CHAPTER 6c. NUMERICAL INTERPOLATION
|
|
|
- Nathan Lester
- 6 years ago
- Views:
Transcription
1 CHAPTER 6c. NUMERICAL INTERPOLATION A. J. Clark School o Egieerig Departmet o Civil ad Evirometal Egieerig y Dr. Irahim A. Assakka Sprig ENCE - Computatio Methods i Civil Egieerig II Departmet o Civil ad Evirometal Egieerig Uiversity o Marylad, College Park Method o Udetermied Coeiciets Eample 6 Develop a ourth-order iterpolatio polyomial or the ollowig set o data, or which we kow their origial uctio, that is, (). 4 () ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 7
2 Fiite Dierece Iterpolatio I the values o the idepedet variales are equally spaced, a iite dierece scheme ca e used to develop a iterpolatio polyomial. I is the icremet o which the values o the idepedet variale are recorded, the the irst iite dierece o the depedet variale y () is ( ) () ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 74 Fiite Dierece Iterpolatio The secod iite dierece ca e epressed as [ ] [ ( ) ( ) ] -[ ( ) ] I geeral, the iite dierece is [ ] () () ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 75
3 Fiite Dierece Iterpolatio Rule The ith iite dierece, where i is less tha or equal to, o a th-degree polyomial is a polyomial o degree ( ). This rule implies that the th dierece would e a costat. Assume the ollowig th-order polyomial: () L (4) does ot equal zero. ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 76 Fiite Dierece Iterpolatio For ( ), the polyomial give y Eq. 4 ecomes ( ) ( ) ( ) ( ) L ( ) This polyomial ca e rearrage ito ( ) ( ) L (5) (6) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 77
4 Fiite Dierece Iterpolatio The irst iite dierece ca e costructed y sutractig Eq. 4 rom Eq.6 as ollows: () L ( ) ( ) L Or ( ) ( ) L (7) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 78 Fiite Dierece Iterpolatio For ( ), Eq. 5 ecomes ( ) ( ) ( ) L (8) Equatios 8, 6, ad 4 ca e sustituted ito Equatio to compute a secod iite dierece, with the ollowig result: ( )( ) L (9) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 79 4
5 Fiite Dierece Iterpolatio Equatio 9 is a polyomial o order (-). By iductio,the ith iite dierece is give y i i ( )( ) L ( i ) L i () Equatio is a polyomial o order ( ). The th iite dierece equals the ollowig costat: L! () ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 8 Fiite Dierece Iterpolatio Eample Suppose that, the Eq. 4 ecomes Or () () () ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 8 5
6 6 Slide No. 8 ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Fiite Dierece Iterpolatio Eample (cot d): For ( ), ( ) ca e deied as ollows: () Slide No. 8 ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Fiite Dierece Iterpolatio Eample (cot d): For ( ), ( ) ca e deied as ollows: (4) 4
7 Fiite Dierece Iterpolatio Eample (cot d): Recall Eq.,, ad ( ) () ( ) ( ) ( ) The irst dierece is computed as ollows: ( ) (5) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 84 Fiite Dierece Iterpolatio Eample (cot d): Similarly, recall Eq.,,, ad 4 [ ] [ ( ) ( ) ] [ ( ) ] - () ( ) ( 4 ) ( ) ( ) The secod iite dierece is computed as ollows: ( ) ( ) ( ) (6) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 85 7
8 Fiite Dierece Iterpolatio First ad Secod Fiite Dierece The a quadratic polyomial o the orm () The irst ad secod iite dierece are give as ad ( ) ( ) (8) (7) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 86 Fiite Dierece Iterpolatio Eample Develop a iterpolatio polyomial or the ollowig data usig the iite dierece approach. Estimate the () or.7 () 5 8 ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 87 8
9 Fiite Dierece Iterpolatio Eample (cot d): () 5 8 () ( ) 5 ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 88 Fiite Dierece Iterpolatio Eample (cot d): The irst iite dierece ca e otaied usig Eq. as ( ) 5 This value ca e equated to Eq. 7 as ollows: ( ) ()() () () (9) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 89 9
10 Fiite Dierece Iterpolatio Eample (cot d): The secod iite dierece ca e otaied usig Eq. as [ ( ) ( ) ]-[ ( ) ] [ 8 5] [ 5 ] This value ca e equated to Eq. 8 as ollows: () () ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 9 Fiite Dierece Iterpolatio Eample (cot d): Solvig Eq., gives.5 Sustitutig this result ito Eq. 9, gives Or.5 (.5) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 9
11 Fiite Dierece Iterpolatio Eample (cot d): Usig these values o ad ad ay oe o the three poits, a value o ca e calculated rom Eq. as ollows: () () 8.5().5() ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 9 Fiite Dierece Iterpolatio Eample (cot d): Thus, the iterpolatio polyomial is () ().5.5 (.7) ca e estimated: (.7).5(.7).5(.7) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 9
12 Fiite Dierece Iterpolatio Fiite-dierece Tales We see i the precedig eample that the iite dierece approach ca e used to costruct a iterpolatio polyomial. However, the procedure ca e tedious ad requires a lot o computatios. Thereore, we eed some sort o scheme to orgaize the procedure usig iitedierece tales. ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 94 Fiite Dierece Iterpolatio Fiite-dierece Tale The iite-diereces ca e orgaized ito a tale as show i the et viewgraph. The iite-dierece tale ca e used to derive a iterpolatio polyomial. As we will see, iite-dierece tales ca e used with Newto ormula to derive iterpolatio polyomials. ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 95
13 Fiite-dierece Tale () () () ( ) () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) : : : : ( -) [ ( -)] [ ( -)] [ ( -)] ( -) [ ( -)] [ ( -)] [ ( -)] () ( ) ( ) : [ ( -)] ::: () ( ) ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 96 Fiite Dierece Iterpolatio Eample Costruct a iite-dierece tale or the ollowig set o data: 4 () ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 97
14 Fiite Dierece Iterpolatio Eample (cot d): () ENCE CHAPTER 6c. NUMERICAL INTERPOLATION Slide No. 98 4
CHAPTER 6d. NUMERICAL INTERPOLATION
CHAPER 6d. NUMERICAL INERPOLAION A. J. Clark School o Egieerig Departmet o Civil ad Evirometal Egieerig by Dr. Ibrahim A. Assakka Sprig ENCE - Computatio Methods i Civil Egieerig II Departmet o Civil ad
Numerical Integration Formulas
Numerical Itegratio Formulas Berli Che Departmet o Computer Sciece & Iormatio Egieerig Natioal Taiwa Normal Uiversity Reerece: 1. Applied Numerical Methods with MATLAB or Egieers, Chapter 19 & Teachig
CS321. 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
TAYLOR AND MACLAURIN SERIES
Calculus TAYLOR AND MACLAURIN SERIES Give a uctio ( ad a poit a, we wish to approimate ( i the eighborhood o a by a polyomial o degree. c c ( a c( a c( a P ( c ( a We have coeiciets to choose. We require
CS537. 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
Taylor Polynomials and Approximations - Classwork
Taylor Polyomials ad Approimatios - Classwork Suppose you were asked to id si 37 o. You have o calculator other tha oe that ca do simple additio, subtractio, multiplicatio, or divisio. Fareched\ Not really.
g () n = g () n () f, f n = f () n () x ( n =1,2,3, ) j 1 + j 2 + +nj n = n +2j j n = r & j 1 j 1, j 2, j 3, j 4 = ( 4, 0, 0, 0) f 4 f 3 3!
Higher Derivative o Compositio. Formulas o Higher Derivative o Compositio.. Faà di Bruo's Formula About the ormula o the higher derivative o compositio, the oe by a mathematicia Faà di Bruo i Italy o about
Topic 9 - Taylor and MacLaurin Series
Topic 9 - Taylor ad MacLauri Series A. Taylors Theorem. The use o power series is very commo i uctioal aalysis i act may useul ad commoly used uctios ca be writte as a power series ad this remarkable result
Modification of Weerakoon-Fernando s Method with Fourth-Order of Convergence for Solving Nonlinear Equation
ISSN: 50-08 Iteratioal Joural o AdvacedResearch i Sciece, Egieerig ad Techology Vol. 5, Issue 8, August 018 Modiicatio o Weerakoo-Ferado s Method with Fourth-Order o Covergece or Solvig Noliear Equatio
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
A NUMERICAL METHOD OF SOLVING CAUCHY PROBLEM FOR DIFFERENTIAL EQUATIONS BASED ON A LINEAR APPROXIMATION
U.P.B. Sci. Bull., Series A, Vol. 79, Iss. 4, 7 ISSN -77 A NUMERICAL METHOD OF SOLVING CAUCHY PROBLEM FOR DIFFERENTIAL EQUATIONS BASED ON A LINEAR APPROXIMATION Cristia ŞERBĂNESCU, Marius BREBENEL A alterate
Solving third order boundary value problem with fifth order block method
Matematical Metods i Egieerig ad Ecoomics Solvig tird order boudary value problem wit it order bloc metod A. S. Abdulla, Z. A. Majid, ad N. Seu Abstract We develop a it order two poit bloc metod or te
Lecture 11. Solution of Nonlinear Equations - III
Eiciecy o a ethod Lecture Solutio o Noliear Equatios - III The eiciecy ide o a iterative ethod is deied by / E r r: rate o covergece o the ethod : total uber o uctios ad derivative evaluatios at each step
Castiel, Supernatural, Season 6, Episode 18
13 Differetial Equatios the aswer to your questio ca best be epressed as a series of partial differetial equatios... Castiel, Superatural, Seaso 6, Episode 18 A differetial equatio is a mathematical equatio
The Discrete-Time Fourier Transform (DTFT)
EEL: Discrete-Time Sigals ad Systems The Discrete-Time Fourier Trasorm (DTFT) The Discrete-Time Fourier Trasorm (DTFT). Itroductio I these otes, we itroduce the discrete-time Fourier trasorm (DTFT) ad
Phys. 201 Mathematical Physics 1 Dr. Nidal M. Ershaidat Doc. 12
Physics Departmet, Yarmouk Uiversity, Irbid Jorda Phys. Mathematical Physics Dr. Nidal M. Ershaidat Doc. Fourier Series Deiitio A Fourier series is a expasio o a periodic uctio (x) i terms o a iiite sum
Mathematical Expressions for Estimation of Errors in the Formulas which are used to obtaining intermediate values of Biological Activity in QSAR
Iteratioal Joural o Iovatio ad Applied Studies ISSN 08-934 Vol. No. 3 Mar. 03, pp. 7-79 03 Iovative Space o Scietiic Researc Jourals ttp//www.issr-jourals.org/ijias/ Matematical Expressios or Estimatio
Representing Functions as Power Series. 3 n ...
Math Fall 7 Lab Represetig Fuctios as Power Series I. Itrouctio I sectio.8 we leare the series c c c c c... () is calle a power series. It is a uctio o whose omai is the set o all or which it coverges.
Hyperbolic Heat Equation in Bar and Finite Difference Schemes of Exact Spectrum
atest reds o heoretical ad Applied Mechaics Fluid Mechaics ad Heat & Mass raser Hyperbolic Heat Equatio i Bar ad Fiite Dierece Schemes o Exact Spectrum A BUIKIS H KAIS Abstract he solutios o correspodig
SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE)
QUESTION BANK 8 SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayaavaam Road 5758 QUESTION BANK (DESCRIPTIVE) Subject with Code : (6HS6) Course & Brach: B.Tech AG Year & Sem: II-B.Tech&
Mon Apr Second derivative test, and maybe another conic diagonalization example. Announcements: Warm-up Exercise:
Math 2270-004 Week 15 otes We will ot ecessarily iish the material rom 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
ECE Spring Prof. David R. Jackson ECE Dept. Notes 20
ECE 6341 Sprig 016 Prof. David R. Jackso ECE Dept. Notes 0 1 Spherical Wave Fuctios Cosider solvig ψ + k ψ = 0 i spherical coordiates z φ θ r y x Spherical Wave Fuctios (cot.) I spherical coordiates we
A widely used display of protein shapes is based on the coordinates of the alpha carbons - - C α
Nice plottig of proteis: I A widely used display of protei shapes is based o the coordiates of the alpha carbos - - C α -s. The coordiates of the C α -s are coected by a cotiuous curve that roughly follows
Chapter 2: Numerical Methods
Chapter : Numerical Methods. Some Numerical Methods for st Order ODEs I this sectio, a summar of essetial features of umerical methods related to solutios of ordiar differetial equatios is give. I geeral,
GRAPHING LINEAR EQUATIONS. Linear Equations ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.
GRAPHING LINEAR EQUATIONS Quadrat II Quadrat I ORDERED PAIR: The first umer i the ordered pair is the -coordiate ad the secod umer i the ordered pair is the y-coordiate. (,1 ) Origi ( 0, 0 ) _-ais Liear
Root Finding COS 323
Root Fidig COS 33 Why Root Fidig? Solve or i ay equatio: b where? id root o g b 0 Might ot be able to solve or directly e.g., e -0. si3-0.5 Evaluatig might itsel require solvig a dieretial equatio, ruig
Computation Sessional. Numerical Differentiation and Integration
CE 6: Egieerig Computatio Sessioal Numerical Dieretiatio ad Itegratio ti di ad gradiet commad di() Returs te dierece betwee adjacet elemets i. Typically used or uequally spaced itervals = gradiet(, ) Determies
FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES
LECTURE Third Editio FAILURE CRITERIA: MOHR S CIRCLE AND PRINCIPAL STRESSES A. J. Clark School of Egieerig Departmet of Civil ad Evirometal Egieerig Chapter 7.4 b Dr. Ibrahim A. Assakkaf SPRING 3 ENES
This chapter describes different methods to discretize the diffusion equation. f z 2 = 0. y ) x f
Chapter 8 Diusio Equatio This chapter describes dieret methods to discretize the diusio equatio 2 t α x 2 + 2 y 2 + 2 z 2 = 0 which represets a combied boudary ad iitial value problem, i.e., requires to
PRELIMINARY MATHEMATICS LECTURE 5
SCHOOL OF ORIENTAL AND AFRICAN STUDIES UNIVERSITY OF LONDON DEPARTMENT OF ECONOMICS 5 / - 6 5 MSc Ecoomics PRELIMINARY MATHEMATICS LECTURE 5 Course website: http://mercur.soas.ac.uk/users/sm97/teachig_msc_premath.htm
MATH 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 ) +
Maclaurin and Taylor series
At the ed o the previous chapter we looed at power series ad oted that these were dieret rom other iiite series as they were actually uctios o a variable R: a a + + a + a a Maclauri ad Taylor series +
AE/ME 339 Computational Fluid Dynamics (CFD)
AE/ME 339 Computatioal Fluid Dyamics (CFD 0//004 Topic0_PresCorr_ Computatioal Fluid Dyamics (AE/ME 339 Pressure Correctio Method The pressure correctio formula (6.8.4 Calculatio of p. Coservatio form
Where do eigenvalues/eigenvectors/eigenfunctions come from, and why are they important anyway?
Where do eigevalues/eigevectors/eigeuctios come rom, ad why are they importat ayway? I. Bacgroud (rom Ordiary Dieretial Equatios} Cosider the simplest example o a harmoic oscillator (thi o a vibratig strig)
Linear Differential Equations of Higher Order Basic Theory: Initial-Value Problems d y d y dy
Liear Differetial Equatios of Higher Order Basic Theory: Iitial-Value Problems d y d y dy Solve: a( ) + a ( )... a ( ) a0( ) y g( ) + + + = d d d ( ) Subject to: y( 0) = y0, y ( 0) = y,..., y ( 0) = y
McGill University Math 354: Honors Analysis 3 Fall 2012 Solutions to selected problems. x y
McGill Uiversity Math 354: Hoors Aalysis 3 Fall 212 Assigmet 3 Solutios to selected problems Problem 1. Lipschitz uctios. Let M K be the set o all uctios cotiuous uctios o [, 1] satisyig a Lipschitz coditio
Polynomial Generalizations and Combinatorial Interpretations for Sequences Including the Fibonacci and Pell Numbers
Ope Joural o Discrete Mathematics,,, - http://dxdoiorg/46/odm6 Published Olie Jauary (http://wwwscirporg/oural/odm) Polyomial Geeralizatios ad Combiatorial Iterpretatios or Seueces Icludig the Fiboacci
Solving Third Order Boundary Value Problem Using. Fourth Order Block Method
Applied Matematical Scieces, Vol. 7,, o. 5, 69-65 HIKARI Ltd, www.m-ikari.com Solvig Tird Order Boudar Value Problem Usig Fourt Order Block Metod Amad Sa Abdulla, *Zaaria Abdul Maid, ad Norazak Seu, Istitute
DECOMPOSITION 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
CS475 Parallel Programming
CS475 Parallel Programmig Dieretiatio ad Itegratio Wim Bohm Colorado State Uiversity Ecept as otherwise oted, the cotet o this presetatio is licesed uder the Creative Commos Attributio.5 licese. Pheomea
Interpolation. 1. What is interpolation?
Iterpoltio. Wht is iterpoltio? A uctio is ote give ol t discrete poits such s:.... How does oe id the vlue o t other vlue o? Well cotiuous uctio m e used to represet the + dt vlues with pssig through the
Fourier Series and Transforms
Fourier Series ad rasorms Orthogoal uctios Fourier Series Discrete Fourier Series Fourier rasorm Chebyshev polyomials Scope: wearetryigto approimate a arbitrary uctio ad obtai basis uctios with appropriate
Probability, Statistics, and Reliability for Engineers and Scientists MULTIPLE RANDOM VARIABLES
CHATER robability, Statistics, and Reliability or Engineers and Scientists MULTILE RANDOM VARIABLES Second Edition A. J. Clark School o Engineering Department o Civil and Environmental Engineering 6a robability
NUMERICAL DIFFERENTIAL 1
NUMERICAL DIFFERENTIAL Ruge-Kutta Metods Ruge-Kutta metods are ver popular ecause o teir good eiciec; ad are used i most computer programs or dieretial equatios. Te are sigle-step metods as te Euler metods.
Chapter 9: Numerical Differentiation
178 Chapter 9: Numerical Differetiatio Numerical Differetiatio Formulatio of equatios for physical problems ofte ivolve derivatives (rate-of-chage quatities, such as velocity ad acceleratio). Numerical
-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
Euler-MacLaurin using the ZETA-matrix
Idetities ivolvig biomial-coeiciets, Beroulli- ad Stirligumbers Gottried Helms - Uiv Kassel 2' 26-23 Euler-MacLauri usig the ZETA-matrix Abstract: The represetatio o the Euler-MacLauri-summatio ormula
FIR Filter Design: Part I
EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I. Itroductio FIR Filter Desig: Part I I this set o otes, we cotiue our exploratio o the requecy respose o FIR ilters. First, we cosider some
3.3 Rules for Differentiation Calculus. Drum Roll please [In a Deep Announcer Voice] And now the moment YOU VE ALL been waiting for
![3.3 Rules for Differentiation Calculus. Drum Roll please [In a Deep Announcer Voice] And now the moment YOU VE ALL been waiting for](/thumbs/90/104334191.jpg "3.3 Rules for Differentiation Calculus. Drum Roll please [In a Deep Announcer Voice] And now the moment YOU VE ALL been waiting for")
. Rules or Dieretiatio Calculus. RULES FOR DIFFERENTIATION Drum Roll please [I a Deep Aoucer Voice] A ow the momet YOU VE ALL bee waitig or Rule #1 Derivative o a Costat Fuctio I c is a costat value, the
An Extension of the Szász-Mirakjan Operators
A. Şt. Uiv. Ovidius Costaţa Vol. 7(), 009, 37 44 A Extesio o the Szász-Mirakja Operators C. MORTICI Abstract The paper is devoted to deiig a ew class o liear ad positive operators depedig o a certai uctio
Northwest High School s Algebra 2/Honors Algebra 2 Summer Review Packet
Northwest High School s Algebra /Hoors Algebra Summer Review Packet This packet is optioal! It will NOT be collected for a grade et school year! This packet has bee desiged to help you review various mathematical
Polynomials with Rational Roots that Differ by a Non-zero Constant. Generalities
Polyomials with Ratioal Roots that Differ by a No-zero Costat Philip Gibbs The problem of fidig two polyomials P(x) ad Q(x) of a give degree i a sigle variable x that have all ratioal roots ad differ by
Numerical Derivatives by Symbolic Tools in MATLAB
Numerical Derivatives by Symbolic ools i MALAB Mig-Gog Lee * ad Rei-Wei Sog ad Hsuag-Ci Cag mglee@cu.edu.tw * Departmet o Applied Matematics Cug Hua Uiversity Hsicu, aiwa Abstract: Numerical approaces
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,
On maximally nonlinear and extremal balanced Boolean functions
O imally oliear ad extremal balaced Boolea uctios Michel Mitto DCSSI/SDS/Crypto.Lab. 18, rue du docteur Zameho 9131 Issy-les-Moulieaux cedex, Frace e-mail: michelmitto@compuserve.com Abstract. We prove
Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
Worksheet: Taylor Series, Lagrange Error Bound ilearnmath.net
Taylor s Thorm & Lagrag Error Bouds Actual Error This is th ral amout o rror, ot th rror boud (worst cas scario). It is th dirc btw th actual () ad th polyomial. Stps:. Plug -valu ito () to gt a valu.
International 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
9.3 Taylor s Theorem: Error Analysis for Series. Tacoma Narrows Bridge: November 7, 1940
9. Taylor s Theorem: Error Aalysis or Series Tacoma Narrows Bridge: November 7, 940 Last time i BC So the Taylor Series or l x cetered at x is give by ) l x ( ) ) + ) ) + ) ) 4 Use the irst two terms o
Quiz. Use either the RATIO or ROOT TEST to determine whether the series is convergent or not.
Quiz. Use either the RATIO or ROOT TEST to determie whether the series is coverget or ot. e .6 POWER SERIES Defiitio. A power series i about is a series of the form c 0 c a c a... c a... a 0 c a where
TIME-PERIODIC SOLUTIONS OF A PROBLEM OF PHASE TRANSITIONS
Far East Joural o Mathematical Scieces (FJMS) 6 Pushpa Publishig House, Allahabad, Idia Published Olie: Jue 6 http://dx.doi.org/.7654/ms99947 Volume 99, umber, 6, Pages 947-953 ISS: 97-87 Proceedigs o
μ are complex parameters. Other
A New Numerical Itegrator for the Solutio of Iitial Value Problems i Ordiary Differetial Equatios. J. Suday * ad M.R. Odekule Departmet of Mathematical Scieces, Adamawa State Uiversity, Mubi, Nigeria.
Mth 95 Notes Module 1 Spring Section 4.1- Solving Systems of Linear Equations in Two Variables by Graphing, Substitution, and Elimination
Mth 9 Notes Module Sprig 4 Sectio 4.- Solvig Sstems of Liear Equatios i Two Variales Graphig, Sustitutio, ad Elimiatio A Solutio to a Sstem of Two (or more) Liear Equatios is the commo poit(s) of itersectio
Lecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
Sequences, 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
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
Chapter 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,
Polynomials. Computer Programming for Engineers (2014 Spring)
Computer Programmig for Egieers (014 Sprig) Polyomials Hyougshick Kim Departmet of Computer Sciece ad Egieerig College of Iformatio ad Commuicatio Egieerig Sugkyukwa Uiversity Polyomials A th order polyomial
Recursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
CSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
Accuracy. Computational Fluid Dynamics. Computational Fluid Dynamics. Computational Fluid Dynamics
http://www.d.edu/~gtryggva/cfd-course/ Computatioal Fluid Dyamics Lecture Jauary 3, 7 Grétar Tryggvaso It is clear that although the umerical solutio is qualitatively similar to the aalytical solutio,
Notes 19 Bessel Functions
ECE 638 Fall 17 David R. Jackso Notes 19 Bessel Fuctios Notes are from D. R. Wilto, Dept. of ECE 1 Cylidrical Wave Fuctios Helmholtz equatio: ψ + k ψ = I cylidrical coordiates: ψ 1 ψ 1 ψ ψ ρ ρ ρ ρ φ z
CHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
We are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
Newton s Method. y f x 1 x x 1 f x 1. Letting y 0 and solving for x produces. x x 1 f x 1. x 1. x 2 x 1 f x 1. f x 1. x 3 x 2 f x 2 f x 2.
460_008.qd //04 :7 PM Page 9 SECTION.8 Newto s Method 9 (a) a a Sectio.8 (, ( )) (, ( )) Taget lie c Taget lie c b (b) The -itercept o the taget lie approimates the zero o. Figure.60 b Newto s Method Approimate
CALCULATING FIBONACCI VECTORS
THE GENERALIZED BINET FORMULA FOR CALCULATING FIBONACCI VECTORS Stuart D Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithacaedu ad Dai Novak Departmet
Solution 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
An Improved Self-Starting Implicit Hybrid Method
IOSR Joural o Matematics (IOSR-JM e-issn: 78-78, p-issn:9-76x. Volume 0, Issue Ver. II (Mar-Apr. 04, PP 8-6 www.iosrourals.org A Improved Sel-Startig Implicit Hbrid Metod E. O. Adeea Departmet o Matematics/Statistics,
TECHNIQUES OF INTEGRATION
7 TECHNIQUES OF INTEGRATION Simpso s Rule estimates itegrals b approimatig graphs with parabolas. Because of the Fudametal Theorem of Calculus, we ca itegrate a fuctio if we kow a atiderivative, that is,
SHAFTS: STATICALLY INDETERMINATE SHAFTS
LECTURE SHAFTS: STATICALLY INDETERMINATE SHAFTS Third Editio A.. Clark School of Egieerig Departmet of Civil ad Eviromet Egieerig 7 Chapter 3.6 by Dr. Ibrahim A. Assakkaf SPRING 2003 ENES 220 Mechaics
STAT 516 Answers Homework 6 April 2, 2008 Solutions by Mark Daniel Ward PROBLEMS
STAT 56 Aswers Homework 6 April 2, 28 Solutios by Mark Daiel Ward PROBLEMS Chapter 6 Problems 2a. The mass p(, correspods to either o the irst two balls beig white, so p(, 8 7 4/39. The mass p(, correspods
Eigenvalues and Eigenvectors
5 Eigevalues ad Eigevectors 5.3 DIAGONALIZATION DIAGONALIZATION Example 1: Let. Fid a formula for A k, give that P 1 1 = 1 2 ad, where Solutio: The stadard formula for the iverse of a 2 2 matrix yields
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
CHAPTER 5: FOURIER SERIES PROPERTIES OF EVEN & ODD FUNCTION PLOT PERIODIC GRAPH
CHAPTER : FOURIER SERIES PROPERTIES OF EVEN & ODD FUNCTION POT PERIODIC GRAPH PROPERTIES OF EVEN AND ODD FUNCTION Fuctio is said to be a eve uctio i: Fuctio is said to be a odd uctio i: Fuctio is said
I v r e s n a d di t r th o e r m e s o f r monoton e app o r i x m t a ion b a t 1. I r n o n a
Iverse ad direct orems or mootoe aroxima tio ukeia Abdulla Al- B ermai B abylo Uiversity ciece College For Wome C omuter ciece Deartmet A bstract We rove that i s icreasig uctio o i or e ac h re a icreasig
CHAPTER 6b. NUMERICAL INTERPOLATION
CHAPTER 6. NUMERICAL INTERPOLATION A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering y Dr. Irhim A. Asskk Spring ENCE - Computtion s in Civil Engineering II Deprtment o Civil
(8) 1f = f. can be viewed as a real vector space where addition is defined by ( a1+ bi
Geeral Liear Spaes (Vetor Spaes) ad Solutios o ODEs Deiitio: A vetor spae V is a set, with additio ad salig o elemet deied or all elemets o the set, that is losed uder additio ad salig, otais a zero elemet
Section 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
Root Finding COS 323
Root Fidig COS 323 Remider Sig up for Piazza Assigmet 0 is posted, due Tue 9/25 Last time.. Floatig poit umbers ad precisio Machie epsilo Sources of error Sesitivity ad coditioig Stability ad accuracy
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
CHAPTER 3a. INTRODUCTION TO NUMERICAL METHODS
CHAPTER 3a. INTRODUCTION TO NUMERICAL METHODS A. J. Clark School of Engineering Department of Civil and Environmental Engineering by Dr. Ibrahim A. Assakkaf Spring 1 ENCE 3 - Computation in Civil Engineering
d y f f dy Numerical Solution of Ordinary Differential Equations Consider the 1 st order ordinary differential equation (ODE) . dx
umerical Solutio o Ordiar Dieretial Equatios Cosider te st order ordiar dieretial equatio ODE d. d Te iitial coditio ca be tae as. Te we could use a Talor series about ad obtai te complete solutio or......!!!
Joint Probability Distributions and Random Samples. Jointly Distributed Random Variables. Chapter { }
UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
Assignment 5: Solutions
McGill Uiversity Departmet of Mathematics ad Statistics MATH 54 Aalysis, Fall 05 Assigmet 5: Solutios. Let y be a ubouded sequece of positive umbers satisfyig y + > y for all N. Let x be aother sequece
Fuzzy n-normed Space and Fuzzy n-inner Product Space
Global Joural o Pure ad Applied Matheatics. ISSN 0973-768 Volue 3, Nuber 9 (07), pp. 4795-48 Research Idia Publicatios http://www.ripublicatio.co Fuzzy -Nored Space ad Fuzzy -Ier Product Space Mashadi
In algebra one spends much time finding common denominators and thus simplifying rational expressions. For example:
74 The Method of Partial Fractios I algebra oe speds much time fidig commo deomiators ad thus simplifyig ratioal epressios For eample: + + + 6 5 + = + = = + + + + + ( )( ) 5 It may the seem odd to be watig
AP 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)
PARTIAL DIFFERENTIAL EQUATIONS SEPARATION OF VARIABLES
Diola Bagayoko (0 PARTAL DFFERENTAL EQUATONS SEPARATON OF ARABLES. troductio As discussed i previous lectures, partial differetial equatios arise whe the depedet variale, i.e., the fuctio, varies with
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
Abstract Vector Spaces. Abstract Vector Spaces
Astract Vector Spaces The process of astractio is critical i egieerig! Physical Device Data Storage Vector Space MRI machie Optical receiver 0 0 1 0 1 0 0 1 Icreasig astractio 6.1 Astract Vector Spaces