Some New Iterative Methods for Solving Nonlinear Equations

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Some New Iterative Methods for Solving Nonlinear Equations"

Transcription

1 World Applied Scieces Joural 0 (6): , 01 ISSN IDOSI Publicatios, 01 DOI: /idosi.wasj Some New Iterative Methods for Solvig Noliear Equatios Muhammad Aslam Noor, Khalida Iayat Noor ad Kshif Aftab Departmet of Mathematics, COMSATS Istitute of Iformatio Techology, Park Road, Islamabad, Pakista Abstract: I this paper, we suggest ad aalyze some ew iterative methods for solvig oliear equatios by usig a ew series expasio of the oliear fuctio. Some special cases are also discussed. These ew methods ca be viewed as sigificat modificatio ad improvemet of the Newto method. Several examples are give to illustrate the efficiecy ad robustess of these methods. Kew words: Householder method Iterative method Covergece Noliear equatio INTRODUCTION f(x) = 0 (1) It is well kow a wide class of liear ad oliear Various umerical methods have bee developed problems which arise i differet braches of usig the Taylor series ad other techiques. I this mathematical such as physical, biomedical, regioal, paper, we use aother series of the oliear fuctio f(x) optimizatio, ecology, ecoomics ad egieerig which ca be obtaied by usig the trapezoidal rule ad scieces ca be formulated i terms of oliear the Fudametal Theorem of Calculus. To be more equatios. Iterative methods for fidig the approximate precise, we assume that is a simple root of (1) ad is solutios of the oliear equatio f(x) = 0 are beig a iitial guess sufficietly close to Now usig the developed usig several differet techiques icludig trapezoidal rule ad fudametal theorem of calculus, oe Taylor series, quadrature formulas, homotopy ad ca show that the fuctio f(x) ca be approximated by decompositio techiques, see [1-5, 9-11, 13-17, 19, 1] the series ad the refereces therei. Ispired ad motivated by the ogoig research activities i this area, we x f( x) = f + [ ( x) + ] suggest ad aalyze a ew iterative method for () solvig oliear equatios. To derive these iterative methods, we show that the oliear fuctio ca where f'(x) is the differetial of f. be approximated by a ew series which ca be obtaied by usig the trapezoidal rule for From (1) ad (), we have approximatig the itegral i cojuctio with the fudametal theorem of calculus. This ew expasio is x = x. used to suggest these ew iterative methods for (3) solvig oliear equatios.. We also cosider the covergece aalysis of these methods. Several examples Usig (3), oe ca suggest the followig iterative are give to illustrate the efficiecy ad compariso with method for solvig the oliear equatios (1). other methods. Algorithm 1: For a give iitial choice x 0, fid the Iterative Methods: It is well kow that a wide class of approximate solutio x +1 by the iterative scheme problems, which arise i various fiekds of pure ad applied scieces ca be formulated i terms of oliear f( x) ( x+ 1) x+ 1 = x ( x+ 1 x), equatios of the type. = 0,1,,... Correspodig Author: M. Aslam Noor, Mathematics Departmet, COMSATS Istitute of Iformatio Techology, Park Road, Islamabad, Pakista. 870

2 Algorithm 1 is a implicit iterative method. Algorithm 5: For a give iitial choice x 0, fid To implemet Algorithm 1, we use the predictor-corrector the approximate solutio x +1 by the iterative techique. Usig the Newto method as a predictor ad schemes: Algorithm 1, as a corrector, we suggest ad aalyze the followig two-step iterative method for solvig the f( x) ( x) x+ 1 = x = 0,1,,... oliear equatio (1) ad this is the mai motivatio of [ ( x)] f( x) ( x) this ote. which is kow as the Halley method. It ca Algorithm : For a give iitial choice x 0, fid the easily show that Halley method has cubic approximate solutio x +1 by the iterative schemes: covergece. I a similar way, oe ca obtai several kow ad ew iterative methods form these y f( x) = x f f x = x ( y x ), = 0,1,, f( x) f( x) ( y) x+ 1 = y +, = 0,1,,... ( x) ( x) ( x) x = f. + ( x) algorithms. We ow cosider the covergece aalysis of Algorithm. I a similar way, oe ca prove the covergece of Algorithm 3 ad algorithm 4. From Algorithm, we ca deduce the followig Theorem.1: Let I be a simple zero of iterative method for solvig the oliear equatios sufficietly differetiable fuctio f : I R R for a ope f(x) = 0 which appears to be ew oe. iterval I. If x 0 is sufficietly close to the the iterative method defied by Algorithm has secod-order Algorithm 3: For a give iitial choice x 0, fid the covergece. approximate solutio x +1 by the iterative scheme Proof: Let be a simple zero of f. The by expadig f(x) Algorithm 3 is called the modified Householder method for solvig the oliear equatios (1), which does ot ivolve the secod derives. From (1) ad (), we ca have ad f'(x ) about we have f( x) = e + ce + ce 3 + ce 4 + O( e), ad ( x) = 1+ ce + 3ce 3 + 4ce 4 + 5ce 5 + O( e), (4) (5) This fixed poit formulatio eables us to suggest the followig iterative method for solvig the oliear equatio. Algorithm 4: For a give iitial choice x, fid the 0 approximate solutio x by the iterative schemes: +1 y f( x) = x f f x 1 + = x = 0,1,,... + Usig the Taylor series expasio of f'(y ), oe obtai the followig iterative method for solvig the oliear equatio f(x) = 0. 1 f ( k) ck =, k,3, k! = where ad e = x. Now, from (4) ad (5), we have f( x) = e ce + ( c c3) e+ (7c3 c 4c 3 c4) e + O( e). ( x) (6) From (6), we have = y ce ( c c ) e (7cc 4c 3 c) e O( e). (7) From (7), we have 871

3 ( y) = 1 + ce + 4( cc 3 c) e + ( 11cc 3+ 6cc 4+ 8 c) e + O( e). (8) From (5) ad (8), we have ( y) = ce + ( 3c3+ 6 c) e + ( 16c 4c4+ 16 cc 3) e + O( e). (9) From (7), we have = y x e c e ( c c ) e (7c c 4c 3 c ) e O( e ). (10) From (9) ad (10), we have ( y x) e 3 ce (5c 10 c ) e ( 30cc 30c 7 c) e O( e). = ( x) (11) Thus, from (6) ad (11), we have = x ce ( c 6 c ) e O( e ), (1) which implies that = e ce ( c 6 c ) e O( e). (10) This shows that Algorithm is secod-order coverget. Numerical Results: We preset some examples to illustrate the efficiecy of the ew developed two-step iterative methods, see Table 1. We compare the Newto method (NM), 15 Algorithm (NR1) ad Algorithm 3 (NR).. We used = 10. The followig stoppig criteria is used for computer programs: () i x x <, ( ii) f < The examples are the same as i Chu []. 1 f ( x) = si x x + 1, f ( x) = x e 3x + f ( x) = cos x x, f ( x) = ( x 1) x 5 6 f ( x) = x 10, f ( x) = xe si x + 3cos x x + 7x 30 f ( x) = e 1. 3 x As for the covergece criteria, it was From the Table 1, we see that our method is required that the distace of two cosecutive comparable with the Newto Method. I fact, our approximatios 15 for the zero was les tha 10. methods ca be cosidered as sigificat improvemet of Also displayed is the umber of iteratios to the Newto Method ad ca be cosidered as alterative approximate the zero (IT), the approximate zero x ad method to other secod order coverget methods of the value f(x ) solvig oliear equatios. 87

4 Table 1: (Numerical Examples ad Compariso) Method IT x f(x ) f 1, x 0 = 1 NM e e-6 NR e e-8 NR e e-3 f, x 0 = NM e e-8 NR e-3.11e-16 NR e e-19 f 3, x 0 = 1.7 NM e-3.34e-16 NR e e-7 NR e e-7 f 4, x 0 = 3.5 NM 8.06e-4 8.8e- NR e e-30 NR e e-8 f 5, x 0 = 1.5 NM e e-8 NR e-45.30e-3 NR e e-19 f 6, x 0 = NM e-40.73e-1 NR e-3 5.8e-17 NR e e- f 7, x 0 = 3.5 NM e e-5 NR e e-19 NR e e-16 CONCLUSION Islamabad, Pakista (CIIT), for providig excellet research facilities. Authors are also grateful to Prof. Dr. I this paper, we have used a ew series of the Syed Tauseeh Mohyud Di, Editor--Chief for valueable fuctio f(x), which is obtaied by usig the trapezoidal suggestios ad commets. rule ad fudamet theorem of calculus. This series is used to suggest ad aalyzed a ew iterative method for REFERENCES solvig the oliear equatios. It is a iterestig problem to use this expasio of the fuctio to 1. Richard L. Burde ad J. Douglas Faires, 001. suggest ad cosider some ew iterative method for Numerical Aalysis, PWS publishig compay solvig the variatioal iequalities ad related problems, Bosta. see [6-9, 17-0] ad the referece therei. I our other. Chu, C., 005. Iterative methods improvig papers, we will the homotopy perturbatio method ad Newto s method by the decompositio method, some decompositios method to derive several iterative Computers Math. Appl., 50: methods for solvig the oliear equatios. It is a 3. Householder, A.S., The Numerical Treatmet of iterestig problem to derive the iterative methods for a Sigle Noliear Equatio, McGraw-Hill, New York. solvig system of oliear equatios. 4. Aslam Noor, M., 007. New family of iterative methods for oliear equatios, Appl. Math. ACKNOWLEDGEMENT Computatio, 190: Noor, M.A., New classes of iterative methods for The authors would like to thak Dr. S. M. Juai Zaidi, oliear equatios, Appl. Math. Computatio. Rector, COMSATS Istitute of Iformatio Techology, 191(007),

5 6. Noor, M.A., Geeral variatioal iequalities, 15. Noor, M.A. ad W.A. Kha, 01. Fourth-order Appl. Math. Letters, 1: iterative method free from secod derivative for 7. Noor, M.A., 004. Some developmets i geeral solvig oliear equatios, Appl. Math. Sci., variatioal iequalities, Appl. Math. Comput., 6(93): : Noor, M.A., W.A. Kha ad S. Youus, Noor, M.A., 009. Exteded geeral variatioal Homotopy perturbatio techiques for the solutio iequalities, Appl. Math. Letters, : of certai oliear equatios, Appl. Math. Sci., 9. Noor, M.A., 011. O iterative methods for oliear 6(130): equatios usig homotopy perturbatio techique, 17. Noor, M.A. ad K.I. Noor, 013. Auxiliary priciple Appl. Math. Iform Sci., 4: techique for solvig split feasibility problems, Appl. 10. Noor, M.A., 010. Some iterative methods for solvig Math. Iform. Sci., 7(1): 1-7. oliear equatios usig homotopy perturbatio 18. Noor, M.A., K.I. Noor ad Th. M. Rassias, method, Iter. J. Computer Math., 87: Some aspects of variatioal iequalities, J. Comput. 11. Noor, M.A. ad K.I. Noor, 006. Iterative schemes for Appl. Math., 47: solvig oliear equatios, Appl. Math. 19. Noor, M.A., K.I. Noor, E. Al-Said ad M. Waseem, Computatio, 183: Some ew iterative methods for oliear 1. Noor, K.I. ad M.A. Noor, 007. Predicot-corrector equatios, Math. Prob. Eg. 010(010), Article ID, Halley method for oliear equatios, Appl. Math : 1. Comput., 188: Noor, M.A., K.I. Noor ad E. Al-Said, 01. Iterative 13. Noor, K.I., M.A. Noor ad S. Momai, 007. Modified methods for solvig ocovex equilibrium Householder iterative method for oliear variatioal iequalities, Appl. Math. Iform. Sci., equatios, Appl. Math. Computatio, 190: (): Noor, M.A. ad W.A. Kha, 01. New iterative 1. Traub, J.F., Iterative Methods for Solutio of methods for solvig oliear equatios by usig Equatios, Pretice-Hall, Eglewood Cliffs, NJ, homotopy perturbatio method, Appl. Math. Comput., 19:

Two-step Extrapolated Newton s Method with High Efficiency Index

Two-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 information

Research Article A New Second-Order Iteration Method for Solving Nonlinear Equations

Research 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 information

Some Variants of Newton's Method with Fifth-Order and Fourth-Order Convergence for Solving Nonlinear Equations

Some Variants of Newton's Method with Fifth-Order and Fourth-Order Convergence for Solving Nonlinear Equations Copyright, Darbose Iteratioal Joural o Applied Mathematics ad Computatio Volume (), pp -6, 9 http//: ijamc.darbose.com Some Variats o Newto's Method with Fith-Order ad Fourth-Order Covergece or Solvig

More information

Solving a Nonlinear Equation Using a New Two-Step Derivative Free Iterative Methods

Solving 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 information

A collocation method for singular integral equations with cosecant kernel via Semi-trigonometric interpolation

A 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 information

MATH 10550, EXAM 3 SOLUTIONS

MATH 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 information

Comparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series

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 information

Notes on iteration and Newton s method. Iteration

Notes 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 information

Using An Accelerating Method With The Trapezoidal And Mid-Point Rules To Evaluate The Double Integrals With Continuous Integrands Numerically

Using An Accelerating Method With The Trapezoidal And Mid-Point Rules To Evaluate The Double Integrals With Continuous Integrands Numerically ISSN -50 (Paper) ISSN 5-05 (Olie) Vol.7, No., 017 Usig A Acceleratig Method With The Trapezoidal Ad Mid-Poit Rules To Evaluate The Double Itegrals With Cotiuous Itegrads Numerically Azal Taha Abdul Wahab

More information

NEW CLOSE FORM APPROXIMATIONS OF ln(1 + x) Sanjay Kumar Khattri. 1. Introduction

NEW CLOSE FORM APPROXIMATIONS OF ln(1 + x) Sanjay Kumar Khattri. 1. Introduction THE TEACHING OF MATHEMATICS 009 Vol XII pp 7 4 NEW CLOSE FORM APPROXIMATIONS OF l + x) Sajay Kumar Khattri Abstract Based o Newto-Cotes ad Gaussia quadrature rules we develop several closed form approximatios

More information

HOUSEHOLDER S APPROXIMANTS AND CONTINUED FRACTION EXPANSION OF QUADRATIC IRRATIONALS. Vinko Petričević University of Zagreb, Croatia

HOUSEHOLDER S APPROXIMANTS AND CONTINUED FRACTION EXPANSION OF QUADRATIC IRRATIONALS. Vinko Petričević University of Zagreb, Croatia HOUSEHOLDER S APPROXIMANTS AND CONTINUED FRACTION EXPANSION OF QUADRATIC IRRATIONALS Viko Petričević Uiversity of Zagre, Croatia Astract There are umerous methods for ratioal approximatio of real umers

More information

The Adomian Polynomials and the New Modified Decomposition Method for BVPs of nonlinear ODEs

The Adomian Polynomials and the New Modified Decomposition Method for BVPs of nonlinear ODEs Mathematical Computatio March 015, Volume, Issue 1, PP.1 6 The Adomia Polyomials ad the New Modified Decompositio Method for BVPs of oliear ODEs Jusheg Dua # School of Scieces, Shaghai Istitute of Techology,

More information

Householder s approximants and continued fraction expansion of quadratic irrationals

Householder s approximants and continued fraction expansion of quadratic irrationals Householder s approximats ad cotiued fractio expasio of quadratic irratioals Viko Petričević Departmet of Mathematics, Uiversity of Zagre Bijeička cesta 30, 0000 Zagre, Croatia E-mail: vpetrice@mathhr

More information

Section A assesses the Units Numerical Analysis 1 and 2 Section B assesses the Unit Mathematics for Applied Mathematics

Section 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 information

An Algebraic Elimination Method for the Linear Complementarity Problem

An Algebraic Elimination Method for the Linear Complementarity Problem Volume-3, Issue-5, October-2013 ISSN No: 2250-0758 Iteratioal Joural of Egieerig ad Maagemet Research Available at: wwwijemret Page Number: 51-55 A Algebraic Elimiatio Method for the Liear Complemetarity

More information

Generalization of Samuelson s inequality and location of eigenvalues

Generalization of Samuelson s inequality and location of eigenvalues Proc. Idia Acad. Sci. Math. Sci.) Vol. 5, No., February 05, pp. 03. c Idia Academy of Scieces Geeralizatio of Samuelso s iequality ad locatio of eigevalues R SHARMA ad R SAINI Departmet of Mathematics,

More information

62. 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 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 information

MAT1026 Calculus II Basic Convergence Tests for Series

MAT1026 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 information

A STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE METHOD

A STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE METHOD IRET: Iteratioal oural of Research i Egieerig ad Techology eissn: 39-63 pissn: 3-7308 A STUDY ON MHD BOUNDARY LAYER FLOW OVER A NONLINEAR STRETCHING SHEET USING IMPLICIT FINITE DIFFERENCE METHOD Satish

More information

POWER SERIES SOLUTION OF FIRST ORDER MATRIX DIFFERENTIAL EQUATIONS

POWER 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 information

Stability Analysis of the Euler Discretization for SIR Epidemic Model

Stability Analysis of the Euler Discretization for SIR Epidemic Model Stability Aalysis of the Euler Discretizatio for SIR Epidemic Model Agus Suryato Departmet of Mathematics, Faculty of Scieces, Brawijaya Uiversity, Jl Vetera Malag 6545 Idoesia Abstract I this paper we

More information

HOMOTOPY PERTURBATION METHOD FOR VISCOUS HEATING IN PLANE COUETTE FLOW

HOMOTOPY PERTURBATION METHOD FOR VISCOUS HEATING IN PLANE COUETTE FLOW Yu, Y.-S. et al.: Homotopy Perturbatio Method for Viscous Heatig THERMAL SCIENCE, Year 13, Vol. 17, No. 5, pp. 1355-136 1355 HOMOTOPY PERTURBATION METHOD FOR VISCOUS HEATING IN PLANE COUETTE FLOW by Yi-Sha

More information

Some Approximate Fixed Point Theorems

Some Approximate Fixed Point Theorems It. Joural of Math. Aalysis, Vol. 3, 009, o. 5, 03-0 Some Approximate Fixed Poit Theorems Bhagwati Prasad, Bai Sigh ad Ritu Sahi Departmet of Mathematics Jaypee Istitute of Iformatio Techology Uiversity

More information

On general Gamma-Taylor operators on weighted spaces

On general Gamma-Taylor operators on weighted spaces It. J. Adv. Appl. Math. ad Mech. 34 16 9 15 ISSN: 347-59 Joural homepage: www.ijaamm.com IJAAMM Iteratioal Joural of Advaces i Applied Mathematics ad Mechaics O geeral Gamma-Taylor operators o weighted

More information

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A)

REGRESSION (Physics 1210 Notes, Partial Modified Appendix A) REGRESSION (Physics 0 Notes, Partial Modified Appedix A) HOW TO PERFORM A LINEAR REGRESSION Cosider the followig data poits ad their graph (Table I ad Figure ): X Y 0 3 5 3 7 4 9 5 Table : Example Data

More information

Numerical Methods in Fourier Series Applications

Numerical Methods in Fourier Series Applications Numerical Methods i Fourier Series Applicatios Recall that the basic relatios i usig the Trigoometric Fourier Series represetatio were give by f ( x) a o ( a x cos b x si ) () where the Fourier coefficiets

More information

Stopping oscillations of a simple harmonic oscillator using an impulse force

Stopping oscillations of a simple harmonic oscillator using an impulse force It. J. Adv. Appl. Math. ad Mech. 5() (207) 6 (ISSN: 2347-2529) IJAAMM Joural homepage: www.ijaamm.com Iteratioal Joural of Advaces i Applied Mathematics ad Mechaics Stoppig oscillatios of a simple harmoic

More information

ON POINTWISE BINOMIAL APPROXIMATION

ON POINTWISE BINOMIAL APPROXIMATION Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No. 1 2011, 57-66 ON POINTWISE BINOMIAL APPROXIMATION BY w-functions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece

More information

Carleton College, Winter 2017 Math 121, Practice Final Prof. Jones. Note: the exam will have a section of true-false questions, like the one below.

Carleton College, Winter 2017 Math 121, Practice Final Prof. Jones. Note: the exam will have a section of true-false questions, like the one below. Carleto College, Witer 207 Math 2, Practice Fial Prof. Joes Note: the exam will have a sectio of true-false questios, like the oe below.. True or False. Briefly explai your aswer. A icorrectly justified

More information

A NEW APPROACH TO SOLVE AN UNBALANCED ASSIGNMENT PROBLEM

A NEW APPROACH TO SOLVE AN UNBALANCED ASSIGNMENT PROBLEM A NEW APPROACH TO SOLVE AN UNBALANCED ASSIGNMENT PROBLEM *Kore B. G. Departmet Of Statistics, Balwat College, VITA - 415 311, Dist.: Sagli (M. S.). Idia *Author for Correspodece ABSTRACT I this paper I

More information

PAijpam.eu ON TENSOR PRODUCT DECOMPOSITION

PAijpam.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 information

Discrete Orthogonal Moment Features Using Chebyshev Polynomials

Discrete Orthogonal Moment Features Using Chebyshev Polynomials Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical

More information

About the use of a result of Professor Alexandru Lupaş to obtain some properties in the theory of the number e 1

About the use of a result of Professor Alexandru Lupaş to obtain some properties in the theory of the number e 1 Geeral Mathematics Vol. 5, No. 2007), 75 80 About the use of a result of Professor Alexadru Lupaş to obtai some properties i the theory of the umber e Adrei Verescu Dedicated to Professor Alexadru Lupaş

More information

Most text will write ordinary derivatives using either Leibniz notation 2 3. y + 5y= e and y y. xx tt t

Most 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 information

Name: Math 10550, Final Exam: December 15, 2007

Name: Math 10550, Final Exam: December 15, 2007 Math 55, Fial Exam: December 5, 7 Name: Be sure that you have all pages of the test. No calculators are to be used. The exam lasts for two hours. Whe told to begi, remove this aswer sheet ad keep it uder

More information

Decoupling Zeros of Positive Discrete-Time Linear Systems*

Decoupling Zeros of Positive Discrete-Time Linear Systems* Circuits ad Systems,,, 4-48 doi:.436/cs..7 Published Olie October (http://www.scirp.org/oural/cs) Decouplig Zeros of Positive Discrete-Time Liear Systems* bstract Tadeusz Kaczorek Faculty of Electrical

More information

w (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.

w (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ. 2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For

More information

Numerical Methods for Ordinary Differential Equations

Numerical Methods for Ordinary Differential Equations Numerical Methods for Ordiary Differetial Equatios Braislav K. Nikolić Departmet of Physics ad Astroomy, Uiversity of Delaware, U.S.A. PHYS 460/660: Computatioal Methods of Physics http://www.physics.udel.edu/~bikolic/teachig/phys660/phys660.html

More information

IP Reference guide for integer programming formulations.

IP Reference guide for integer programming formulations. IP Referece guide for iteger programmig formulatios. by James B. Orli for 15.053 ad 15.058 This documet is iteded as a compact (or relatively compact) guide to the formulatio of iteger programs. For more

More information

*X203/701* X203/701. APPLIED MATHEMATICS ADVANCED HIGHER Numerical Analysis. Read carefully

*X203/701* X203/701. APPLIED MATHEMATICS ADVANCED HIGHER Numerical Analysis. Read carefully X0/70 NATIONAL QUALIFICATIONS 006 MONDAY, MAY.00 PM.00 PM APPLIED MATHEMATICS ADVANCED HIGHER Numerical Aalysis Read carefully. Calculators may be used i this paper.. Cadidates should aswer all questios.

More information

PLEASE 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)

PLEASE 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 information

INFINITE SEQUENCES AND SERIES

INFINITE SEQUENCES AND SERIES 11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS

More information

q-durrmeyer operators based on Pólya distribution

q-durrmeyer operators based on Pólya distribution Available olie at wwwtjsacom J Noliear Sci Appl 9 206 497 504 Research Article -Durrmeyer operators based o Pólya distributio Vijay Gupta a Themistocles M Rassias b Hoey Sharma c a Departmet of Mathematics

More information

Differentiable Convex Functions

Differentiable 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 information

Differentiable Convex Functions

Differentiable 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 information

(A) 0 (B) (C) (D) (E) 2.703

(A) 0 (B) (C) (D) (E) 2.703 Class Questios 007 BC Calculus Istitute Questios for 007 BC Calculus Istitutes CALCULATOR. How may zeros does the fuctio f ( x) si ( l ( x) ) Explai how you kow. = have i the iterval (0,]? LIMITS. 00 Released

More information

Math 116 Practice for Exam 3

Math 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 information

HOMEWORK #10 SOLUTIONS

HOMEWORK #10 SOLUTIONS Math 33 - Aalysis I Sprig 29 HOMEWORK # SOLUTIONS () Prove that the fuctio f(x) = x 3 is (Riema) itegrable o [, ] ad show that x 3 dx = 4. (Without usig formulae for itegratio that you leart i previous

More information

Modified Ratio Estimators Using Known Median and Co-Efficent of Kurtosis

Modified Ratio Estimators Using Known Median and Co-Efficent of Kurtosis America Joural of Mathematics ad Statistics 01, (4): 95-100 DOI: 10.593/j.ajms.01004.05 Modified Ratio s Usig Kow Media ad Co-Efficet of Kurtosis J.Subramai *, G.Kumarapadiya Departmet of Statistics, Podicherry

More information

INEQUALITIES BJORN POONEN

INEQUALITIES 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 information

Teaching Mathematics Concepts via Computer Algebra Systems

Teaching Mathematics Concepts via Computer Algebra Systems Iteratioal Joural of Mathematics ad Statistics Ivetio (IJMSI) E-ISSN: 4767 P-ISSN: - 4759 Volume 4 Issue 7 September. 6 PP-- Teachig Mathematics Cocepts via Computer Algebra Systems Osama Ajami Rashaw,

More information

Oscillation and Property B for Third Order Difference Equations with Advanced Arguments

Oscillation and Property B for Third Order Difference Equations with Advanced Arguments Iter atioal Joural of Pure ad Applied Mathematics Volume 3 No. 0 207, 352 360 ISSN: 3-8080 (prited versio); ISSN: 34-3395 (o-lie versio) url: http://www.ijpam.eu ijpam.eu Oscillatio ad Property B for Third

More information

Sequences of Definite Integrals, Factorials and Double Factorials

Sequences of Definite Integrals, Factorials and Double Factorials 47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology

More information

Complex Analysis Spring 2001 Homework I Solution

Complex 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 information

Intermittent demand forecasting by using Neural Network with simulated data

Intermittent demand forecasting by using Neural Network with simulated data Proceedigs of the 011 Iteratioal Coferece o Idustrial Egieerig ad Operatios Maagemet Kuala Lumpur, Malaysia, Jauary 4, 011 Itermittet demad forecastig by usig Neural Network with simulated data Nguye Khoa

More information

SOLVING NONLINEAR EQUATIONS USING A NEW TENTH-AND SEVENTH-ORDER METHODS FREE FROM SECOND DERIVATIVE M.A. Hafiz 1, Salwa M.H.

SOLVING NONLINEAR EQUATIONS USING A NEW TENTH-AND SEVENTH-ORDER METHODS FREE FROM SECOND DERIVATIVE M.A. Hafiz 1, Salwa M.H. International Journal of Differential Equations and Applications Volume 12 No. 4 2013, 169-183 ISSN: 1311-2872 url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijdea.v12i4.1344 PA acadpubl.eu SOLVING

More information

HOUSEHOLDER S APPROXIMANTS AND CONTINUED FRACTION EXPANSION OF QUADRATIC IRRATIONALS. Vinko Petričević University of Zagreb, Croatia

HOUSEHOLDER S APPROXIMANTS AND CONTINUED FRACTION EXPANSION OF QUADRATIC IRRATIONALS. Vinko Petričević University of Zagreb, Croatia HOUSEHOLDER S APPROXIMANTS AND CONTINUED FRACTION EXPANSION OF QUADRATIC IRRATIONALS Viko Petričević Uiversity of Zagre, Croatia Astract There are umerous methods for ratioal approximatio of real umers

More information

Implicit function theorem

Implicit function theorem Jovo Jaric Implicit fuctio theorem The reader kows that the equatio of a curve i the x - plae ca be expressed F x, =., this does ot ecessaril represet a fuctio. Take, for example F x, = 2x x =. (1 either

More information

ROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND

ROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND Pacific-Asia Joural of Mathematics, Volume 5, No., Jauary-Jue 20 ROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND SHAKEEL JAVAID, Z. H. BAKHSHI & M. M. KHALID ABSTRACT: I this paper, the roll cuttig problem

More information

Beyond simple iteration of a single function, or even a finite sequence of functions, results

Beyond simple iteration of a single function, or even a finite sequence of functions, results A Primer o the Elemetary Theory of Ifiite Compositios of Complex Fuctios Joh Gill Sprig 07 Abstract: Elemetary meas ot requirig the complex fuctios be holomorphic Theorem proofs are fairly simple ad are

More information

Ellipsoid Method for Linear Programming made simple

Ellipsoid 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 information

Section 1 of Unit 03 (Pure Mathematics 3) Algebra

Section 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 information

Recurrence Relations

Recurrence 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 information

Solutions to Final Exam Review Problems

Solutions to Final Exam Review Problems . Let f(x) 4+x. Solutios to Fial Exam Review Problems Math 5C, Witer 2007 (a) Fid the Maclauri series for f(x), ad compute its radius of covergece. Solutio. f(x) 4( ( x/4)) ( x/4) ( ) 4 4 + x. Sice the

More information

On the Weak Localization Principle of the Eigenfunction Expansions of the Laplace-Beltrami Operator by Riesz Method ABSTRACT 1.

On the Weak Localization Principle of the Eigenfunction Expansions of the Laplace-Beltrami Operator by Riesz Method ABSTRACT 1. Malaysia Joural of Mathematical Scieces 9(): 337-348 (05) MALAYSIA JOURAL OF MATHEMATICAL SCIECES Joural homepage: http://eispemupmedumy/joural O the Weak Localizatio Priciple of the Eigefuctio Expasios

More information

Integer Linear Programming

Integer Linear Programming Iteger Liear Programmig Itroductio Iteger L P problem (P) Mi = s. t. a = b i =,, m = i i 0, iteger =,, c Eemple Mi z = 5 s. t. + 0 0, 0, iteger F(P) = feasible domai of P Itroductio Iteger L P problem

More information

Seunghee Ye Ma 8: Week 5 Oct 28

Seunghee Ye Ma 8: Week 5 Oct 28 Week 5 Summary I Sectio, we go over the Mea Value Theorem ad its applicatios. I Sectio 2, we will recap what we have covered so far this term. Topics Page Mea Value Theorem. Applicatios of the Mea Value

More information

(a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b)

(a) (b) All real numbers. (c) All real numbers. (d) None. to show the. (a) 3. (b) [ 7, 1) (c) ( 7, 1) (d) At x = 7. (a) (b) Chapter 0 Review 597. E; a ( + )( + ) + + S S + S + + + + + + S lim + l. D; a diverges by the Itegral l k Test sice d lim [(l ) ], so k l ( ) does ot coverge absolutely. But it coverges by the Alteratig

More information

A numerical Technique Finite Volume Method for Solving Diffusion 2D Problem

A numerical Technique Finite Volume Method for Solving Diffusion 2D Problem The Iteratioal Joural Of Egieerig d Sciece (IJES) Volume 4 Issue 10 Pages PP -35-41 2015 ISSN (e): 2319 1813 ISSN (p): 2319 1805 umerical Techique Fiite Volume Method for Solvig Diffusio 2D Problem 1 Mohammed

More information

(p, q)-baskakov-kantorovich Operators

(p, q)-baskakov-kantorovich Operators Appl Math If Sci, No 4, 55-556 6 55 Applied Mathematics & Iformatio Scieces A Iteratioal Joural http://ddoiorg/8576/amis/433 p, q-basaov-katorovich Operators Vijay Gupta Departmet of Mathematics, Netaji

More information

Sequences and Limits

Sequences and Limits Chapter Sequeces ad Limits Let { a } be a sequece of real or complex umbers A ecessary ad sufficiet coditio for the sequece to coverge is that for ay ɛ > 0 there exists a iteger N > 0 such that a p a q

More information

The Choquet Integral with Respect to Fuzzy-Valued Set Functions

The Choquet Integral with Respect to Fuzzy-Valued Set Functions The Choquet Itegral with Respect to Fuzzy-Valued Set Fuctios Weiwei Zhag Abstract The Choquet itegral with respect to real-valued oadditive set fuctios, such as siged efficiecy measures, has bee used i

More information

The Jordan Normal Form: A General Approach to Solving Homogeneous Linear Systems. Mike Raugh. March 20, 2005

The 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 information

The Gamma function Michael Taylor. Abstract. This material is excerpted from 18 and Appendix J of [T].

The Gamma function Michael Taylor. Abstract. This material is excerpted from 18 and Appendix J of [T]. The Gamma fuctio Michael Taylor Abstract. This material is excerpted from 8 ad Appedix J of [T]. The Gamma fuctio has bee previewed i 5.7 5.8, arisig i the computatio of a atural Laplace trasform: 8. ft

More information

On Exact Finite-Difference Scheme for Numerical Solution of Initial Value Problems in Ordinary Differential Equations.

On Exact Finite-Difference Scheme for Numerical Solution of Initial Value Problems in Ordinary Differential Equations. O Exact Fiite-Differece Sceme for Numerical Solutio of Iitial Value Problems i Ordiar Differetial Equatios. Josua Suda, M.Sc. Departmet of Matematical Scieces, Adamawa State Uiversit, Mubi, Nigeria. E-mail:

More information

G. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan

G. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity

More information

PUTNAM TRAINING INEQUALITIES

PUTNAM TRAINING INEQUALITIES PUTNAM TRAINING INEQUALITIES (Last updated: December, 207) Remark This is a list of exercises o iequalities Miguel A Lerma Exercises If a, b, c > 0, prove that (a 2 b + b 2 c + c 2 a)(ab 2 + bc 2 + ca

More information

REVIEW 1, MATH n=1 is convergent. (b) Determine whether a n is convergent.

REVIEW 1, MATH n=1 is convergent. (b) Determine whether a n is convergent. REVIEW, MATH 00. Let a = +. a) Determie whether the sequece a ) is coverget. b) Determie whether a is coverget.. Determie whether the series is coverget or diverget. If it is coverget, fid its sum. a)

More information

A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca

A 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 information

Topics in Probability Theory and Stochastic Processes Steven R. Dunbar. Stirling s Formula Derived from the Gamma Function

Topics in Probability Theory and Stochastic Processes Steven R. Dunbar. Stirling s Formula Derived from the Gamma Function Steve R. Dubar Departmet of Mathematics 23 Avery Hall Uiversity of Nebraska-Licol Licol, NE 68588-3 http://www.math.ul.edu Voice: 42-472-373 Fax: 42-472-8466 Topics i Probability Theory ad Stochastic Processes

More information

Diploma Programme. Mathematics HL guide. First examinations 2014

Diploma Programme. Mathematics HL guide. First examinations 2014 Diploma Programme First eamiatios 014 33 Topic 6 Core: Calculus The aim of this topic is to itroduce studets to the basic cocepts ad techiques of differetial ad itegral calculus ad their applicatio. 6.1

More information

The Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1

The Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1 460_0906.qxd //04 :8 PM Page 69 SECTION 9.6 The Ratio ad Root Tests 69 Sectio 9.6 EXPLORATION Writig a Series Oe of the followig coditios guaratees that a series will diverge, two coditios guaratee that

More information

On the Derivation and Implementation of a Four Stage Harmonic Explicit Runge-Kutta Method *

On the Derivation and Implementation of a Four Stage Harmonic Explicit Runge-Kutta Method * Applied Mathematics, 05, 6, 694-699 Published Olie April 05 i SciRes. http://www.scirp.org/joural/am http://dx.doi.org/0.46/am.05.64064 O the Derivatio ad Implemetatio of a Four Stage Harmoic Explicit

More information

Definitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients.

Definitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients. Defiitios ad Theorems Remember the scalar form of the liear programmig problem, Miimize, Subject to, f(x) = c i x i a 1i x i = b 1 a mi x i = b m x i 0 i = 1,2,, where x are the decisio variables. c, b,

More information

Period Function of a Lienard Equation

Period 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 information

Eigenvalue localization for complex matrices

Eigenvalue localization for complex matrices Electroic Joural of Liear Algebra Volume 7 Article 1070 014 Eigevalue localizatio for complex matrices Ibrahim Halil Gumus Adıyama Uiversity, igumus@adiyama.edu.tr Omar Hirzallah Hashemite Uiversity, o.hirzal@hu.edu.jo

More information

Fourier Series and the Wave Equation

Fourier Series and the Wave Equation Fourier Series ad the Wave Equatio We start with the oe-dimesioal wave equatio u u =, x u(, t) = u(, t) =, ux (,) = f( x), u ( x,) = This represets a vibratig strig, where u is the displacemet of the strig

More information

Ans: a n = 3 + ( 1) n Determine whether the sequence converges or diverges. If it converges, find the limit.

Ans: a n = 3 + ( 1) n Determine whether the sequence converges or diverges. If it converges, find the limit. . Fid a formula for the term a of the give sequece: {, 3, 9, 7, 8 },... As: a = 3 b. { 4, 9, 36, 45 },... As: a = ( ) ( + ) c. {5,, 5,, 5,, 5,,... } As: a = 3 + ( ) +. Determie whether the sequece coverges

More information

Analytic Continuation

Analytic 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 information

Analysis of Algorithms. Introduction. Contents

Analysis of Algorithms. Introduction. Contents Itroductio The focus of this module is mathematical aspects of algorithms. Our mai focus is aalysis of algorithms, which meas evaluatig efficiecy of algorithms by aalytical ad mathematical methods. We

More information

Mechanical Efficiency of Planetary Gear Trains: An Estimate

Mechanical Efficiency of Planetary Gear Trains: An Estimate Mechaical Efficiecy of Plaetary Gear Trais: A Estimate Dr. A. Sriath Professor, Dept. of Mechaical Egieerig K L Uiversity, A.P, Idia E-mail: sriath_me@klce.ac.i G. Yedukodalu Assistat Professor, Dept.

More information

f x x c x c x c... x c...

f x x c x c x c... x c... CALCULUS BC WORKSHEET ON POWER SERIES. Derive the Taylor series formula by fillig i the blaks below. 4 5 Let f a a c a c a c a4 c a5 c a c What happes to this series if we let = c? f c so a Now differetiate

More information

Monte Carlo Integration

Monte Carlo Integration Mote Carlo Itegratio I these otes we first review basic umerical itegratio methods (usig Riema approximatio ad the trapezoidal rule) ad their limitatios for evaluatig multidimesioal itegrals. Next we itroduce

More information

Math 475, Problem Set #12: Answers

Math 475, Problem Set #12: Answers Math 475, Problem Set #12: Aswers A. Chapter 8, problem 12, parts (b) ad (d). (b) S # (, 2) = 2 2, sice, from amog the 2 ways of puttig elemets ito 2 distiguishable boxes, exactly 2 of them result i oe

More information

A Simplified Binet Formula for k-generalized Fibonacci Numbers

A Simplified Binet Formula for k-generalized Fibonacci Numbers A Simplified Biet Formula for k-geeralized Fiboacci Numbers Gregory P. B. Dresde Departmet of Mathematics Washigto ad Lee Uiversity Lexigto, VA 440 dresdeg@wlu.edu Zhaohui Du Shaghai, Chia zhao.hui.du@gmail.com

More information

AP Calculus Chapter 9: Infinite Series

AP Calculus Chapter 9: Infinite Series AP Calculus Chapter 9: Ifiite Series 9. Sequeces a, a 2, a 3, a 4, a 5,... Sequece: A fuctio whose domai is the set of positive itegers = 2 3 4 a = a a 2 a 3 a 4 terms of the sequece Begi with the patter

More information

MA131 - Analysis 1. Workbook 9 Series III

MA131 - Analysis 1. Workbook 9 Series III MA3 - Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................

More information

Zeros of Polynomials

Zeros 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 information

e to approximate (using 4

e to approximate (using 4 Review: Taylor Polyomials ad Power Series Fid the iterval of covergece for the series Fid a series for f ( ) d ad fid its iterval of covergece Let f( ) Let f arcta a) Fid the rd degree Maclauri polyomial

More information

JANE PROFESSOR WW Prob Lib1 Summer 2000

JANE PROFESSOR WW Prob Lib1 Summer 2000 JANE PROFESSOR WW Prob Lib Summer 000 Sample WeBWorK problems. WeBWorK assigmet Series6CompTests due /6/06 at :00 AM..( pt) Test each of the followig series for covergece by either the Compariso Test or

More information