 3F [] 2 F (3) [] 2F [] 2 F [] 3 2 F [] F [] 2 F (3) [] F [] 3 F [] 2 In [5] the following method is introduced where M N. Now x n+ = x n G(x) = 2MF (x n ) ( ), () 2M k= F x n F (xn) k.5 F (x n) 2M 2M ( 2M k= F x F (x) ). k.5 2M
6 22 Ernest Scheiber The required computtions is given by the code F [] = T5[x ] = x 2MF [x]/sum[f [x F [x]/f [x](k /2)/(2M)], {k,, 2M}] 2MF [x] x [ ] 2M k= F T5[x]/.x D[T5[x], x] + 2MF [x] 2M k= x ( 2 +k )F [x] 2MF [x] D[T5[x], x]/.x D[T5[x], {x, 2}]/.x D[T5[x], {x, 3}]/.x 2M ( 2 +k 2M + ( 2 +k)f [x]f [ [x] )F 2MF [x] 2 x ( ] +k)f [x] 2 2MF [x] ( [ 2M k= F x ( 2 ]) +k)f [x] 2MF 2 [x] ( 3F [] 2 4MF [] 2 + F (3) [] 2MF [] + 3F [] ( [ 2MF [x] 2M k= F x ( 2 ] +k)f [x] 2MF [x] )) F [] 2 4MF [] 3 24M 2 F [] 2 F []F (3) []+6M 2 F []F (3) [] 96M 3 F [] 3 Even the Lguerre method to compute root of polynomil [8], [6] my be presented in the sme mnner. Let be P (x) = m i= (x x i) polynomil hving only simple roots. The Lguerre method is defined by x n+ = x n P (x n) P (x n ) m + m m, n N, x C. P (x n)p (x n) P 2 (x n) The squre root of complex number is chosen to hve non-negtive rel prt. In this cse G(x) = m + m m. P (x)p (x) P 2 (x) The Mthemtic code is P [] = G[x ]:= /(/m + (m )/msqrt[ m/(m )P [x]d[p [x], {x, 2}]/D[P [x], x] 2]) T 6[x ]:=x P [x]/d[p [x], x]g[x] T 6[x]/.x Simplify[D[T 6[x], x]/.x ] Simplify[D[T 6[x], {x, 2}]/.x ] Simplify[D[T 6[x], {x, 3}]/.x ] 3( 2+m)P [] 2 4( +m)p []P (3) [] 4( +m)p [] 2
7 Modified Newton methods through computer lgebr 23 As expected, we found tht the convergence order of the method is 3. References [] Crtn H.: Clcul différentiel. Ed. Hermnn, Pris, 967. [2] Frontini M., Sormnie E.: Some vrints of Newtons method with third-order convergence. Appl. Mth. Comput., 4 (23), [3] Homeier H.H.H.: A modified Newton method for rootfinding with cubic convergence. J. Comput. Appl. Mth., 57 (23), [4] Homeier H.H.H.: On Newton-type methods with cubic convergence. J. Comput. Appl. Mth., 76 (25), [5] Mishr B., Pny A.K., Dutt S.: A New Modified Newton Method use of Hr Wwelet for Solving Nonliner Equtions. rxiv:7.468v, 26. [6] Möller H., Convergence nd visuliztion of Lguerre s rootfinfing lgorithm. rxiv: 5.268v, 25. [7] Özbn A.Y.: Some new vrints of Newtons method. Appl. Mth. Lett., 7 (24), [8] Press, W.H., Teukolsky S.A., Vetterling W.T., Flnnery B.P., Numericl Recipes: The Art of Scientific Computing (3rd ed.). New York: Cmbridge University Press, 27. [9] Scheiber E., On the Convergence Order for Some Methods of Simultneously Root Finding for Polynomils Using Computer Algebr System. Bulletin of the Trnsilvni University of Brşov, 4(53) (2), no., [] Wng P.: A Third-Order Fmily of Newton-Like Itertion Methods for Solving Nonliner Equtions. J. Numericl Mth. Stochstics, 3 (2), no., 3-9. [] Weerkoon S., Fernndo T.G.I.: A vrint of Newtons method with ccelerted third-order convergence, Appl. Mth. Lett., 3 (2), [2] * * *, [3] * * *,
8 24 Ernest Scheiber
New Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationON BERNOULLI BOUNDARY VALUE PROBLEM
LE MATEMATICHE Vol. LXII (2007) Fsc. II, pp. 163 173 ON BERNOULLI BOUNDARY VALUE PROBLEM FRANCESCO A. COSTABILE - ANNAROSA SERPE We consider the boundry vlue problem: x (m) (t) = f (t,x(t)), t b, m > 1
More informationKRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION
Fixed Point Theory, 13(2012), No. 1, 285-291 http://www.mth.ubbcluj.ro/ nodecj/sfptcj.html KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION FULI WANG AND FENG WANG School of Mthemtics nd
More informationKeywords : Generalized Ostrowski s inequality, generalized midpoint inequality, Taylor s formula.
Generliztions of the Ostrowski s inequlity K. S. Anstsiou Aristides I. Kechriniotis B. A. Kotsos Technologicl Eductionl Institute T.E.I.) of Lmi 3rd Km. O.N.R. Lmi-Athens Lmi 3500 Greece Abstrct Using
More information1 Linear Least Squares
Lest Squres Pge 1 1 Liner Lest Squres I will try to be consistent in nottion, with n being the number of dt points, nd m < n being the number of prmeters in model function. We re interested in solving
More informationA Modified ADM for Solving Systems of Linear Fredholm Integral Equations of the Second Kind
Applied Mthemticl Sciences, Vol. 6, 2012, no. 26, 1267-1273 A Modified ADM for Solving Systems of Liner Fredholm Integrl Equtions of the Second Kind A. R. Vhidi nd T. Dmercheli Deprtment of Mthemtics,
More informationCzechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction
Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCK-KURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When rel-vlued
More information1 The Lagrange interpolation formula
Notes on Qudrture 1 The Lgrnge interpoltion formul We briefly recll the Lgrnge interpoltion formul. The strting point is collection of N + 1 rel points (x 0, y 0 ), (x 1, y 1 ),..., (x N, y N ), with x
More informationSome estimates on the Hermite-Hadamard inequality through quasi-convex functions
Annls of University of Criov, Mth. Comp. Sci. Ser. Volume 3, 7, Pges 8 87 ISSN: 13-693 Some estimtes on the Hermite-Hdmrd inequlity through qusi-convex functions Dniel Alexndru Ion Abstrct. In this pper
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationAn iterative method for solving nonlinear functional equations
J. Mth. Anl. Appl. 316 (26) 753 763 www.elsevier.com/locte/jm An itertive method for solving nonliner functionl equtions Vrsh Dftrdr-Gejji, Hossein Jfri Deprtment of Mthemtics, University of Pune, Gneshkhind,
More informationAN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION
Applied Mthemtics E-Notes, 5(005), 53-60 c ISSN 1607-510 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION
More informationSeveral Answers to an Open Problem
Int. J. Contemp. Mth. Sciences, Vol. 5, 2010, no. 37, 1813-1817 Severl Answers to n Open Problem Xinkun Chi, Yonggng Zho nd Hongxi Du College of Mthemtics nd Informtion Science Henn Norml University Henn
More informationMath 270A: Numerical Linear Algebra
Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner
More informationBest Approximation in the 2-norm
Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationarxiv: v1 [math.ra] 1 Nov 2014
CLASSIFICATION OF COMPLEX CYCLIC LEIBNIZ ALGEBRAS DANIEL SCOFIELD AND S MCKAY SULLIVAN rxiv:14110170v1 [mthra] 1 Nov 2014 Abstrct Since Leibniz lgebrs were introduced by Lody s generliztion of Lie lgebrs,
More informationCMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
More informationA unified generalization of perturbed mid-point and trapezoid inequalities and asymptotic expressions for its error term
An. Ştiinţ. Univ. Al. I. Cuz Işi. Mt. (N.S. Tomul LXIII, 07, f. A unified generliztion of perturbed mid-point nd trpezoid inequlities nd symptotic expressions for its error term Wenjun Liu Received: 7.XI.0
More informationGENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS. (b a)3 [f(a) + f(b)] f x (a,b)
GENERALIZATIONS OF WEIGHTED TRAPEZOIDAL INEQUALITY FOR MONOTONIC MAPPINGS AND ITS APPLICATIONS KUEI-LIN TSENG, GOU-SHENG YANG, AND SEVER S. DRAGOMIR Abstrct. In this pper, we estblish some generliztions
More informationModification Adomian Decomposition Method for solving Seventh OrderIntegro-Differential Equations
IOSR Journl of Mthemtics (IOSR-JM) e-issn: 2278-5728, p-issn: 239-765X. Volume, Issue 5 Ver. V (Sep-Oct. 24), PP 72-77 www.iosrjournls.org Modifiction Adomin Decomposition Method for solving Seventh OrderIntegro-Differentil
More informationOrthogonal Polynomials
Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils
More informationEuler, Ioachimescu and the trapezium rule. G.J.O. Jameson (Math. Gazette 96 (2012), )
Euler, Iochimescu nd the trpezium rule G.J.O. Jmeson (Mth. Gzette 96 (0), 36 4) The following results were estblished in recent Gzette rticle [, Theorems, 3, 4]. Given > 0 nd 0 < s
More informationS. S. Dragomir. 2, we have the inequality. b a
Bull Koren Mth Soc 005 No pp 3 30 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Abstrct Compnions of Ostrowski s integrl ineulity for bsolutely
More informationCAAM 453 NUMERICAL ANALYSIS I Examination There are four questions, plus a bonus. Do not look at them until you begin the exam.
Exmintion 1 Posted 23 October 2002. Due no lter thn 5pm on Mondy, 28 October 2002. Instructions: 1. Time limit: 3 uninterrupted hours. 2. There re four questions, plus bonus. Do not look t them until you
More informationPositive Solutions of Operator Equations on Half-Line
Int. Journl of Mth. Anlysis, Vol. 3, 29, no. 5, 211-22 Positive Solutions of Opertor Equtions on Hlf-Line Bohe Wng 1 School of Mthemtics Shndong Administrtion Institute Jinn, 2514, P.R. Chin sdusuh@163.com
More informationSet Integral Equations in Metric Spaces
Mthemtic Morvic Vol. 13-1 2009, 95 102 Set Integrl Equtions in Metric Spces Ion Tişe Abstrct. Let P cp,cvr n be the fmily of ll nonempty compct, convex subsets of R n. We consider the following set integrl
More informationTRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS
TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS S.S. DRAGOMIR AND A. SOFO Abstrct. In this pper by utilising result given by Fink we obtin some new results relting to the trpezoidl inequlity
More informationS. S. Dragomir. 1. Introduction. In [1], Guessab and Schmeisser have proved among others, the following companion of Ostrowski s inequality:
FACTA UNIVERSITATIS NIŠ) Ser Mth Inform 9 00) 6 SOME COMPANIONS OF OSTROWSKI S INEQUALITY FOR ABSOLUTELY CONTINUOUS FUNCTIONS AND APPLICATIONS S S Drgomir Dedicted to Prof G Mstroinni for his 65th birthdy
More informationResearch Article On Existence and Uniqueness of Solutions of a Nonlinear Integral Equation
Journl of Applied Mthemtics Volume 2011, Article ID 743923, 7 pges doi:10.1155/2011/743923 Reserch Article On Existence nd Uniqueness of Solutions of Nonliner Integrl Eqution M. Eshghi Gordji, 1 H. Bghni,
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARI- ABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL NEIL S. BARNETT, PIETRO CERONE, SEVER S. DRAGOMIR
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationOn the Decomposition Method for System of Linear Fredholm Integral Equations of the Second Kind
Applied Mthemticl Sciences, Vol. 2, 28, no. 2, 57-62 On the Decomposition Method for System of Liner Fredholm Integrl Equtions of the Second Kind A. R. Vhidi 1 nd M. Mokhtri Deprtment of Mthemtics, Shhr-e-Rey
More informationNumerical radius Haagerup norm and square factorization through Hilbert spaces
J Mth Soc Jpn Vol 58, No, 006 Numericl rdius Hgerup norm nd squre fctoriztion through Hilbert spces By Tkshi Itoh nd Msru Ngis (Received Apr 7, 004) (Revised Mr, 005) Abstrct We study fctoriztion of bounded
More informationLecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics - A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the
More informationAN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS. I. Fedotov and S. S. Dragomir
RGMIA Reserch Report Collection, Vol., No., 999 http://sci.vu.edu.u/ rgmi AN INEQUALITY OF OSTROWSKI TYPE AND ITS APPLICATIONS FOR SIMPSON S RULE AND SPECIAL MEANS I. Fedotov nd S. S. Drgomir Astrct. An
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion - re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics GENERALIZATIONS OF THE TRAPEZOID INEQUALITIES BASED ON A NEW MEAN VALUE THEOREM FOR THE REMAINDER IN TAYLOR S FORMULA volume 7, issue 3, rticle 90, 006.
More informationJournal of Inequalities in Pure and Applied Mathematics
Journl of Inequlities in Pure nd Applied Mthemtics http://jipmvueduu/ Volume, Issue, Article, 00 SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL NS BARNETT,
More informationThe logarithmic mean is a mean
Mthemticl Communictions 2(1997), 35-39 35 The logrithmic men is men B. Mond, Chrles E. M. Perce nd J. Pečrić Abstrct. The fct tht the logrithmic men of two positive numbers is men, tht is, tht it lies
More informationSTUDY GUIDE FOR BASIC EXAM
STUDY GUIDE FOR BASIC EXAM BRYON ARAGAM This is prtil list of theorems tht frequently show up on the bsic exm. In mny cses, you my be sked to directly prove one of these theorems or these vrints. There
More informationJournal of Computational and Applied Mathematics. On positive solutions for fourth-order boundary value problem with impulse
Journl of Computtionl nd Applied Mthemtics 225 (2009) 356 36 Contents lists vilble t ScienceDirect Journl of Computtionl nd Applied Mthemtics journl homepge: www.elsevier.com/locte/cm On positive solutions
More informationNumerical Linear Algebra Assignment 008
Numericl Liner Algebr Assignment 008 Nguyen Qun B Hong Students t Fculty of Mth nd Computer Science, Ho Chi Minh University of Science, Vietnm emil. nguyenqunbhong@gmil.com blog. http://hongnguyenqunb.wordpress.com
More informationCommunications inmathematicalanalysis Volume 6, Number 2, pp (2009) ISSN
Communictions inmthemticlanlysis Volume 6, Number, pp. 33 41 009) ISSN 1938-9787 www.commun-mth-nl.org A SHARP GRÜSS TYPE INEQUALITY ON TIME SCALES AND APPLICATION TO THE SHARP OSTROWSKI-GRÜSS INEQUALITY
More informationThe Riemann Integral
Deprtment of Mthemtics King Sud University 2017-2018 Tble of contents 1 Anti-derivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Anti-derivtive Function Definition Let f : I R be function
More informationAQA Further Pure 1. Complex Numbers. Section 1: Introduction to Complex Numbers. The number system
Complex Numbers Section 1: Introduction to Complex Numbers Notes nd Exmples These notes contin subsections on The number system Adding nd subtrcting complex numbers Multiplying complex numbers Complex
More informationThe Banach algebra of functions of bounded variation and the pointwise Helly selection theorem
The Bnch lgebr of functions of bounded vrition nd the pointwise Helly selection theorem Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics, University of Toronto Jnury, 015 1 BV [, b] Let < b. For f
More informationRegulated functions and the regulated integral
Regulted functions nd the regulted integrl Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics University of Toronto April 3 2014 1 Regulted functions nd step functions Let = [ b] nd let X be normed
More informationOrthogonal Polynomials and Least-Squares Approximations to Functions
Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny
More informationNEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS. := f (4) (x) <. The following inequality. 2 b a
NEW INEQUALITIES OF SIMPSON S TYPE FOR s CONVEX FUNCTIONS WITH APPLICATIONS MOHAMMAD ALOMARI A MASLINA DARUS A AND SEVER S DRAGOMIR B Abstrct In terms of the first derivtive some ineulities of Simpson
More informationA new algorithm for generating Pythagorean triples 1
A new lgorithm for generting Pythgoren triples 1 RH Dye 2 nd RWD Nicklls 3 The Mthemticl Gzette (1998; 82 (Mrch, No. 493, pp. 86 91 http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf 1 Introduction
More informationON A GENERALIZED STURM-LIOUVILLE PROBLEM
Foli Mthemtic Vol. 17, No. 1, pp. 17 22 Act Universittis Lodziensis c 2010 for University of Łódź Press ON A GENERALIZED STURM-LIOUVILLE PROBLEM GRZEGORZ ANDRZEJCZAK AND TADEUSZ POREDA Abstrct. Bsic results
More information3.4 Numerical integration
3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,
More informationON A CONVEXITY PROPERTY. 1. Introduction Most general class of convex functions is defined by the inequality
Krgujevc Journl of Mthemtics Volume 40( (016, Pges 166 171. ON A CONVEXITY PROPERTY SLAVKO SIMIĆ Abstrct. In this rticle we proved n interesting property of the clss of continuous convex functions. This
More informationACM 105: Applied Real and Functional Analysis. Solutions to Homework # 2.
ACM 05: Applied Rel nd Functionl Anlysis. Solutions to Homework # 2. Andy Greenberg, Alexei Novikov Problem. Riemnn-Lebesgue Theorem. Theorem (G.F.B. Riemnn, H.L. Lebesgue). If f is n integrble function
More informationB.Sc. in Mathematics (Ordinary)
R48/0 DUBLIN INSTITUTE OF TECHNOLOGY KEVIN STREET, DUBLIN 8 B.Sc. in Mthemtics (Ordinry) SUPPLEMENTAL EXAMINATIONS 01 Numericl Methods Dr. D. Mckey Dr. C. Hills Dr. E.A. Cox Full mrks for complete nswers
More informationf (a) + f (b) f (λx + (1 λ)y) max {f (x),f (y)}, x, y [a, b]. (1.1)
TAMKANG JOURNAL OF MATHEMATICS Volume 41, Number 4, 353-359, Winter 1 NEW INEQUALITIES OF HERMITE-HADAMARD TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX M. ALOMARI, M. DARUS
More informationNew Integral Inequalities for n-time Differentiable Functions with Applications for pdfs
Applied Mthemticl Sciences, Vol. 2, 2008, no. 8, 353-362 New Integrl Inequlities for n-time Differentible Functions with Applictions for pdfs Aristides I. Kechriniotis Technologicl Eductionl Institute
More informationWHEN IS A FUNCTION NOT FLAT? 1. Introduction. {e 1 0, x = 0. f(x) =
WHEN IS A FUNCTION NOT FLAT? YIFEI PAN AND MEI WANG Abstrct. In this pper we prove unique continution property for vector vlued functions of one vrible stisfying certin differentil inequlity. Key words:
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution
More informationON AN INTEGRATION-BY-PARTS FORMULA FOR MEASURES
Volume 8 (2007), Issue 4, Article 93, 13 pp. ON AN INTEGRATION-BY-PARTS FORMULA FOR MEASURES A. ČIVLJAK, LJ. DEDIĆ, AND M. MATIĆ AMERICAN COLLEGE OF MANAGEMENT AND TECHNOLOGY ROCHESTER INSTITUTE OF TECHNOLOGY
More informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More informationOn the Generalized Weighted Quasi-Arithmetic Integral Mean 1
Int. Journl of Mth. Anlysis, Vol. 7, 2013, no. 41, 2039-2048 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2013.3499 On the Generlized Weighted Qusi-Arithmetic Integrl Men 1 Hui Sun School
More informationA HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES. 1. Introduction
Ttr Mt. Mth. Publ. 44 (29), 159 168 DOI: 1.2478/v1127-9-56-z t m Mthemticl Publictions A HELLY THEOREM FOR FUNCTIONS WITH VALUES IN METRIC SPACES Miloslv Duchoň Peter Mličký ABSTRACT. We present Helly
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationLecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at Urbana-Champaign. March 20, 2014
Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t Urbn-Chmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method
More informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
More informationOstrowski Grüss Čebyšev type inequalities for functions whose modulus of second derivatives are convex 1
Generl Mthemtics Vol. 6, No. (28), 7 97 Ostrowski Grüss Čebyšev type inequlities for functions whose modulus of second derivtives re convex Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn Abstrct In this pper,
More informationIntroduction to Some Convergence theorems
Lecture Introduction to Some Convergence theorems Fridy 4, 005 Lecturer: Nti Linil Notes: Mukund Nrsimhn nd Chris Ré. Recp Recll tht for f : T C, we hd defined ˆf(r) = π T f(t)e irt dt nd we were trying
More informationSolution to Fredholm Fuzzy Integral Equations with Degenerate Kernel
Int. J. Contemp. Mth. Sciences, Vol. 6, 2011, no. 11, 535-543 Solution to Fredholm Fuzzy Integrl Equtions with Degenerte Kernel M. M. Shmivnd, A. Shhsvrn nd S. M. Tri Fculty of Science, Islmic Azd University
More informationLecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
More informationON THE GENERALIZED SUPERSTABILITY OF nth ORDER LINEAR DIFFERENTIAL EQUATIONS WITH INITIAL CONDITIONS
PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 9811 015, 43 49 DOI: 10.98/PIM15019019H ON THE GENERALIZED SUPERSTABILITY OF nth ORDER LINEAR DIFFERENTIAL EQUATIONS WITH INITIAL CONDITIONS
More informationInstructor: Marios M. Fyrillas HOMEWORK ASSIGNMENT ON INTERPOLATION
AMAT 34 Numericl Methods Instructor: Mrios M. Fyrills Emil: m.fyrills@fit.c.cy Office Tel.: 34559/6 Et. 3 HOMEWORK ASSIGNMENT ON INTERPOATION QUESTION Using interpoltion by colloction-polynomil fit method
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationSOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL
SOME INEQUALITIES FOR THE DISPERSION OF A RANDOM VARIABLE WHOSE PDF IS DEFINED ON A FINITE INTERVAL NS BARNETT P CERONE SS DRAGOMIR AND J ROUMELIOTIS Abstrct Some ineulities for the dispersion of rndom
More informationA Convergence Theorem for the Improper Riemann Integral of Banach Space-valued Functions
Interntionl Journl of Mthemticl Anlysis Vol. 8, 2014, no. 50, 2451-2460 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/10.12988/ijm.2014.49294 A Convergence Theorem for the Improper Riemnn Integrl of Bnch
More informationNumerical Methods I Orthogonal Polynomials
Numericl Methods I Orthogonl Polynomils Aleksndr Donev Cournt Institute, NYU 1 donev@cournt.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fll 2014 Nov 6th, 2014 A. Donev (Cournt Institute) Lecture IX
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationDiscrete Least-squares Approximations
Discrete Lest-squres Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve
More informationTaylor Polynomial Inequalities
Tylor Polynomil Inequlities Ben Glin September 17, 24 Abstrct There re instnces where we my wish to pproximte the vlue of complicted function round given point by constructing simpler function such s polynomil
More informationp(x) = 3x 3 + x n 3 k=0 so the right hand side of the equality we have to show is obtained for r = b 0, s = b 1 and 2n 3 b k x k, q 2n 3 (x) =
Norwegin University of Science nd Technology Deprtment of Mthemticl Sciences Pge 1 of 5 Contct during the exm: Elen Celledoni, tlf. 73593541, cell phone 48238584 PLESE NOTE: this solution is for the students
More informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More informationLecture notes. Fundamental inequalities: techniques and applications
Lecture notes Fundmentl inequlities: techniques nd pplictions Mnh Hong Duong Mthemtics Institute, University of Wrwick Emil: m.h.duong@wrwick.c.uk Februry 8, 207 2 Abstrct Inequlities re ubiquitous in
More informationSOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL (1 + µ(f n )) f(x) =. But we don t need the exact bound.) Set
SOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL 28 Nottion: N {, 2, 3,...}. (Tht is, N.. Let (X, M be mesurble spce with σ-finite positive mesure µ. Prove tht there is finite positive mesure ν on (X, M such
More informationAn optimal 3-point quadrature formula of closed type and error bounds
Revist Colombin de Mtemátics Volumen 8), págins 9- An optiml 3-point qudrture formul of closed type nd error bounds Un fórmul de cudrtur óptim de 3 puntos de tipo cerrdo y error de fronter Nend Ujević,
More informationSolutions of Klein - Gordan equations, using Finite Fourier Sine Transform
IOSR Journl of Mthemtics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 13, Issue 6 Ver. IV (Nov. - Dec. 2017), PP 19-24 www.iosrjournls.org Solutions of Klein - Gordn equtions, using Finite Fourier
More informationA Computational Method for Solving Linear Volterra Integral Equations
Applied Mthemticl Sciences, Vol. 6, 01, no. 17, 807-814 A Computtionl Method for Solving Liner Volterr Integrl Equtions Frshid Mirzee Deprtment of Mthemtics, Fculty of Science Mlyer University, Mlyer,
More informationHandout 4. Inverse and Implicit Function Theorems.
8.95 Hndout 4. Inverse nd Implicit Function Theorems. Theorem (Inverse Function Theorem). Suppose U R n is open, f : U R n is C, x U nd df x is invertible. Then there exists neighborhood V of x in U nd
More informationKey words. Numerical quadrature, piecewise polynomial, convergence rate, trapezoidal rule, midpoint rule, Simpson s rule, spectral accuracy.
O SPECTRA ACCURACY OF QUADRATURE FORMUAE BASED O PIECEWISE POYOMIA ITERPOATIO A KURGAOV AD S TSYKOV Abstrct It is well-known tt te trpezoidl rule, wile being only second-order ccurte in generl, improves
More informationLecture 12: Numerical Quadrature
Lecture 12: Numericl Qudrture J.K. Ryn@tudelft.nl WI3097TU Delft Institute of Applied Mthemtics Delft University of Technology 5 December 2012 () Numericl Qudrture 5 December 2012 1 / 46 Outline 1 Review
More informationA basic logarithmic inequality, and the logarithmic mean
Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, Bbeş-Bolyi University Str. Koglnicenu
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl
More information1. Gauss-Jacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),
1. Guss-Jcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on
More informationPath product and inverse M-matrices
Electronic Journl of Liner Algebr Volume 22 Volume 22 (2011) Article 42 2011 Pth product nd inverse M-mtrices Yn Zhu Cheng-Yi Zhng Jun Liu Follow this nd dditionl works t: http://repository.uwyo.edu/el
More informationThe Modified Heinz s Inequality
Journl of Applied Mthemtics nd Physics, 03,, 65-70 Pulished Online Novemer 03 (http://wwwscirporg/journl/jmp) http://dxdoiorg/0436/jmp03500 The Modified Heinz s Inequlity Tkshi Yoshino Mthemticl Institute,
More informationOn the Continuous Dependence of Solutions of Boundary Value Problems for Delay Differential Equations
Journl of Computtions & Modelling, vol.3, no.4, 2013, 1-10 ISSN: 1792-7625 (print), 1792-8850 (online) Scienpress Ltd, 2013 On the Continuous Dependence of Solutions of Boundry Vlue Problems for Dely Differentil
More informationBounds for the Riemann Stieltjes integral via s-convex integrand or integrator
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 6, Number, 0 Avilble online t www.mth.ut.ee/ct/ Bounds for the Riemnn Stieltjes integrl vi s-convex integrnd or integrtor Mohmmd Wjeeh
More information4181H Problem Set 11 Selected Solutions. Chapter 19. n(log x) n 1 1 x x dx,
48H Problem Set Selected Solutions Chpter 9 # () Tke f(x) = x n, g (x) = e x, nd use integrtion by prts; this gives reduction formul: x n e x dx = x n e x n x n e x dx. (b) Tke f(x) = (log x) n, g (x)
More informationA PROOF OF THE FUNDAMENTAL THEOREM OF CALCULUS USING HAUSDORFF MEASURES
INROADS Rel Anlysis Exchnge Vol. 26(1), 2000/2001, pp. 381 390 Constntin Volintiru, Deprtment of Mthemtics, University of Buchrest, Buchrest, Romni. e-mil: cosv@mt.cs.unibuc.ro A PROOF OF THE FUNDAMENTAL
More informationPresentation Problems 5
Presenttion Problems 5 21-355 A For these problems, ssume ll sets re subsets of R unless otherwise specified. 1. Let P nd Q be prtitions of [, b] such tht P Q. Then U(f, P ) U(f, Q) nd L(f, P ) L(f, Q).
More informationChapter 3 Solving Nonlinear Equations
Chpter 3 Solving Nonliner Equtions 3.1 Introduction The nonliner function of unknown vrible x is in the form of where n could be non-integer. Root is the numericl vlue of x tht stisfies f ( x) 0. Grphiclly,
More information