NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.


 Carmel Short
 1 years ago
 Views:
Transcription
1 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 respect to the independent vrible x, evluted between the limits x = to x = b. The function f(x) in the integrl is referred to s the integrnd. The vlue of the integrl I(f) is number when nd b re numbers. Grphiclly, the vlue of the integrl corresponds to the shded re under the curve of f(x) between nd b. The function to be differentited or integrted will typiclly be in one of the following three forms: A simple continuous function such s polynomil, n exponentil, or trigonometric function. A complicted continuous function tht is difficult or impossible to integrte directly. A tbulted function where vlues of x nd f(x) re given t number of discrete points, s is often the cse with experimentl or field dt. In the first cse, the derivtive or integrl of simple function my be evluted nlyticlly using clculus. For the second cse, nlyticl solutions re often imprcticl, nd sometimes impossible, to obtin. In these instnces, s well s in the third cse of discrete dt, pproximte methods must be employed. This chpter is devoted to the most common pproches for numericl integrtion. We begin with the NewtonCotes formuls. These reltionships re bsed on replcing complicted function or tbulted dt with simple polynomil tht is esy to integrte. Three of the most widely used NewtonCotes formuls re discussed in detil: the trpezoidl rule, Simpson s 1/3 rule, nd Simpson s 3/8 rule. All these formuls re designed for cses where the dt to be integrted is evenly spced. Newton s cotes formuls cn be used for tbulted dt s well s in integrting functions numericlly. where F n (x) is the polynomil of the form f(x) dx F n (x) dx F n (x) = o + 1 x + x n x n Two dditionl techniques when the function is given re lso considered. The first is bsed on Richrdson s extrpoltion, the second method is clled Guss qudrture. 1
2 Newton s cotes integrtion formuls Closed nd open forms of the NewtonCotes formuls re vilble. The closed forms re those where the dt points t the beginning nd end of the limits of integrtion re known (prt of figure below). The open forms hve integrtion limits tht extend beyond the rnge of the dt (prt b of sme figure). Open NewtonCotes formuls re not generlly used for definite integrtion. This chpter emphsizes the closed forms..1 The Trpezoidl rule The trpezoidl rule is the first of the NewtonCotes closed integrtion formuls. It corresponds to the cse where the polynomil is firstorder. recll tht Newton s form of interpolting polynomils with two points x = nd x = b yields ( ) f(b) f() F 1 (x) = f() + f[, b](x ) = f() + (x ) b The re under this stright line is n estimte of the integrl of f(x) between the limits nd b: I(f) ( f(b) f() f() + b ) (x ) dx = f() + f(b) (b ) When we employ the integrl under strightline segment to pproximte the integrl under curve, we obviously cn incur n error tht my be substntil. An estimte for the locl trunction error of single ppliction of the trpezoidl rule is E t = 1 1 f (η)(b ) 3 where η lies somewhere in the intervl from to b. This eqution indictes tht if the function being integrted is liner, the trpezoidl rule will be exct. Otherwise, for functions with second nd higherorder derivtives (tht is, with curvture), some error cn occur.
3 Note: The error in the trpezoidl rule cn be derived using the following theorem: Let p n (x) be n interpolting polynomil for given set of dt points y k = f(x k ), for distinct grid points x k, k = 1,...n + 1. Let = min(x k ) nd b = mx(x k ), nd suppose tht f(x) is n + 1 times differentible. Then for ech x ɛ [, b], there exist η ɛ [, b] such tht f(x) = p n (x) + f (n+1) (η) (n + 1)! (x x 1)...(x x n+1 ) We cn then clculte the error in the integrl for n + 1 = s E = f (η)!.1.1 The MultipleAppliction Trpezoidl Rule (x )(x b) = f (η) (b )3 1 One wy to improve the ccurcy of the trpezoidl rule is to divide the integrtion intervl from to b into number of segments nd pply the method to ech segment. The res of individul segments cn then be dded to yield the integrl for the entire intervl. The resulting equtions re clled multipleppliction, or composite, integrtion formuls. There re n segments of equl width x x 1 f(x) dx + x3 x f(x) dx xn+1 x n f(x) dx f(x 1) + f(x ) h + f(x ) + f(x 3 ) h +... f(x n) + f(x n+1 ) h or, grouping terms [ ] h n f() + f(b) + f(x i ) An error for the multipleppliction trpezoidl rule cn be obtined by summing the individul errors for ech segment to give (b )3 E t = 1n 3 i= n f (η i ) where f (η i ) is the second derivtive t point η i locted in segment i. This result cn be simplified by estimting the men or verge vlue of the second derivtive for the entire intervl s i=1 3
4 f n f (η i ) (b )3 E t f 1n Thus, if the number of segments is doubled, the trunction error will be qurtered. Notes: i=1 For individul pplictions with nicely behved functions, the multiplesegment trpezoidl rule is just fine for ttining the type of ccurcy required in mny engineering pplictions. If high ccurcy is required, the multiplesegment trpezoidl rule demnds gret del of computtionl effort. Although this effort my be negligible for single ppliction, it could be very importnt when () numerous integrls re being evluted or (b) where the function itself is time consuming to evlute. For such cses, more efficient pproches my be necessry. Finlly, roundoff errors cn limit our bility to determine integrls. This is due both to the mchine precision s well s to the numerous computtions involved in simple techniques like the multiplesegment trpezoidl rule. Simpson s rule Aside from pplying the trpezoidl rule with finer segmenttion, nother wy to obtin more ccurte estimte of n integrl is to use higherorder polynomils to connect the points. For exmple, if there is n extr point midwy between f() nd f(b), the three points cn be connected with prbol. If there re two points eqully spced between f() nd f(b), the four points cn be connected with thirdorder polynomil. The formuls tht result from tking the integrls under these polynomils re clled Simpsons rules...1 Simpson s 1/3 rule n Simpsons 1/3 rule results when secondorder interpolting polynomil is substituted into f(x) dx F (x) dx If nd b re designted s x 1 nd x 3 nd F (x) is represented by secondorder Lgrnge polynomil, the integrl becomes I(f) x3 x 1 [ (x x )(x x 3 ) (x 1 x )(x 1 x 3 ) f(x 1) + (x x 1)(x x 3 ) (x x 1 )(x x 3 ) f(x ) + (x x ] 1)(x x ) (x 3 x 1 )(x 3 x ) f(x 3) dx 4
5 After integrtion nd lgebric mnipultion, the following formul results: I(f) h 3 [f(x 1) + 4f(x ) + f(x 3 )] = h + b [f() + 4f( 3 ) + f(b)] where, for this cse, h = (b )/. This eqution is known s Simpson s 1/3 rule. The lbel 1/3 stems from the fct tht h is divided by 3. It cn be shown tht singlesegment ppliction of Simpson s 1/3 rule hs trunction error of (b )5 E t = 880 f (4) (η) where η lies somewhere in the intervl from to b. Thus, Simpsons 1/3 rule is more ccurte thn the trpezoidl rule. However, comprison with the error from the trpezoidl rule indictes tht it is more ccurte thn expected. Rther thn being proportionl to the third derivtive, the error is proportionl to the fourth derivtive. Consequently, Simpson s 1/3 rule is thirdorder ccurte even though it is bsed on only three points. In other words, it yields exct results for cubic polynomils even though it is derived from prbol! Proof: To prove this result we extend the expnsion beyond the intervl for resons tht will be pprent shortly between x 1 =, x = + h, x 3 = + h = b, x 4 = + 3h f(x) = p (x) + f[x 1, x, x 3, x 4 ](x x 1 )(x x )(x x 3 ) + f (4) (η) (x x 1 )(x x )(x x 3 )(x x 4 ) 4 where one cn show tht E t = p (x) = f(x 1 ) + f[x 1, x ](x x 1 ) + f[x 1, x, x 3 ](x x 1 )(x x ) +h +h since h = (b )/ then f[x 1, x, x 3, x 4 ](x x 1 )(x x )(x x 3 ) dx = 0 f (4) (η) (x x 1 )(x x )(x x 3 )(x x 4 ) dx = h f (4) (η) E t = (b ) f (4) (b )5 (η) = 880 f (4) (η).. The MultipleAppliction Simpson s 1/3 Rule 5
6 Just s with the trpezoidl rule, Simpson s rule cn be improved by dividing the integrtion intervl into number of segments of equl width x3 x 1 f(x) dx + h = b n x5 x 3 f(x) dx xn+1 x n f(x) dx Substituting Simpson s 1/3 rule for the individul integrl then combining yields I(f) h n n f() + 4 f(x i ) + f(x j ) + f(b) 3 i=,4,6 j=3,5,7 Notice tht, n even number of segments must be utilized to implement the method. An error estimte for the multipleppliction Simpson s rule is obtined in the sme fshion s for the trpezoidl rule by summing the individul errors for the segments nd verging the derivtive to yield..3 Simpson s 3/8 rule E t (b )5 180n 4 f (4) In similr mnner to the derivtion of the trpezoidl nd Simpson s 1/3 rule, thirdorder Lgrnge polynomil cn be fit to four points nd integrted: to yield f(x) dx F 3 (x) dx I(f) 3h 8 [f(x 1) + 3f(x ) + 3f(x 3 ) + f(x 4 )] where h = (b )/3. This eqution is clled Simpson s 3/8 rule becuse h is multiplied by 3/8. Simpson s 3/8 rule hs n error of (b )5 E t = 6480 f (4) (η) Becuse the denomintor is lrger thn tht for Simpson s (1/3), the 3/8 rule is somewht more ccurte thn the 1/3 rule. Simpson s 1/3 rule is usully the method of preference becuse it ttins thirdorder ccurcy with three points rther thn the four points required for the 3/8 version. However, the 3/8 rule hs utility when the number of segments is odd. 6
7 ..4 The MultipleAppliction Simpson s 3/8 Rule When the domin [, b] is divided into n subintervls (where n is t lest 6 nd divisible by 3) Simpson s (3/8) rule cn be generlized to n n X X 3h I(f ) f () + 3 [f (xi ) + f (xi+1 )] + f (xj ) + f (b) 8 i=,5,8 j=4,7,10 3 Guss Qudrture Guss qudrture is technique tht is bsed on evluting the integrl using weighted ddition of the vlues of f (x) t different points (clled Guss points) within the intervl [, b]. Guss qudrture is n open method where the guss points re not eqully spced nd do not include the end points. The loction of the points nd the weights re determined in such wy to minimize the error (in wy tht the right side of the eqution below is exctly equl to the left side when f (x) = 1, x, x,...). Z b f (x)dx I(f ) = n X Ci f (xi ) i=1 The tble below lists the vlues of the coefficients Ci nd the loction of the Guss points xi for i = 1,...6 for = nd b = 1 7
8 exmple: for n =, = nd b = 1 where f(x)dx C 1 f(x 1 ) + C f(x ) cse f(x) = 1 (1)dx = = C 1 + C cse f(x) = x (x)dx = 0 = C 1 x 1 + C x cse f(x) = x (x )dx = 3 = C 1x 1 + C x cse f(x) = x 3 (x 3 )dx = 0 = C 1 x C x 3 Since some of these equtions re not liner, multiple solutions cn exist. If however, we dd nother condition, sy x 1 nd x should be symmetriclly locted bout the origin (x = x 1 ) then the solution is unique nd note: C 1 = 1 C = 1 x 1 = 3 = x = 1 3 = If or b re different thn 1 nd 1 respectively, chnge of vrible cn be mde to chnge the originl bounds to 1 nd 1. x = 1 [t(b ) + + b] dx = 1 (b )dt where f(x) dx = f(x) dx = b ( ) t(b ) + + b (b ) f dt h(t) dt b ( ) t(b ) + + b h(t) = f C i h(t i ) i 4 Richrdson extrpoltion This technique is quite similr to the one used in the previous chpter. estimtes of n integrl to compute third more ccurte estimte. It is bsed on the use of two When n integrl is numericlly evluted with method whose trunction error cn be written in terms of even powers of h, strting with h p, where p is n even number, then the true (unknown) vlue of the integrl I(f) cn be expressed s the sum of the estimte I pp (h) nd the error ccording to f(x)dx = I pp (h) + Ch p + Dh p
9 Assume the expnsion coefficients to be constnt. We cn get two estimtes of the sme integrl by reducing the size h to h/ f(x)dx = I pp (h/) + C ( ) p ( ) p+ h h + D +... Combining the lst two equtions we get third pproximtion of the integrl of order h p+ I(f) p I pp (h/) I pp (h) p 1 For exmple, consider the estimte nd error ssocited with the ppliction of the composite trpezoidl method I n + E(h) where I(f) is the exct vlue of the integrl, I n is the pproximtion from n nsegment ppliction of the trpezoidl rule with step size h = (b )/n, nd E(h) is the trunction error. (b ) E(h) = f 1 h If we mke n dditionl estimte by reducing the size of the step size from h to h/ doubling by tht the number of intervls between nd b we get fter elimintion of the error, n pproximtion of order h 4 5 Romberg integrtion I(f) I n I n 1 Romberg integrtion is technique tht is bsed on the successive ppliction of the composite trpezoidl rule to obtin more ccurte estimte of the vlue of n integrl. The method cn be explined following the digrm in the figure below The vlue of the integrl is clculted with the composite trpezoidl method severl times. In the first time, the number of subintervls is n nd in ech clcultion tht follows the number of subintervls is doubled. 9
10 Richrdson s extrpoltion formul, is used for obtining improved estimtes for the vlue of the integrl from the vlues listed in the first column nd the process is repeted in the third nd the rest of the columns To pply Romberg integrtion use the following steps 1. Using the composite trpezoidl method find n estimte of I(f) using n, 1 n, n, 3 n,... intervls. These estimtes re lbeled I i,j, where j is the column number nd i is the row number, of squre mtric or dimension k k I i,j = (j) I i+1,j I i,j (j) 1 10
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationChapter 3 Solving Nonlinear Equations
Chpter 3 Solving Nonliner Equtions 3.1 Introduction The nonliner function of unknown vrible x is in the form of where n could be noninteger. Root is the numericl vlue of x tht stisfies f ( x) 0. Grphiclly,
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 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 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 informationChapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...
Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting
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 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 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 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 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 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 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 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 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 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 information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
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 informationCBE 291b  Computation And Optimization For Engineers
The University of Western Ontrio Fculty of Engineering Science Deprtment of Chemicl nd Biochemicl Engineering CBE 9b  Computtion And Optimiztion For Engineers Mtlb Project Introduction Prof. A. Jutn Jn
More informationIf u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du
Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find ntiderivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible
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 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 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 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 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 informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
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 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 informationMA 124 January 18, Derivatives are. Integrals are.
MA 124 Jnury 18, 2018 Prof PB s oneminute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
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 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 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 2Dimensionl Gussin Qudrture 20 4
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 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 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 informationx = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b
CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick
More informationAN INTEGRAL INEQUALITY FOR CONVEX FUNCTIONS AND APPLICATIONS IN NUMERICAL INTEGRATION
Applied Mthemtics ENotes, 5(005), 5360 c ISSN 1607510 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 informationThe goal of this section is to learn how to use a computer to approximate definite integrals, i.e. expressions of the form. Z b
Lecture notes for Numericl Anlysis Integrtion Topics:. Problem sttement nd motivtion 2. First pproches: Riemnn sums 3. A slightly more dvnced pproch: the Trpezoid rule 4. Tylor series (the most importnt
More information1.9 C 2 inner variations
46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for
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 informationIntegration Techniques
Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u
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 informationTangent Line and Tangent Plane Approximations of Definite Integral
RoseHulmn 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 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 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 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 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 informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls 5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the lefthnd
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 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 informationSections 5.2: The Definite Integral
Sections 5.2: The Definite Integrl In this section we shll formlize the ides from the lst section to functions in generl. We strt with forml definition.. The Definite Integrl Definition.. Suppose f(x)
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 information31.2. Numerical Integration. Introduction. Prerequisites. Learning Outcomes
Numericl Integrtion 3. Introduction In this Section we will present some methods tht cn be used to pproximte integrls. Attention will be pid to how we ensure tht such pproximtions cn be gurnteed to be
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 informationf(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as f(x) dx = lim f(x i ) x; 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 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 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 informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
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 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 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 informationIndefinite Integral. Chapter Integration  reverse of differentiation
Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the
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 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 informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
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 informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/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 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 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 informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More information