Lecture 14: Quadrature
|
|
- Easter Mason
- 6 years ago
- Views:
Transcription
1 Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl method is commonly referred to s qudrture In Clculus one discusses mny techniques for determining n ntiderivtive of fx) The integrl is then computed by evluting the ntiderivtive t the endpoints of the intervl However, mny integrnds of interest in science nd engineering do not hve known ntiderivtive Moreover, in some pplictions the integrnd is only known t few points in the intervl We would like to be ble to determine ccurte pproximtions of integrls lso in these situtions We will pproximte integrls 1) by sums fx j )w j ) These sums re referred to s qudrture rules The x j re the nodes nd the w j the weights of the qudrture rule We re interested in determining weights so tht the qudrture rules gives ccurte pproximtions of the integrl 1 for lrge clsses of integrnds fx) The nodes often cnnot be chosen freely, becuse the integrnd my only be known t certin points For instnce, the function fx) might not be explicitly known, only mesured vlues fx j ) t the nodes x j my be vilble We remrk tht you lredy encountered the pproximtion of integrls by sums in Clculus There, however, one ws less concerned with how smll the error E N f) = fx)dx fx j )w j 3) is for smll vlues of N; it ws sufficient tht the error E N f) converged to zero s N incresed to infinity In this lecture, we would like the error E N f) to be smll lredy for smll to modest number of terms N in the qudrture rule ) Method of undetermined coefficients Let the nodes x j, 1 j N, be given Throughout this lecture the nodes will be ordered so tht x 1 < x < < x N b 4) We would like to determine the weights w j so tht qudrture rule is exct for polynomils of s high degree s possible, ie, we would like the weights be such tht px)dx = for ll polynomils px) of s high degree s possible px j )w j 5) 1
2 The powers {1, x, x, x 3, } form bsis for polynomils Our objective therefore cn be expressed s follows: We would like the equlity 5) to hold for s mny powers px) = x j, j =, 1,,, s possible This requirement gives rise to the liner system of equtions for the weights, px) = 1 : w 1 + w + + w N = px) = x : x 1 w 1 + x w + + x N w N = px) = x : x 1w 1 + x w + + x N w N = px) = x N 1 : x N 1 1 w 1 + x N 1 w + + w N x N 1 N This system conveniently cn be expressed in the form x 1 x x N x N 1 1 x N 1 x N 1 N w 1 w w N = dx = b, xdx = 1 b ), x 1 dx = 3 b3 3 ), = xn 1 dx = 1 N bn N ) b 1 b ) 1 N bn N ) 6) 7) We recognize the mtrix s the trnspose of Vndermonde mtrix Vndermonde mtrices were encountered in the fll semester in connection with polynomil interpoltion, when we showed tht Vndermonde mtrices determined by distinct nodes re nonsingulr Determinnts re invrint under trnsposition This follows from the fct tht determinnts cn be expnded either by rows or by columns We conclude tht the mtrix in 7) is nonsingulr when the nodes x j re distinct Hence the bove liner system of equtions hs unique solution Thus, given n distinct nodes, we cn determine n unique weights w j, such tht the qudrture rule ) integrtes ll polynomils of degree strictly less thn n exctly Exmple 1 Consider the midpoint qudrture rule fx)dx fx 1 )w 1, x 1 = 1 + b) We only hve one weight to determine The liner system of equtions 6) reduces to px) = 1 : w 1 = dx = b This determines the weight w 1 Exmple Apply the midpoint rule to integrte the function fx) = exp 1 x/) ) 8) on the intervl [, 1] The midpoint rule pproximtes the integrnd by the constnt function f1/) = exp 7/8) on the intervl [, b] nd integrtes the ltter function exctly The blue grph of Figure 1 shows fx) nd the vlue of the integrl is the re below this grph The dshed red line displyes the constnt function tht is integrted exctly by the midpoint rule The vlue determined by the midpoint rule is the re below the dshed red line Sometimes N-node qudrture rules determined by solving the liner system of equtions 7) lso integrte powers x k for k N exctly The following exmple illustrtes this
3 Figure 1: Integrnd of Exmple nd pproximtion used by the midpoint rule Exmple 3 Consider the midpoint qudrture rule of Exmple 1 Appliction of this rule to fx) = x yields x 1 w 1 = 1 + b)b ) = 1 b ) The right-hnd side equls xdx, ie, the midpoint rule integrtes not only constnts but lso liner polynomils exctly Exercise 1 Determine the weights of the -node qudrture rule fx)dx fx 1 )w 1 + fx )w, x 1 =, x = b This rule is known s the trpezoidl rule By construction, it integrtes liner polynomils exctly Does it integrte polynomils of higher degree exctly? Justify your nswer Exercise Determine the weights of the 3-node qudrture rule 1 fx)dx fx 1 )w 1 + fx )w + fx 3 )w 3, x 1 =, x = 1/, x 3 = 1 This rule is known s Simpson s rule It integrtes qudrtic polynomils exctly by construction Does it integrte polynomils of higher degree exctly? Justify your nswer Wht is the nlogous qudrture rule for the intervl [1, ]? Hint: Do not recompute the qudrture rule, just mke chnge of vribles Integrting Lgrnge polynomils When discussing polynomil interpoltion lst fll, we considered different polynomil bses, such s monomils, Lgrnge polynomils, nd Newton polynomils We cn lso in the present context use bses different 3
4 from the monomil one For instnce, the nodes x 1, x,,x N determine the Lgrnge polynomils l k x) = N j k x x j x k x j, k = 1,,, N, 9) which form bsis for ll polynomils of degree t most N 1 We therefore my require tht n N-node qudrture rule integrtes the Lgrnge polynomils 9) exctly Substituting px) = l k x) into 5) yields l k x)dx = l k x j )w j = l k x k )w k = w k, k = 1,,,N, where the simplifictions of the right-hnd side follow from the fct tht { 1, k = j, l k x j ) =, k j Thus, the weights cn be obtined by integrting the Lgrnge polynomils However, the evlution of these integrls is tedious unless N is smll Composite qudrture rules nd Tylor expnsion Assume tht the midpoint rule of Exmple does not yield sufficiently ccurte pproximtion of the integrl We my then subdivide the intervl [, 1] into subintervls nd pply the midpoint rule on ech subintervl This defines the composite midpoint rule Figure : Integrnd of Exmple 4 nd pproximtion used by the composite midpoint rule obtined by dividing the intervl [, 1] into two subintervls of equl length 4
5 Exmple 4 Divide the intervl [, 1] into the subintervls [, 1/] nd [1/, 1] nd pply the midpoint rule on ech subintervl to the integrnd fx) defined by 8) This yields the composite midpoint rule 1 fx)dx fx 1 )w 1 + fx )w with x 1 = 1/4, x = 3/4, nd w 1 = w = 1/ The integrl is pproximted by the re below the dshed curve of Figure Compring the grphs of Figures 1 nd suggests tht subdivision of the intervl should increse the ccurcy of the computed pproximtion While the computtion of the weights by solving the liner system of equtions 7) is esily done in MATLAB or Octve, this pproch does not shed ny light on how the error behves when we increse the number of nodes Further insight on the behvior of qudrture rules cn be gined by expnding the integrnd into Tylor series Consider the pproximtion of the integrl fx)dx by the midpoint rule nd use the Tylor expnsion of fx) t x = h/ Here x h/ should be thought of s firly smll, nd we ssume tht fx) is continuously differentible s mny times s required Then ) ) h h fx) = f + f x h ) + f ) h! x h ) + f ) h x h 3! Integrting the left-hnd nd right-hnd sides from to h yields ) h ) h fx)dx = f dx + f x h + f ) h x h ) 3 dx + f ) h 3! 4! ) dx + f h! ) 3 + f ) h x h 4 + 4! ) ) x h ) 4 dx + x h ) dx The integrl over odd powers of x h/ vnishes due to symmetry The right-hnd side therefore simplifies to ) h fx)dx = f h + f ) h x h ) dx + f ) h x h 4 dx +! 4! ) ) h = f h + f ) h h 3 + f ) h h The midpoint rule pplied to the integrl on the left-hnd side gives the first term in the right-hnd side The remining terms express the qudrture error Hence, this error is given by ) h fx)dx f h = f ) h h 3 + f ) h h 5 + 1) 1 8 Since ll derivtives of order nd higher of liner function vnish, the right-hnd side vnishes for such functions It follows tht the midpoint rule is exct for liner functions We know this lredy, nd here it is consequence of the expnsion of the integrl in powers of h Exercise 3 Use Tylor expnsions round x = nd x = h to determine how the pproximtion of the integrl fx)dx determined by the trpezoidl rule depends on h Thus, express the qudrture error fx)dx h f) + fh)) 5
6 s function of h in similrly mnner s 1) Consider the pproximtion of the integrl 1) by the N-point composite midpoint rule with the nodes nd weights x j = + j 1 ) h, h = b N 1, w j = h, j = 1,,,N 11) Thus, the qudrture rule is given by M h f) = h fx j ) Anlogously to 1), we obtin for ech subintervl [x j h/, x j + h/] of length h the expression xj+h/ x j h/ nd summing over j = 1,,,N yields fx) M h f) = fx)dx fx j )h = f x j ) h 3 +, 1) 1 f x j ) h 3 + = 1 1 N f x j ) Nh ) The verge of the second derivtive vlues 1 N N f x j ) is not smller thn the smllest of the vlues f x j ) Similrly, the verge is not lrger thn lrgest of the vlues f x j ) This is expressed by the inequlities min 1 j N f x j ) 1 N f x j ) mx 1 j N f x j ) We ssumed f x) to be continuous Therefore there is vlue ξ in [, b], such tht f ξ) = 1 N f x j ) Substituting this expression into the right-hnd side of 13) yields Multiplying the expression for h in 11) by the denomintor gives fx) M h f) = f ξ) Nh ) which when substituted into 14) yields Nh = b + h, fx) M h f) = f ξ) b 1 h + 15) The right-hnd side indictes tht we cn expect the error to decrese by fctor 4 when h is hlved nd the number of nodes x j is doubled) 6
7 Generlly, one does not know in dvnce how mny nodes to use in order to chieve desired ccurcy We therefore my be interested in hlving h until consecutive pproximtions M h f), M h/ f), M h/4 f),, of the integrl do not vry much when h is further reduced Exercise 4 In pplictions with complicted functions, the evlution of the function vlues my dominte the rithmetic work required to evlute qudrture rules The computtion of M h f) requires the evlution of N function vlues How mny of these function vlues cn be used gin when evluting M h/ f)? Exercise 5 Consider the N-point composite trpezoidl rule for the pproximtion of 1), 1 T h f) = h fx 1) + fx ) + fx 3 ) + + fx N 1 ) + 1 ) fx N) with ) Use the representtion x j = + j 1)h, T h f) = N 1 h = b, j = 1,,, N N 1 h fx j) + fx j+1 )) nd the result of Exercise 3 to determine n expnsion of the qudrture error similr to 13) b) Wht is the nlog of formul 15)? How much is the error reduced when h is hlved? c) Assume tht T h f) hs been evluted, nd we would like to compute T h/ f) How mny dditionl function evlutions re required? Singulr integrnds, infinite intervls, nd dptive qudrture rules The composite midpoint nd trpezoidl rules cn be used lso for functions tht re not twice continuously differentible However, the error then my converge to zero slower when h is reduced nd mny nodes might be required to give smll qudrture error We discuss n pproch to remedy this sitution Consider the evlution of the integrl 1 fx) lnx)dx 16) where fx) is ssumed to be smooth function We note tht the trpezoidl rule cnnot be used since the integrnd is infinite t x = Nevertheless, the integrl exists for mny smooth functions fx) We therefore my consider using composite midpoint rule; see Exercise 6 Alterntively, the method of undetermined coefficients cn be pplied to the function fx) only Thus, we would like to determine qudrture rule of the form fx j )w j with given nodes, nd we seek to determine weights w j so tht the reltion px) lnx)dx = px j )w j 17) 7
8 holds for ll polynomils of s high degree s possible This yields the liner system of equtions, which is nlogous to 6), px) = 1 : w 1 + w + + w N = px) = x : x 1 w 1 + x w + + x N w N = px) = x : x 1w 1 + x w + + x N w N = px) = x N 1 : x N 1 1 w 1 + x N 1 w + + w N x N 1 N lnx)dx, b xlnx)dx, b x lnx)dx, = xn 1 lnx)dx The right-hnd side expressions cn be evluted by integrtion by prts Note tht the lloction of nodes x j is quite rbitrry; in prticulr, we my put node t the origin Exercise 6 Compute n pproximtion of 16) with fx) = expx ) by the -node composite midpoint rule Use the MATLAB function qud to determine the exct vlue Note tht the MATLAB function for the integrnd hs to written to llow vector rguments, eg, f=expx^)*logx); How lrge is the error? The MATLAB function qud pplies composite Simpson rule The MATLAB function qud is n exmple of n dptive qudrture rule Adptive rules re composite rules tht estimte the qudrture error by compring the result obtined with different mesh sizes h New nodes re llocted in subintervls [x j, x j+1 ] in which the qudrture error is deemed to be lrger thn prescribed tolernce Exercise 7 Determine -point qudrture rule with the sme nodes s in Exercise 6 by solving liner system of equtions of the form 18) Apply it to the function fx) of Exercise 6 Is this rule more ccurte thn the one in Exmple 6? Exercise 8 Evlute the integrl exp x 3 )dx with t lest 5 correct deciml digits using the MATLAB function qud This function requires the intervl of integrtion to be finite Therefore the integrl hs to be split, exp x 3 )dx = c exp x 3 )dx + c exp x 3 )dx, where the constnt c > is chosen lrge enough so tht the second integrl cn be neglected nd the first integrl is evluted with the function qud For instnce, we my choose c lrge enough so tht c exp x 3 )dx Determine such vlue of c nd justify your choice The error bound for the integrl evluted with qud then should not be lrger thn In this lecture, we hve fixed the nodes nd then determined the weights so tht we integrte polynomils of s high degree s possible A clever choice of nodes s well s weights in n N-point qudrture rule mkes it possible to integrte polynomils of degree up to N 1 exctly The midpoint rule is n exmple These rules re known s Gussin qudrture rules The nodes nd weights generlly re not known in closed form, but they cn be computed quite efficiently by solving n eigenvlue problem symmetric N N tridigonl mtrix This is often done with the QR-lgorithm or modifiction thereof 18) 8
Numerical 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 informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More 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 informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 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 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& 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 informationCMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
More 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 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 informationIII. Lecture on Numerical Integration. File faclib/dattab/lecture-notes/numerical-inter03.tex /by EC, 3/14/2008 at 15:11, version 9
III Lecture on Numericl Integrtion File fclib/dttb/lecture-notes/numerical-inter03.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 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 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 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 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 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 Three-Point
More informationAPPROXIMATE INTEGRATION
APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose nti-derivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution
More informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
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 informationMatrices, Moments and Quadrature, cont d
Jim Lmbers MAT 285 Summer Session 2015-16 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 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 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 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 information1. Gauss-Jacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),
1. Guss-Jcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on
More 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 informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More informationTHE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS
THE HANKEL MATRIX METHOD FOR GAUSSIAN QUADRATURE IN 1 AND 2 DIMENSIONS CARLOS SUERO, MAURICIO ALMANZAR CONTENTS 1 Introduction 1 2 Proof of Gussin Qudrture 6 3 Iterted 2-Dimensionl Gussin Qudrture 20 4
More information6.5 Numerical Approximations of Definite Integrals
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 6.5 Numericl Approximtions of Definite Integrls Sometimes the integrl of function cnnot be expressed with elementry functions, i.e., polynomil,
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 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 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 informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationZ b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but...
Chpter 7 Numericl Methods 7. Introduction In mny cses the integrl f(x)dx cn be found by finding function F (x) such tht F 0 (x) =f(x), nd using f(x)dx = F (b) F () which is known s the nlyticl (exct) solution.
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 informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationSection 6.1 Definite Integral
Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined
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 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 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 informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
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 informationConstruction of Gauss Quadrature Rules
Jim Lmbers MAT 772 Fll Semester 2010-11 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 Newton-Cotes qudrture
More informationSuppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.
Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot
More informationDiscrete Least-squares Approximations
Discrete Lest-squres Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve
More informationTrapezoidal Rule, n = 1, x 0 = a, x 1 = b, h = b a. f (x)dx = h 2 (f (x 0) + f (x 1 )) h3
Trpezoidl Rule, n = 1, x 0 =, x 1 = b, h = b f (x)dx = h 2 (f (x 0) + f (x 1 )) h3 12 f (ξ). Simpson s Rule: n = 3, x 0 =, x 1 = +b 2, x 2 = b, h = b 2. Qudrture Rule, double node t x 1 P 3 (x)dx = f (x
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 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 informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
More informationDOING PHYSICS WITH MATLAB MATHEMATICAL ROUTINES
DOIG PHYSICS WITH MATLAB MATHEMATICAL ROUTIES COMPUTATIO OF OE-DIMESIOAL ITEGRALS In Cooper School of Physics, University of Sydney in.cooper@sydney.edu.u DOWLOAD DIRECTORY FOR MATLAB SCRIPTS mth_integrtion_1d.m
More informationOrthogonal Polynomials and Least-Squares Approximations to Functions
Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny
More informationLecture 17. Integration: Gauss Quadrature. David Semeraro. University of Illinois at Urbana-Champaign. March 20, 2014
Lecture 17 Integrtion: Guss Qudrture Dvid Semerro University of Illinois t Urbn-Chmpign Mrch 0, 014 Dvid Semerro (NCSA) CS 57 Mrch 0, 014 1 / 9 Tody: Objectives identify the most widely used qudrture method
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
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 informationWe know that if f is a continuous nonnegative function on the interval [a, b], then b
1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going
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 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 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 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 informationThe Riemann Integral
Deprtment of Mthemtics King Sud University 2017-2018 Tble of contents 1 Anti-derivtive Function nd Indefinite Integrls 2 3 4 5 Indefinite Integrls & Anti-derivtive Function Definition Let f : I R be function
More 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 informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More 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 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 informationc n φ n (x), 0 < x < L, (1) n=1
SECTION : Fourier Series. MATH4. In section 4, we will study method clled Seprtion of Vribles for finding exct solutions to certin clss of prtil differentil equtions (PDEs. To do this, it will be necessry
More informationCalculus II: Integrations and Series
Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]
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 informationLECTURE. INTEGRATION AND ANTIDERIVATIVE.
ANALYSIS FOR HIGH SCHOOL TEACHERS LECTURE. INTEGRATION AND ANTIDERIVATIVE. ROTHSCHILD CAESARIA COURSE, 2015/6 1. Integrtion Historiclly, it ws the problem of computing res nd volumes, tht triggered development
More informationAN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION
Applied Mthemtics E-Notes, 5(005), 53-60 c ISSN 1607-510 Avilble free t mirror sites of http://www.mth.nthu.edu.tw/ men/ AN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION
More 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 informationSection 7.1 Integration by Substitution
Section 7. Integrtion by Substitution Evlute ech of the following integrls. Keep in mind tht using substitution my not work on some problems. For one of the definite integrls, it is not possible to find
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 informationCOSC 3361 Numerical Analysis I Numerical Integration and Differentiation (III) - Gauss Quadrature and Adaptive Quadrature
COSC 336 Numericl Anlysis I Numericl Integrtion nd Dierentition III - Guss Qudrture nd Adptive Qudrture Edgr Griel Fll 5 COSC 336 Numericl Anlysis I Edgr Griel Summry o the lst lecture I For pproximting
More informationMath 131. Numerical Integration Larson Section 4.6
Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n
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 informationQUADRATURE is an old-fashioned word that refers to
World Acdemy of Science Engineering nd Technology Interntionl Journl of Mthemticl nd Computtionl Sciences Vol:5 No:7 011 A New Qudrture Rule Derived from Spline Interpoltion with Error Anlysis Hdi Tghvfrd
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 informationTangent Line and Tangent Plane Approximations of Definite Integral
Rose-Hulmn Undergrdute Mthemtics Journl Volume 16 Issue 2 Article 8 Tngent Line nd Tngent Plne Approximtions of Definite Integrl Meghn Peer Sginw Vlley Stte University Follow this nd dditionl works t:
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 informationBest Approximation in the 2-norm
Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationNumerical Integration. Newton Cotes Formulas. Quadrature. Newton Cotes Formulas. To approximate the integral b
Numericl Integrtion Newton Cotes Formuls Given function f : R R nd two rel numbers, b R, < b, we clculte (pproximtely) the integrl I(f,, b) = f (x) dx K. Frischmuth (IfM UR) Numerics for CSE 08/09 8 /
More informationTHE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS.
THE EXISTENCE-UNIQUENESS THEOREM FOR FIRST-ORDER DIFFERENTIAL EQUATIONS RADON ROSBOROUGH https://intuitiveexplntionscom/picrd-lindelof-theorem/ This document is proof of the existence-uniqueness theorem
More informationMAT 168: Calculus II with Analytic Geometry. James V. Lambers
MAT 68: Clculus II with Anlytic Geometry Jmes V. Lmbers Februry 7, Contents Integrls 5. Introduction............................ 5.. Differentil Clculus nd Quotient Formuls...... 5.. Integrl Clculus nd
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 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 informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationAnti-derivatives/Indefinite Integrals of Basic Functions
Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second
More informationLab 11 Approximate Integration
Nme Student ID # Instructor L Period Dte Due L 11 Approximte Integrtion Ojectives 1. To ecome fmilir with the right endpoint rule, the trpezoidl rule, nd Simpson's rule. 2. To compre nd contrst the properties
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 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 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 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 informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly well-defined, is too restrictive for mny purposes; there re functions which
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 informationMAT 772: Numerical Analysis. James V. Lambers
MAT 772: Numericl Anlysis Jmes V. Lmbers August 23, 2016 2 Contents 1 Solution of Equtions by Itertion 7 1.1 Nonliner Equtions....................... 7 1.1.1 Existence nd Uniqueness................ 7 1.1.2
More informationCalculus I-II Review Sheet
Clculus I-II 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 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 informationReversing the Chain Rule. As we have seen from the Second Fundamental Theorem ( 4.3), the easiest way to evaluate an integral b
Mth 32 Substitution Method Stewrt 4.5 Reversing the Chin Rule. As we hve seen from the Second Fundmentl Theorem ( 4.3), the esiest wy to evlute n integrl b f(x) dx is to find n ntiderivtive, the indefinite
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 nti-derivtive. Differentiting integrls. Tody we provide the connection
More information