CMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature


 Ambrose Carson
 4 years ago
 Views:
Transcription
1 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 tht polynomil p interpoltes f t the point x [, b] if f x = p x. In the context of tody s lecture, we im to use interpoltion s wy to construct good polynomil pproximtions to f. The next lecture we will put tody s results into the broder context of the course: we will show how the pproximte solutions constructed by the finite element method cn be relted to interpolting polynomils, nd so the ccurcy of interpolting polynomils will led to error bounds for the finite element method. We hve three gols tody: 1 construct interpolting polynomils p n ; 2 bound the error fx p n x; 3 integrte interpolting polynomils to pproximte the integrl of f 1. Constructing Interpolting Polynomils. We seek to solve the following problem: Polynomil interpoltion problem. Given function f C[, b] nd points x 0,..., x n [, b], construct polynomil p n of degree not exceeding n such tht p n x j = fx j, j = 0,..., n. In ny numericl nlysis course, one lerns tht unique solution p n to this problem lwys exists, nd be constructed in vrious wys. Here we shll just detil one elegnt wy to develop the interpolnt, clled the Lgrnge form. The ide behind the Lgrnge form is simple. Consider the functions L k x = i=0 i k x x i x k x i. Note tht ech L k is degreen polynomil. For ech vlue of k, the product contins n terms of the form x x j /x k x j. Ech of these terms is degree1 polynomil. The product of n degree1 polynomils is degreen polynomil. Moreover, these polynomils hve very specil property: by construction, L k tkes the vlue 1 t x k nd hs roots t ech of the points x j, j k: L k x j = { 1, j = k 0, j k. These n+1 polynomils L 0,..., L n form bsis for the n+1 dimensionl vector spce of polynomils hving degree n or less. 1
2 These Lgrnge bsis functions mke it trivil to construct the solution p n to the polynomil interpoltion function: p n x = fx k L k x. k=0 Since p n is the sum of degreen polynomils, it too is degreen polynomil. The property tht L k x i = 0 for i k ensures tht t the interpoltion point x j, p n x j = fx k L k x j = fx j L j x j = fx j. k=0 Thus the polynomil p n psses through f t the designted points. But how ccurtely does p n pproximte f t the other points in the intervl [, b] where we hve not specified the interpoltion condition? 2. Interpoltion Error Anlysis. We now seek to chrcterize the mximum error mx fx p nx. x [,b] The chrcteriztion of this error is one of the most fundmentl results in numericl nlysis. Theorem Interpoltion Error Bound. Suppose f C n+1 [, b] nd let p n P n denote the polynomil tht interpoltes f t the points x 0,..., x n [, b] for j = 0,..., n. The for every x [, b] there exists ξ [, b] such tht fx p n x = f n+1 ξ n + 1! x x j. This result yields bound for the worst error over the intervl [, b]: mx fx p nx mx x [,b] ξ [,b] f n+1 ξ mx n + 1! x [,b] x x j. 1 Proof. Consider some rbitrry point x [, b]. We seek descriptive expression for the error f x p n x. If x = x j for some j {0,..., n}, then f x p n x = 0 nd there is nothing to prove. Thus, suppose for the rest of the proof tht x is not one of the interpoltion points. To describe f x p n x, we shll build the polynomil of degree n + 1 tht interpoltes f t x 0,..., x n, nd lso x. Of course, this polynomil will give zero error t x, since it interpoltes f there. From this polynomil we cn extrct formul for f x p n x by mesuring how much the degree n + 1 interpolnt improves upon the degreen interpolnt p n t x. Since we wish to understnd the reltionship of the degree n + 1 interpolnt to p n, we shll write tht degree n + 1 interpolnt in mnner tht explicitly incorportes p n. Given this setting, use of the Newton form of the interpolnt is nturl; i.e., we write the degree n + 1 polynomil s p n x + γ x x j 2
3 for some constnt γ chosen to mke the interpolnt exct t x. For convenience, we write wx x x j nd then denote the error of this degree n + 1 interpolnt by φx fx p n x + γwx. To mke the polynomil p n x + γwx interpolte f t x, we shll pick γ such tht φ x = 0. The fct tht x {x j } n ensures tht w x 0, nd so we cn force φ x = 0 by setting γ = f x p n x. w x Furthermore, since fx j = p n x j nd wx j = 0 t ll the n + 1 interpoltion points x 0,..., x n, we lso hve φx j = fx j p n x j γwx j = 0. Thus, φ is function with t lest n + 2 zeros in the intervl [, b]. Rolle s Theorem 1 tells us tht between every two consecutive zeros of φ, there is some zero of φ. Since φ hs t lest n + 2 zeros in [, b], φ hs t lest n + 1 zeros in this sme intervl. We cn repet this rgument with φ to see tht φ must hve t lest n zeros in [, b]. Continuing in this mnner with higher derivtives, we eventully conclude tht φ n+1 must hve t lest one zero in [, b]; we denote this zero s ξ, so tht φ n+1 ξ = 0. We now wnt more concrete expression for φ n+1. Note tht φ n+1 x = f n+1 x p n+1 n x γw n+1 x. Since p n is polynomil of degree n or less, p n+1 n 0. Now observe tht w is polynomil of degree n + 1. We could write out ll the coefficients of this polynomil explicitly, but tht is bit tedious, nd we do not need ll of them. Simply observe tht we cn write wx = x n+1 + qx, for some q P n, nd this polynomil q will vnish when we tke n + 1 derivtives: d n+1 w n+1 x = xn+1 dxn+1 + q n+1 x = n + 1! + 0. Assembling the pieces, φ n+1 x = f n+1 x γ n + 1!. Since φ n+1 ξ = 0, we conclude tht γ = f n+1 ξ n + 1!. Substituting this expression into 0 = φ x = f x p n x λw x, we obtin f x p n x = f n+1 ξ n + 1! x x j. 1 Recll the Men Vlue Theorem from clculus: Given d > c, suppose f C[c, d] is differentible on c, d. Then there exists some η c, d such tht fd fc/d c = f η. Rolle s Theorem is specil cse: If fd = fc, then there is some point η c, d such tht f η = 0. 3
4 This error bound hs strong prllels to the reminder term in Tylor s formul. Recll tht for sufficiently smooth h, the Tylor expnsion of f bout the point x 0 is given by fx = fx 0 + x x 0 f x x x 0 k k! f k x 0 + f k+1 ξ k + 1! x x 0 k. Ignoring the reminder term t the end, note tht the Tylor expnsion gives polynomil model of f, but one bsed on locl informtion bout f nd its derivtives, s opposed to the polynomil interpolnt, which is bsed on globl informtion, but only bout f, not its derivtives. Rerrnging this expression, we hve fx fx 0 + x x 0 f x x x 0 k f k x 0 = f k+1 ξ k! k + 1! x x 0 k, perfect nlogue of the interpoltion error formul we hve just proved. 3. Interpoltory Qudrture Formuls. The finite element method requires computtions like f, φ k = 1 0 fxφ k x dx to construct the lod vector. It my be inconvenient or even impossible for some f to compute this inner product. For such cses we wish to pproximte the integrl. We shll consider the generic problem of pproximting If = fx dx. Polynomil interpoltion provides simple wy to pproximte the integrl: Construct the polynomil interpolnt p n to f t designted points; Approximte fx dx by p nx dx. If we construct p n using the Lgrnge form described bove, this procedure becomes very simple: Construct the interpolting polynomil p n x = fx j L j x; Integrte the interpolting polynomil to obtin I n f, pproximting the exct integrl If: I n f = p n x dx = fx j L j x dx = fx j L j x dx. 4
5 Notice tht the integrls tht remin depend on the Lgrnge bsis functions L j but not on f. We will cll these integrls the weights of the qudrture rule: w j = L j x dx. Then the qudrture rule tkes the simple form I n f = w j fx j. The points x 0,..., x n re clled the nodes of the qudrture rule. When you choose evenly spced points over [, b], you recover fmilir rules tht you hve lredy encountered in clculus: n = 0 x 0 = + b/2, L 0 x = 1 gives fx dx b f b; n = 1 x 0 =, x 1 = b gives the trpezoid rule: fx dx b 2 f + fb ; n = 2 x 0 =, x 1 = + b/2, x 2 = b gives Simpson s rule: fx dx b 6 f + 4f b + fb. The first rule pproximtes f with n interpolting constnt; the trpezoid rule pproximtes f with n interpolting liner polynomil; Simpson s rule pproximtes f with n interpolting qudrtic. How does one quntify the error If I n f? Simply integrte the error formul for polynomil interpoltion! One must then clculte: If I n f = = fx p n x dx f n+1 ξx n + 1! x x j dx. The error nlysis for the trpezoid rule where x 0 = nd x 1 = b follows from ppliction of the men vlue theorem for integrls: fx dx p 1 x dx = = 1 2 f η 1 2 f ξxx x b dx x x b dx = 1 2 f η b b2 1 6 b3 = 1 12 f ηb 3 5
6 for some η [, b]. As we expect, if fx is liner polynomil, then f x = 0 for ll x, nd hence the trpezoid rule will be exct. The nlysis for Simpson s rule is bit more complicted. One cn ctully show something stronger thn wht you might expect from integrting the polynomil interpoltion error: fx dx p 2 x dx = 1 b f 4 η for some η [, b]. Notice tht this bound involves f 4, rther thn the expected f 3 : Simpson s rule will be exct for cubic polynomils, not just qudrtics! If you wnt greter ccurcy thn these bounds suggest, you could simply increse the degree n, nd there re some settings in which this mkes gret sense: but one must be creful bout how to select the nodes x 0,..., x n, nd uniformly spced points re not the best choice. Composite rules. As n lterntive to integrting highdegree polynomil, one cn pursue simpler pproch tht is often very effective, especilly for problems tht re not prticulrly smooth e.g., our ht functions: Brek the intervl [, b] into subintervls, nd pply the trpezoid rule or Simpson s rule on ech subintervl. Applying the trpezoid rule gives fx dx = xj x j 1 fx dx x j x j 1 fx j 1 + fx j. 2 The stndrd implementtion ssumes tht f is evluted t uniformly spced points between nd b, x j = + jh for j = 0,..., n nd h = b /n, giving the following fmous formultion: fx dx h 2 n 1 f + 2 f + jh + fb. Of course, one cn redily djust this rule to cope with irregulrly spced points. The error in the composite trpezoid rule cn be derived by summing up the error in ech ppliction of the trpezoid rule: fx dx h 2 n 1 f + 2 f + jh + fb = 1 12 f η j x j x j 1 3 = h3 12 f η j for η j [x j 1, x j ]. We cn simplify these f terms by noting tht 1 n n f η j is the verge of n vlues of f evluted t points in the intervl [, b]. Nturlly, this verge cnnot exceed the mximum or minimum vlue tht f ssumes on [, b], so there exist points ξ 1, ξ 2 [, b] such tht f ξ 1 1 f η j f ξ 2. n Thus the intermedite vlue theorem gurntees the existence of some η [, b] such tht f η = 1 f η j. n 6
7 The composite trpezoid error bound thus simplifies to fx dx h 2 n 1 f + 2 f + jh + fb = h2 12 b f η. Similr nlysis cn be performed to derive the composite Simpson s rule. We now must ensure tht n is even, since ech intervl on which we pply the stndrd Simpson s rule hs width 2h. Simple lgebr leds to the formul fx dx h n/2 f n/2 1 f + 2j 1h + 2 f + 2jh + fb. Derivtion of the error formul for the composite Simpson s rule follows the sme strtegy s the nlysis of the composite trpezoid rule. One obtins fx dx h n/2 f for some η [, b]. n/2 1 f + 2j 1h + 2 f + 2jh + fb = h4 180 b f 4 η 7
Numerical 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 informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationLecture 23: Interpolatory Quadrature
Lecture 3: Interpoltory Qudrture. Qudrture. The computtion of continuous lest squres pproximtions to f C[, b] required evlutions of the inner product f, φ j = fxφ jx dx, where φ j is polynomil bsis function
More informationNUMERICAL INTEGRATION
NUMERICAL INTEGRATION How do we evlute I = f (x) dx By the fundmentl theorem of clculus, if F (x) is n ntiderivtive of f (x), then I = f (x) dx = F (x) b = F (b) F () However, in prctice most integrls
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 informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
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 relvlues nd smooth The pproximtion of n integrl by numericl
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 informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
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 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 informationTheoretical foundations of Gaussian quadrature
Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of
More informationLecture 4: Piecewise Cubic Interpolation
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics  A Peirce UBC Lecture 4: Piecewise Cubic Interpoltion Compiled 5 September In this lecture we consider piecewise cubic interpoltion
More informationLecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at UrbanaChampaign. March 20, 2014
Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t UrbnChmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method
More informationIII. Lecture on Numerical Integration. File faclib/dattab/lecturenotes/numericalinter03.tex /by EC, 3/14/2008 at 15:11, version 9
III Lecture on Numericl Integrtion File fclib/dttb/lecturenotes/numericalinter03.tex /by EC, 3/14/008 t 15:11, version 9 1 Sttement of the Numericl Integrtion Problem In this lecture we consider the
More informationAdvanced Computational Fluid Dynamics AA215A Lecture 3 Polynomial Interpolation: Numerical Differentiation and Integration.
Advnced Computtionl Fluid Dynmics AA215A Lecture 3 Polynomil Interpoltion: Numericl Differentition nd Integrtion Antony Jmeson Winter Qurter, 2016, Stnford, CA Lst revised on Jnury 7, 2016 Contents 3 Polynomil
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 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 informationNumerical Integration
Chpter 5 Numericl Integrtion Numericl integrtion is the study of how the numericl vlue of n integrl cn be found. Methods of function pproximtion discussed in Chpter??, i.e., function pproximtion vi the
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationNumerical Integration
Chpter 1 Numericl Integrtion Numericl differentition methods compute pproximtions to the derivtive of function from known vlues of the function. Numericl integrtion uses the sme informtion to compute numericl
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
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 informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
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 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 informationNumerical Analysis. Doron Levy. Department of Mathematics Stanford University
Numericl Anlysis Doron Levy Deprtment of Mthemtics Stnford University December 1, 2005 D. Levy Prefce i D. Levy CONTENTS Contents Prefce i 1 Introduction 1 2 Interpoltion 2 2.1 Wht is Interpoltion?............................
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Tody we provide the connection
More informationLecture 20: Numerical Integration III
cs4: introduction to numericl nlysis /8/0 Lecture 0: Numericl Integrtion III Instructor: Professor Amos Ron Scribes: Mrk Cowlishw, Yunpeng Li, Nthnel Fillmore For the lst few lectures we hve discussed
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 informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationIntroduction to Numerical Analysis
Introduction to Numericl Anlysis Doron Levy Deprtment of Mthemtics nd Center for Scientific Computtion nd Mthemticl Modeling (CSCAMM) University of Mrylnd June 14, 2012 D. Levy CONTENTS Contents 1 Introduction
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More information1. GaussJacobi 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. GussJcobi 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 informationChapter 3 Polynomials
Dr M DRAIEF As described in the introduction of Chpter 1, pplictions of solving liner equtions rise in number of different settings In prticulr, we will in this chpter focus on the problem of modelling
More informationNumerical Methods I Orthogonal Polynomials
Numericl Methods I Orthogonl Polynomils Aleksndr Donev Cournt Institute, NYU 1 donev@cournt.nyu.edu 1 MATHGA 2011.003 / CSCIGA 2945.003, Fll 2014 Nov 6th, 2014 A. Donev (Cournt Institute) Lecture IX
More informationChapter 5. Numerical Integration
Chpter 5. Numericl Integrtion These re just summries of the lecture notes, nd few detils re included. Most of wht we include here is to be found in more detil in Anton. 5. Remrk. There re two topics with
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationLecture Note 4: Numerical differentiation and integration. Xiaoqun Zhang Shanghai Jiao Tong University
Lecture Note 4: Numericl differentition nd integrtion Xioqun Zng Sngi Jio Tong University Lst updted: November, 0 Numericl Anlysis. Numericl differentition.. Introduction Find n pproximtion of f (x 0 ),
More informationNumerical Analysis. 10th ed. R L Burden, J D Faires, and A M Burden
Numericl Anlysis 10th ed R L Burden, J D Fires, nd A M Burden Bemer Presenttion Slides Prepred by Dr. Annette M. Burden Youngstown Stte University July 9, 2015 Chpter 4.1: Numericl Differentition 1 ThreePoint
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationConstruction of Gauss Quadrature Rules
Jim Lmbers MAT 772 Fll Semester 201011 Lecture 15 Notes These notes correspond to Sections 10.2 nd 10.3 in the text. Construction of Guss Qudrture Rules Previously, we lerned tht NewtonCotes qudrture
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 informationChapter 2. Numerical Integration also called quadrature. 2.2 Trapezoidal Rule. 2.1 A basic principle Extending the Trapezoidal Rule DRAWINGS
S Cpter Numericl Integrtion lso clled qudrture Te gol of numericl integrtion is to pproximte numericlly. f(x)dx Tis is useful for difficult integrls like sin(x) ; sin(x ); x + x 4 Or worse still for multipledimensionl
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
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 informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH34032: Green s Functions, Integrl Equtions nd the Clculus of Vritions 1 Section 1 Function spces nd opertors Here we gives some brief detils nd definitions, prticulrly relting to opertors. For further
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 informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
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 informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationLECTURE 19. Numerical Integration. Z b. is generally thought of as representing the area under the graph of fèxè between the points x = a and
LECTURE 9 Numericl Integrtion Recll from Clculus I tht denite integrl is generlly thought of s representing the re under the grph of fèxè between the points x = nd x = b, even though this is ctully only
More informationBest Approximation in the 2norm
Jim Lmbers MAT 77 Fll Semester 111 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 informationNumerical Integration. 1 Introduction. 2 Midpoint Rule, Trapezoid Rule, Simpson Rule. AMSC/CMSC 460/466 T. von Petersdorff 1
AMSC/CMSC 46/466 T. von Petersdorff 1 umericl Integrtion 1 Introduction We wnt to pproximte the integrl I := f xdx where we re given, b nd the function f s subroutine. We evlute f t points x 1,...,x n
More information1 Error Analysis of Simple Rules for Numerical Integration
cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion
More informationCOT4501 Spring Homework VII
COT451 Spring 1 Homework VII The ssignment is due in clss on Thursdy, April 19, 1. There re five regulr problems nd one computer problem (using MATLAB). For written problems, you need to show your work
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO  Ares Under Functions............................................ 3.2 VIDEO  Applictions
More informationRiemann Integrals and the Fundamental Theorem of Calculus
Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More informationMatrices, Moments and Quadrature, cont d
Jim Lmbers MAT 285 Summer Session 201516 Lecture 2 Notes Mtrices, Moments nd Qudrture, cont d We hve described how Jcobi mtrices cn be used to compute nodes nd weights for Gussin qudrture rules for generl
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 informationOrthogonal Polynomials and LeastSquares Approximations to Functions
Chpter Orthogonl Polynomils nd LestSqures Approximtions to Functions **4/5/3 ET. Discrete LestSqures Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny
More informationAPPROXIMATE INTEGRATION
APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose ntiderivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be
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 informationNumerical Integration
Numericl Integrtion Wouter J. Den Hn London School of Economics c 2011 by Wouter J. Den Hn June 3, 2011 Qudrture techniques I = f (x)dx n n w i f (x i ) = w i f i i=1 i=1 Nodes: x i Weights: w i Qudrture
More informationAdvanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015
Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n
More informationDiscrete Leastsquares Approximations
Discrete Lestsqures 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 informationChapter 1. Basic Concepts
Socrtes Dilecticl Process: The Þrst step is the seprtion of subject into its elements. After this, by deþning nd discovering more bout its prts, one better comprehends the entire subject Socrtes (469399)
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationTHE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCEUNIQUENESS THEOREM FOR FIRSTORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrdlindeloftheorem/ This document is proof of the existenceuniqueness theorem
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More informationNumerical quadrature based on interpolating functions: A MATLAB implementation
SEMINAR REPORT Numericl qudrture bsed on interpolting functions: A MATLAB implementtion by Venkt Ayylsomyjul A seminr report submitted in prtil fulfillment for the degree of Mster of Science (M.Sc) in
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationLecture 14 Numerical integration: advanced topics
Lecture 14 Numericl integrtion: dvnced topics Weinn E 1,2 nd Tiejun Li 2 1 Deprtment of Mthemtics, Princeton University, weinn@princeton.edu 2 School of Mthemticl Sciences, Peking University, tieli@pku.edu.cn
More informationPart IB Numerical Analysis
Prt IB Numericl Anlysis Theorems with proof Bsed on lectures by G. Moore Notes tken by Dexter Chu Lent 2016 These notes re not endorsed by the lecturers, nd I hve modified them (often significntly) fter
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationIntegrals  Motivation
Integrls  Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is nonliner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but
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 informationDOING PHYSICS WITH MATLAB MATHEMATICAL ROUTINES
DOIG PHYSICS WITH MATLAB MATHEMATICAL ROUTIES COMPUTATIO OF OEDIMESIOAL ITEGRALS In Cooper School of Physics, University of Sydney in.cooper@sydney.edu.u DOWLOAD DIRECTORY FOR MATLAB SCRIPTS mth_integrtion_1d.m
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationThe Riemann Integral
Deprtment of Mthemtics King Sud University 20172018 Tble of contents 1 Antiderivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Antiderivtive Function Definition Let f : I R be function
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationHow can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
More informationRiemann is the Mann! (But Lebesgue may besgue to differ.)
Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixedpoint itertion to converge when solving the eqution
More informationCalculus III Review Sheet
Clculus III Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More information1 Part II: Numerical Integration
Mth 4 Lb 1 Prt II: Numericl Integrtion This section includes severl techniques for getting pproimte numericl vlues for definite integrls without using ntiderivtives. Mthemticll, ect nswers re preferble
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationLecture 1: Introduction to integration theory and bounded variation
Lecture 1: Introduction to integrtion theory nd bounded vrition Wht is this course bout? Integrtion theory. The first question you might hve is why there is nything you need to lern bout integrtion. You
More information