Numerical integration


 Matilda O’Neal’
 4 years ago
 Views:
Transcription
1 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 is s follows. The rst section covers qudrture procedures, which re the dominnt wy to solve models. The second section covers (pseudo) Monte Crlo integrtion techniques. The lst section discusses qusi Monte Crlo integrtion. 2.2 Qudrture techniques Suppose we wnt to clculte I = ; (2.) where f(x) is sclr function. This could be di cult problem, e.g., becuse the functionl form is nsty or becuse we do not even hve functionl form, but only set of function vlues. Qudrture techniques re numericl integrtion techniques for which the formul of the numericl integrl cn be written s I = w i f i ; (2.2) i=
2 ii 2. Numericl integrtion where f i is the function vlue of f t node x i nd w i is weight. We will discuss two types of qudrture techniques. The rst is NewtonCotes. NewtonCotes is not very creful bout choosing the loction of the nodes, but is clever bout choosing the weights. The second is Gussin Qudrture. This procedure is clever bout choosing the weights s well s the nodes. To implement qudrture methods you cn forget the detils of the derivtion. All you hve to remember is how to construct wht kind of weights nd this is esy NewtonCotes Qudrture Consider the integrtion problem given in Eqution 2. nd suppose tht one hs three function vlues t three nodes. Given this informtion, how would one clculte the integrl? Well, one could clculte n pproximting function nd clculte the integrl for this pproximting function. Since we hve been given three points, we cn clculte secondorder polynomil, P 2 (x), nd get n estimte for the integrl using P 2 (x)dx: (2.3) Since integrting polynomils is esy, this procedure is strightforwrd. But one still hs to nd the pproximtion nd do the integrtion. It turns out tht these procedures cn be stndrdized. Tht is, one cn nd the weights in (2.2) independent of the functionl form of f. They do depend to some extend on the loction of the nodes. We ssume tht x 0 =, x = ( + b)=2, nd x 2 = b. Tht is, we hve equidistnt nodes nd the rst (lst) node is the left (right) boundry. We hve two segments of equl length, h nd we cn write x = x 0 + h nd x 2 = x 0 + 2h. Recll from the chpter on function pproximtion, tht using Lgrnge interpoltion the secondorder polynomil cn be written s P 2 (x) = f 0 L 0 (x) + f L (x) + f 2 L 2 (x): (2.4) This mens tht our pproximting integrl is given by P 2 (x)dx = (f 0 L 0 (x) + f L (x) + f 2 L 2 (x)) dx = f 0 L 0 (x)dx + f L (x)dx + f 2 L 2 (x)dx The righthnd side lredy hs the qudrture form s in Eqution (2.2). The weights re the integrls. Key is tht the integrls, i.e., the weights, do not depend on the function vlues. The nodes re pinned down by the vlue of x 0 nd h. The beuty is tht the integrls do not depend on x 0
3 2.2 Qudrture techniques iii either. Try to do the integrtion for one of them to ensure yourself tht this is true. In prticulr, it is not di cult to show tht Simpson qudrture Combining the results we get L 0 (x)dx = 3 h L (x)dx = 4 3 h L 2 (x)dx = 3 h P 2 (x)dx = 3 f f + 3 f 2 h: This will give you n ccurte nswer if the function f cn be pproximted well with secondorder polynomil over the intervl [; b]. It will give you n exct nswer for ny secondorder polynomil. But clerly this will not give you n ccurte nswer if the function you re integrting is more complex. Sticking to the originl ide of integrting pproximting polynomils, there re two wys to proceed. The rst is to extend the ide to higherorder polynomils. The other is to use the sme ide but to smller intervls. In prticulr, suppose tht one hs n + equidistnt nodes nd the distnce between the nodes is h. The totl number of nodes must be odd so tht there re n=2 segments of length 2h. On ech of these segments of length 2h one then pplies the bove procedure. This would give 3 f f + 3 f 2 h + 3 f f f 4 h f n f n + 3 f n h = 3 f f f f f f n f n + 3 f n h Gussin qudrture In constructing the Simpson weights no thought went into choosing the loction of the nodes. We simply strted with equidistnt nodes nd clculted the formuls for the weights. Writing the code to implement Newton Cotes is so esy, becuse the weights only depend on h nd not on x 0. But
4 iv 2. Numericl integrtion by being smrt bout choosing the nodes we cn do even better. Tht is we cn get more ccurte nswer with the sme number of points. To be precise, if n is the number of nodes, then the following is true. With NewtonCotes qudrture we get n exctly correct nswer if the function we re integrting is polynomil of order n, wheres with Gussin qudrture we get n exctly correct nswer if the function we re interested in is polynomil of order 2n. For exmple, if we hve 5 nodes then we get n exct nswer for ll polynomils of order 9 (or less) nd we get n ccurte nswer for functions tht cn be pproximted well with 9 th order polynomil. Given tht we cn cover quite few functions with 9 th order polynomils, you better be impressed bout the power nd simplicity of Gussin qudrture. To understnd the procedure, suppose we wnt to integrte sclr function de ned on [ ; ] using the qudrture formul with n nodes. Thus, Z! i f( i ): (2.5) Note tht we hve 2n free prmeters, nmely the! i s (the weights) nd the i s (the nodes). We wnt to get the correct nswer for ny polynomil of order 2n. To ccomplish this, we choose the vlues of! i nd i so tht by construction we get the correct nswer for ll the bsis functions, tht is, for, x, x 2,, nd x 2n. But if we get the correct nswer for ll bsis functions, we get the correct nswer for ny combintion. Tht is, one gets the correct nswer for ny polynomil of order 2n. To see why, suppose tht we hve found the! i s nd the i s such tht pplying the formul in Eqution 2.5 for f(x) = x 4 gives the right nswer, tht is Z x 4 dx = i=! i 4 i : (2.6) But this mens tht we lso get the right nswer for f(x) = x 4 for ny vlue of. To see why our pproximtion now would be! i 4 i =! i 4 i : (2.7) i= Tht is our pproximtion is the nswer for f(x) = x 4 times. Since the integrl of f(x) is indeed equl times the integrl of f(x) we get the right nswer. Similrly, we get the right nswer for ny combintion of polynomil bsis functions. But we still hve to nd the! i s nd the i s tht give us the correct nswer for the bsis functions. Tht will be the cse if the following is true: Z x j dx =! i j i j = 0; ; ; 2n (2.8) i= i= i=
5 2.2 Qudrture techniques v This is system of 2n equtions in 2n unknowns. The importnt thing to relize is tht the solution to this system of equtions does not depend on f. Tht is, independent of the prticulr function one is considering, one uses the sme vlues for the! i s nd the i s. In fct, there re stndrd subroutines vilble to solve for the qudrture nodes nd weights. GussLegendre The procedure discussed bove tht clculted the integrl of function over the intervl [ ; ] is clled GussLegendre. So in prctice one would do the following. One would use numericl procedure to generte the! i s nd the i s. The generted vlues will stisfy (2.8), but you don t hve to worry bout how the lgorithm mkes tht hppen. Let the solution be! GL i nd GL i. The only thing tht you hve to do is to obtin function vlues t the indicted nodes nd clculte the pproximtion to the integrl using the qudrture formul, tht is GussHermite Z i=! GL i f( GL i ): (2.9) GussLegendre will give n ccurte nswer if f(x) cn be pproximted well with polynomil. Now suppose tht one wnts to integrte function f(x) tht cn be written s g(x) W (x) nd one knows tht g(x) cn be pproximted well with polynomil but g(x) W (x) cnnot. In this cse, it would not be smrt to use GussLegendre. Insted one would wnt to djust the procedure to tke this into ccount. There re di erent Gussin qudrture procedure tht do exctly this for di erent weighting functions, W (x), nd di erent domins. An importnt one is GussHermite for which the weighting function is e x2 nd the domin is the rel line. For Guss Hermite the weights nd the nodes re chosen to stisfy Z x j e x2 dx =! i j i j = 0; ; ; 2n (2.0) Let the solution be! GH i by Z i= nd GH i. So the pproximtion would be given g(x)e x2 dx i=! GH i g( gh;i ): (2.) Mke sure you understnd why there is n = in Eqution (2.0) nd n in Eqution (2.). In the rst eqution we re choosing the! i s nd the i s so tht our pproximting formul, i.e. Eqution (2.) will give the correct nswer for prticulr choices of g(x), nmely polynomil bsis functions. But unless g(x) is polynomil, the qudrture formul is n pproximtion.
6 vi 2. Numericl integrtion GussChebyshev Another Gussin qudrture procedure is GussChebyshev tht dels with Z g(x) dx ( x 2 ) =2 Qudrture nodes The nodes tht solve the problems discussed here turn out to be the zeros of the bsis functions of the corresponding Orthogonl polynomil. Tht is, the Chebyshev nodes tht solve Z x j ( x 2 ) dx = X n! =2 i j i j = 0; ; ; 2n i= re exctly the sme s the Chebyshev nodes discussed in the chpter on pproximting functions, lthough it goes bit to fr to explin why Chnge in vrible Suppose one wnts to clculte the expecttion of h (y), i.e., E[h(y)] ; where y is rndom vrible with N(; 2 ) distribution. Tht is one wnts to clculte Z (y ) 2 p h(y) exp dy Also, suppose tht h(y) is function tht one expects cn be pproximted well with polynomil. This problem resembles GussHermite qudrture problem but not exctly. One might be tempted to mke it GussHermite problem simply by de ning (y ) 2 h(y) = h(y) exp p exp ( y 2 ) nd considering the identicl integrl Z h(y) exp y 2 dy: But note tht it ws given tht h(y) could be pproximted well with polynomil, not tht h(y) cn be. In fct, given tht exp( y 2 ) is not like polynomil, h(y) my be pproximted very poorly with polynomil. So the pproprite wy to go is to do chnge of vribles. This is very esy but don t forget the Jcobin. Tht is, if y = (x) then g(y)dy = Z (b) () g((x)) 0 (x)dx
7 2.3 Monte Crlo Integrtion vii The trnsformtion we use here is x = y p 2 or y = p 2x + This gives E [h(y)] = = = Z Z Z (y ) 2 p h(y) exp dy p 2 h(p 2x + ) exp x 2 p 2dy p h( p 2x + ) exp x 2 dy Wht to do in prctice? So wht would you do in prctice if one wnts to evlute E[h(y)]. First, one obtins n GussHermite qudrture weights nd nodes using numericl lgorithm. Second, one gets n pproximtion using E [h(y)] p p! GH i h 2 GH i + i= (2.2) nd do not forget to divide by p! Well, how often do you get something in life so complex s n integrl so esily? 2.3 Monte Crlo Integrtion The ide behind Monte Crlo integrtion is very simple. Consider rndom vrible x with CDF F (x). Then one cn pproximte the integrl of the function h(x) with P T t= h(x)df (x) h(x t) ; (2.3) T where fx t g T t= is series drwn from rndom number genertor corresponding to the distribution of x. Although very simple there is one importnt disdvntge: it is not very ccurte. Above we sw tht we cn get n ccurte nswer with just few qudrture nodes for lrge clss of functions. Monte Crlo is subject to smpling vrition nd this only disppers t root n. Suppose we clculte the men of rndom vrible with uniform distribution on the unit intervl. With T = 00 the stndrd error is which is 5.8% of the true men. Even with T = ; 000 we hve stndrd error tht is.8% of the true men.
8 viii 2. Numericl integrtion If one doesn t hve CDF then one cn use uniform distribution. Tht is, h(x)dx = (b ) h(x)f b (x)dx; (2.4) where f b is the density of rndom vrible with uniform distribution over [; b], tht is, f b = (b ). Thus, one could pproximte the integrl with P T t= h(x)dx (b ) h(x t) ; (2.5) T where x t is generted using rndom number genertor for vrible tht is uniform on [; b]. Typiclly one doesn t hve ccess to true rndom numbers nd one only hs ccess to computer progrm tht genertes them. Therefore, these procedures re lso referred to pseudo rndom numbers. The computer progrm genertes dt tht re (if it is good progrm) indistinguishble from true series of rndom numbers. But the function tht genertes the series is deterministic (nd chotic) so tht one should be creful in using theorems for true rndom numbers to think bout things like rtes of convergence. 2.4 Multivrite problems nd qusi Monte Crlo integrtion It is strightforwrd to extend the ide of qudrture techniques to higher dimensionl problems. The number of nodes increses exponentilly, however, which mens tht it becomes computtionlly quickly very expensive. With Monte Crlo integrtion one does not seem to hve this problem. Tht is the men of h(x t ) nd the men of h(x t ; z t ) both converge towrds its men t rte p T. Think of numericl integrtion problem s choosing nodes nd then tking (weighted) verge. By extending the qudrture techniques derived for sclr functions to multivrite problems one doesn t ll in the spce with nodes in the most e cient wy. Building on the ide of pseudo Monte Crlo new techniques hve been developed tht ll in the spce better. The strting point of qusi Monte Crlo integrtion is to generte equidistributed series. A sclr sequence fx t g T t= is equidistributed over [; b] i b lim T! T TX f(x t ) = t= (2.6) for ll Riemnintegrble f(x). Note the similrity with Eqution (2.5). There re severl exmples of equidistributed series. For exmple the se
9 2.4 Multivrite problems nd qusi Monte Crlo integrtion ix quence (; 2; 3; 4; ) is equidistributed modulo for ny irrtionl number. Another exmple is the sequence of prime numbers multiplied by n irrtionl number (2; 3; 5; 7; ). For ddimensionl problem, n equidistributed sequence fx t g T t= D R d stis es (D) TX Z lim f(x t ) = ; (2.7) T! T t= where (D) is the Lebesque mesure of D. 2 Some exmples for equidistributed vectors on the ddimensionl unit hypercube re the following. Weyl: x t = (t p p ; t p p 2 ; ; t p p d ) modulo, (2.8) where p i is the i th positive prime number. Neiderreiter: x t = (t2 =(d+) ; 2 2=(d+) ; ; t2 d=(d+) ) modulo Equidistributed vectors for other hypercubes cn be done using liner trnsformtions. D F rc(x) (or x Modulo ) mens tht we subtrct the lrgest integer tht is less thn x. For exmple, frc(3:564) = 0:564: 2 To see why you hve to multiply the sum with (D) just consider the cse when f(x) = 8x. Then we know the integrl should be equl to (D) which we get becuse P T t= f(x)=t = :
Numerical 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 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 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 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 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 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 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 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 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 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 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 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 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 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 information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
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 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 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 informationand that at t = 0 the object is at position 5. Find the position of the object at t = 2.
7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we
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 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 informationMonte Carlo method in solving numerical integration and differential equation
Monte Crlo method in solving numericl integrtion nd differentil eqution Ye Jin Chemistry Deprtment Duke University yj66@duke.edu Abstrct: Monte Crlo method is commonly used in rel physics problem. The
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 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 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 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 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 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 informationCS667 Lecture 6: Monte Carlo Integration 02/10/05
CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of
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 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 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 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 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 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 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 informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More informationContinuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
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 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 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 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 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 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 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 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 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 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 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 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 information1 1D heat and wave equations on a finite interval
1 1D het nd wve equtions on finite intervl In this section we consider generl method of seprtion of vribles nd its pplictions to solving het eqution nd wve eqution on finite intervl ( 1, 2. Since by trnsltion
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 information( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that
Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we
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 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 informationInfinite Geometric Series
Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to
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 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 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 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 informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
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 informationp(t) dt + i 1 re it ireit dt =
Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)
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 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 informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
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 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 informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
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 informationAP Calculus Multiple Choice: BC Edition Solutions
AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this
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 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 informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationSYDE 112, LECTURES 3 & 4: The Fundamental Theorem of Calculus
SYDE 112, LECTURES & 4: The Fundmentl Theorem of Clculus So fr we hve introduced two new concepts in this course: ntidifferentition nd Riemnn sums. It turns out tht these quntities re relted, but it is
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 informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More information7  Continuous random variables
71 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7  Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin
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 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 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 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 informationName Solutions to Test 3 November 8, 2017
Nme Solutions to Test 3 November 8, 07 This test consists of three prts. Plese note tht in prts II nd III, you cn skip one question of those offered. Some possibly useful formuls cn be found below. Brrier
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 informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
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 informationSection 17.2 Line Integrals
Section 7. Line Integrls Integrting Vector Fields nd Functions long urve In this section we consider the problem of integrting functions, both sclr nd vector (vector fields) long curve in the plne. We
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 informationMATH , Calculus 2, Fall 2018
MATH 362, 363 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
More informationJDEP 384H: Numerical Methods in Business
BT 3.4: Solving Nonliner Systems Chpter 4: Numericl Integrtion: Deterministic nd Monte Crlo Methods Instructor: Thoms Shores Deprtment of Mthemtics Lecture 20, Februry 29, 2007 110 Kufmnn Center Instructor:
More informationB.Sc. in Mathematics (Ordinary)
R48/0 DUBLIN INSTITUTE OF TECHNOLOGY KEVIN STREET, DUBLIN 8 B.Sc. in Mthemtics (Ordinry) SUPPLEMENTAL EXAMINATIONS 01 Numericl Methods Dr. D. Mckey Dr. C. Hills Dr. E.A. Cox Full mrks for complete nswers
More informationLecture 3 Gaussian Probability Distribution
Introduction Lecture 3 Gussin Probbility Distribution Gussin probbility distribution is perhps the most used distribution in ll of science. lso clled bell shped curve or norml distribution Unlike the binomil
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 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 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 informationChapter 5. , r = r 1 r 2 (1) µ = m 1 m 2. r, r 2 = R µ m 2. R(m 1 + m 2 ) + m 2 r = r 1. m 2. r = r 1. R + µ m 1
Tor Kjellsson Stockholm University Chpter 5 5. Strting with the following informtion: R = m r + m r m + m, r = r r we wnt to derive: µ = m m m + m r = R + µ m r, r = R µ m r 3 = µ m R + r, = µ m R r. 4
More information