Theoretical foundations of Gaussian quadrature


 Jean Thomas
 2 years ago
 Views:
Transcription
1 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 vectors: u + v V, which stisfies the properties (A 1 ) ssocitivity: u + (v + w) = (u + v) + w u, v, w in V ; (A 2 ) existence of zero vector: 0 V such tht u + 0 = u u V ; (A ) existence of n opposite element: u V ( u) V such tht u + ( u) = 0; (A 4 ) commuttivity: u + v = v + u u, v in V ; (B) Multipliction of number nd vector: αu V for α R, which stisfies the properties (B 1 ) α(u + v) = αu + αv α R, u, v V ; (B 2 ) (α + β)u = αu + βu α, β R, u V ; (B ) (αβ)u = α(βu) α, β R, u V ; (B 4 ) 1u = u u V. Definition 2. An inner product liner spce is liner spce V with n opertion (, ) stisfying the properties () (u, v) = (v, u) u, v in V ; (b) (u + v, w) = (u, w) + (v, w) u, v, w in V ; (c) (αu, v) = α(u, v) α R, u, v, w in V ; (d) (u, u) 0 u V ; moreover, (u, u) = 0 if nd only if u = 0. Exmple. The stndrd inner product of the vectors u = (u 1, u 2,..., u d ) R d nd v = (v 1, v 2,..., v d ) R d is given by (u, v) = d u i v i. 1
2 Exmple. Let G be symmetric positivedefinite mtrix, for exmple G = (g ij ) = Then one cn define sclr product corresponding to G by (u, v) := d d u i g ij v j. j=1 Remrk. In n inner product liner spce, one cn define the norm of vector by The fmous CuchySchwrz inequlity reds u := (u, u). (u, v) u v. Think bout the mening of this inequlity in R. Exercise. Find the norm of the vector u = (, 0, 4) using the stndrd inner product in R nd then by using the inner product in R defined through the mtrix G. A very importnt exmple. Consider the set of ll polynomils of degree no greter thn 4, where the opertions ddition of vectors nd multipliction of number nd vector re defined in the stndrd wy, nmely: if P nd Q re such polynomils, P (x) = p 4 x 4 + p x + p 2 x 2 + p 1 x + p 0, Q(x) = q 4 x 4 + q x + q 2 x 2 + q 1 x + q 0, then their sum, P + Q is given by (P + Q)(x) = (p 4 + q 4 )x 4 + (p + q )x + (p 2 + q 2 )x 2 + (p 1 + q 1 )x + (p 0 + q 0 ), nd, for α R, the product αp is defined by (αp )(x) = (αp 4 )x 4 + (αp )x + (αp 2 )x 2 + (αp 1 )x + (αp 0 ). Then this set of polynomils is vector spce of dimension 5. One cn tke for bsis in this spce the set of polynomils E 0 (x) := 1, E 1 (x) := x, E 2 (x) := x 2, E (x) := x, E 4 (x) := x 4. This, however, is only one of the infinitely mny bses in this spce. For exmple, the set of vectors G 0 (x) := x 1, G 1 (x) := x + 1, G 2 (x) := x 2 + x +, 2
3 G (x) := x + x 2 4, G 4 (x) := x 4 x 2x is perfectly good bsis. (Note tht I clled G 0, G 1,..., G n vectors to emphsize tht wht is importnt for us is the structure of vector spce nd not so much the fct tht these vectors re polynomils.) Any vector (i.e., polynomil of degree 4) cn be represented in unique wy in ny bsis, for exmple, the polynomil P (x) = x 4 5x 2 + x + 7 cn be written s P = E 4 5 E 2 + E E 0, or, lterntively, s P = G 4 G + 4 G 2 11 G G 0. 2 Inner product in the spce of polynomils One cn define n inner product structure in the spce of polynomils in mny different wys. Let V n (, b) stnd for the spce of polynomils of degree n defined for x [, b]. Most of the theory we will develop works lso if = nd/or b =. Let w : [, b] R be weight function, i.e., function stisfying the following properties: () the integrl w(x) dx exists; (b) w(x) 0 for ll x [, b], nd w(x) cn be zero only t isolted points in [, b] (in prticulr, w(x) cnnot be zero in n intervl of nonzero length). We define sclr product in V n (, b) by (P, Q) := P (x) Q(x) w(x) dx ; (1) if the intervl (, b) is of infinite length, then one hs to tke w such tht this integrl exists for ll P nd Q in V n (, b). Let V n (, b; w) stnds for the inner product liner spce of polynomils of degree n defined on [, b], nd sclr product defined by (1). Exmple. The Legendre polynomils re fmily of polynomils P 0, P 1, P 2,... such tht P n is polynomil of degree n defined for x [ 1, 1], with leding coefficients equl to 1 ( leding re the coefficients of the highest powers of x) nd such tht P n nd P m re orthogonl for n m in the sense of the following inner product: (P n, P m ) = 1 1 P n (x) P m (x) dx. In other words, the polynomils P 0, P 1, P 2,..., P n constitute n orthogonl bsis of the spce V n ( 1, 1; w(x) 1). Here re the first severl Legendre polynomils: P 0 (x) = 1, P 1 (x) = x, P 2 (x) = x 2 1, P (x) = x 5 x, P 4(x) = x x2 + 5,....
4 Sometime Legendre polynomils re normlized in different wy: P 0 (x) = 1, P1 (x) = x, P2 (x) = 1 2 (x2 1), P (x) = 1 2 (5x x), P4 (x) = 1 8 (5x4 0x 2 + ),... ; check tht P n is proportionl to P n for ll the polynomils given here. Exercise. Check tht ech of the first five Legendre polynomils is orthogonl to ll other Legendre polynomils in the exmple bove. Exmple. The Hermite polynomils re fmily of polynomils H 0, H 1, H 2,... such tht H n is polynomil of degree n defined for x R, normlized in such wy tht (H n, H n ) = 2 n n! π nd (H n, H m ) = 0 for n m, where the inner product is defined s follows: (H n, H m ) = H n (x) H m (x) e x2 dx. In other words, the polynomils H 0, H 1, H 2,..., H n constitute n orthogonl bsis of the spce V n (, ; e x2 ). Here re the first five Hermite polynomils: H 0 (x) = 1, H 1 (x) = 2x, H 2 (x) = 4x 2 2, H (x) = 8x 12x, H 4 (x) = 16x 4 48x Gussin qudrture Theorem 1. Let w be weight function on [, b], let n be positive integer, nd let G 0, G 1,..., G n be n orthogonl fmily of polynomils with degree of G k equl to k for ech k = 0, 1,..., n. In other words, G 0, G 1,..., G n form n orthogonl bsis of the inner product liner spce V n (, b; w). Let x 1, x 2,..., x n be the roots of G n, nd define L i (x) := n j=1, j i x x j x i x j for i = 1, 2,..., n. Then the corresponding Gussin qudrture formul is given by where I(f) := f(x) w(x) dx I n (f) := w i := L i (x) w(x) dx. w i f(x i ), The formul I n (f) hs degree of precision exctly 2n 1, which mens tht I n (x k ) = I(x k ) for k = 0, 1,..., 2n 1, but there is polynomil Q of degree 2n for which I n (Q) I(Q). 4
5 Proof: We will prove tht the qudrture formul I n (f) = w i f(x i ) (where the weights w i re given by the formul bove) hs degree of precision exctly 2n 1 in three steps: Step 1: The degree of precision of the qudrture formul is n 1. Let V n 1 stnd for the liner spce of ll polynomils of degree not exceeding n 1 defined for x [, b]. We need to prove tht our qudrture formul is exct for ny R V n 1. The polynomils L i (x), i = 1, 2,..., n re Lgrnge polynomils of degree n 1 such tht L i (x k ) = δ ik for ll k = 1, 2,..., n. This implies tht R(x i ) L i (x) is the Lgrnge polynomil of degree n 1 tht interpoltes the polynomil R t the n points x 1, x 2,..., x n. Reclling tht R is polynomil of degree of degree n 1, we conclude tht R(x) = R(x i ) L i (x) x [, b]. Then we hve I(R) = = R(x) w(x) dx = [ ] R(x i ) L i (x) w(x) dx [ ] R(x i ) L i (x) w(x) dx = R(x i ) w i = I n (R), therefore the qudrture formul is exct for ny polynomil R of degree up to nd including n 1, or, in other words, the degree of precision of the qudrture formul is n 1. Step 2: The degree of precision of the qudrture formul is 2n 1. Let P be polynomil of degree 2n 1, i.e., P V 2n 1. Divide P by G n (which is polynomil of degree n to obtin P (x) = Q(x) G n (x) + R(x), where Q V n 1 nd R V n 1. Since the polynomils G 0, G 1, G 2,..., G n 1 form bsis of the liner spce V n 1, we cn write Q in the form n 1 Q(x) = q i G i (x). 5
6 Then I(P ) = I(Q G n + R) = = [ b n 1 = 0 + [Q(x) G n (x) + R(x)] w(x) dx ] q i G i (x) G n (x) w(x) dx + R(x) w(x) dx = I(R), R(x) w(x) dx becuse ll polynomils G 0, G 1,..., G n 1 re orthogonl to G n (with respect to the sclr product defined with the weight function w): (G i, G n ) = G i (x) G n (x) w(x) dx = 0 i = 0, 1, 2,..., n 1. Now recll tht R is polynomil of degree n 1 to conclude tht I(R) = I n (R) ccording to wht we proved in Step 1. Since x i re roots of the polynomil G n, it follows tht P (x i ) = Q(x i ) G n (x i ) + R(x i ) = 0 + R(x i ) = R(x i ), which implies tht I n (R) = I n (P ). Combining these results, we obtin I(P ) = I n (P ) P V 2n 1. Step : The degree of precision of the qudrture formul is 2n 1. We lredy know tht the degree of precision is t lest 2n 1, so to prove tht it is exctly 2n 1, it is enough to find one polynomil of degree 2n for which the qudrture formul is not exct. Note tht the polynomil G 2 n(x) hs degree exctly 2n, nd tht I(G 2 n) = G 2 n(x) w(x) dx > 0 (2) becuse the integrnd, G 2 n(x) w(x), is nonnegtive nd cn be zero only t isolted points ( polynomil like G 2 n hs t most 2n roots, nd w cn vnish only t isolted points by the definition of weight function). But, since x i re roots of G n for i = 1, 2,..., n, the qudrture formul gives I n (G 2 n) = w i G 2 n(x i ) = 0. () Compring (2) nd (), we see tht the qudrture formul is not exct for G 2 n. 6
7 In the proof of the bove theorem, we implicitly used tht G n hs exctly n simple roots, ll of which re inside the intervl (, b). This property is estblished rigorously in the following lemm. Lemm 1. Let w be weight function on [, b], let n be positive integer, nd let G 0, G 1,..., G n be fmily of polynomils such tht the degree of G k is equl to k for ech k = 0, 1,..., n, nd (G i, G j ) = 0 if i j, where the sclr product is defined by (1). Then for ech k = 0, 1, 2,..., n, the polynomil G k hs exctly k rel roots which re simple nd lie in the intervl (, b). Proof: First we will prove the existence of rel roots for G k, nd then we will count those roots. Step 1: Existence of rel roots of G k. Since the degree of G 0 is zero, it follows tht G 0 (x) = α x [, b] for some constnt α 0. (If you don t understnd why α 0, look t pge 87 of the book.) Then, for k 1, 0 = G 0 (x) G k (x) w(x) dx = α G k (x) w(x) dx, where the first equlity holds due to orthogonlity. However, since w is positive on [, b] except possibly being equl to 0 t isolted points, nd the integrl G k(x) w(x) dx is equl to zero, we conclude tht the function G k must chnge sign in (, b). Since G k is continuous function (ll polynomils re continuous functions), the Intermedite Vlue Theorem gurntees tht G k hs rel root somewhere in (, b). Step 2: Counting the roots of G k. Suppose tht G k chnges sign t exctly j points r 1, r 2,..., r j in (, b) such tht < r 1 < r 2 < < r j < b. Without loss of generlity, ssume tht G k (x) > 0 for x (, r 1 ). Then G k lterntes sign on (r 1, r 2 ), (r 2, r ),..., (r j, b). Define the uxiliry function Z(x) := ( 1) j j (x r j ) V j. By construction, Z(x) nd G k (x) hve the sme sign for ll x [, b]. From this it follows tht Z(x) G k (x) w(x) dx > 0. (4) 7
8 Suppose tht j < k. Since the polynomils G 0, G 1, G 1,..., G j form bsis for V j, there exist constnts z 0, z 1, z 2,..., z j such tht Z(x) = j z i G j (x). Substituting this expression into (4), we obtin [ b j ] Z(x) G k (x) w(x) dx = z i G j (x) G k (x) w(x) dx = j z i G j (x) G k (x) w(x) dx = 0, where we hve used tht j < k, which implies tht the polynomils G j nd G k re orthogonl. The lst equlity clerly contrdicts (4)! Hence the ssumption tht j < k ws wrong, i.e., we must hve tht j k. However, since the degree of the polynomil G k is k, G k cnnot hve more thn k roots, which implies tht j = k. Exmple: As n exmple of ppliction of the bove Theorem, let us rederive the qudrture formul we derived in clss, nmely, 1 ( f(x) dx f 1 ) ( ) 1 + f. 1 This formul cn be obtined by using the Legendre polynomils P k defined on pge. Tke the polynomils P 0 (x) = 1, P 1 (x) = x, nd P 2 (x) = x 2 1, which re n orthogonl bsis in the liner spce V 2 ( 1, 1; w(x) 1). Since n = 2, we expect tht the formul tht we will obtin will hve degree of precision 2n 1 = 2(2) 1 =. As in the Theorem bove, define x 1 = 1 nd x 2 = 1 to be the zeros of the polynomil P 2, define the polynomils (of degree n 1 = 2 1 = 1) nd the weights w 1 = 1 1 L 1 (x) w(x) dx = L 1 (x) = x x 2 x 1 x 2, L 2 (x) = x x 1 x 2 x 1, 1 1 x x 2 x 1 x 2 1 dx = 1 x 1 x 2 nd, similrly, w 2 = 1 1 L 2(x) w(x) dx = = 1, to obtin I 2 (f) = 2 w i f(x i ) = f ( 1 ) ( ) 1 + f ( ) x x 2x. x= 1 = 1, As n exercise, check tht the degree of ccurcy of this qudrture formul is indeed. 8
The 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 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 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 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 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 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 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 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 informationBest Approximation in the 2norm
Jim Lmbers MAT 77 Fll Semester 111 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More 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 informationMath 4310 Solutions to homework 1 Due 9/1/16
Mth 4310 Solutions to homework 1 Due 9/1/16 1. Use the Eucliden lgorithm to find the following gretest common divisors. () gcd(252, 180) = 36 (b) gcd(513, 187) = 1 (c) gcd(7684, 4148) = 68 252 = 180 1
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 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 information20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
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 informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationNumerical Analysis. Doron Levy. Department of Mathematics Stanford University
Numericl Anlysis Doron Levy Deprtment of Mthemtics Stnford University December 1, 2005 D. Levy Prefce i D. Levy CONTENTS Contents Prefce i 1 Introduction 1 2 Interpoltion 2 2.1 Wht is Interpoltion?............................
More 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 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 informationSturmLiouville Eigenvalue problem: Let p(x) > 0, q(x) 0, r(x) 0 in I = (a, b). Here we assume b > a. Let X C 2 1
Ch.4. INTEGRAL EQUATIONS AND GREEN S FUNCTIONS Ronld B Guenther nd John W Lee, Prtil Differentil Equtions of Mthemticl Physics nd Integrl Equtions. Hildebrnd, Methods of Applied Mthemtics, second edition
More informationNotes on length and conformal metrics
Notes on length nd conforml metrics We recll how to mesure the Eucliden distnce of n rc in the plne. Let α : [, b] R 2 be smooth (C ) rc. Tht is α(t) (x(t), y(t)) where x(t) nd y(t) re smooth rel vlued
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 informationMath Solutions to homework 1
Mth 75  Solutions to homework Cédric De Groote October 5, 07 Problem, prt : This problem explores the reltionship between norms nd inner products Let X be rel vector spce ) Suppose tht is norm on X tht
More informationMath 61CM  Solutions to homework 9
Mth 61CM  Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ
More informationBases for Vector Spaces
Bses for Vector Spces 22625 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
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 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 informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly welldefined, is too restrictive for mny purposes; there re functions which
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 informationElementary Linear Algebra
Elementry Liner Algebr Anton & Rorres, 1 th Edition Lecture Set 5 Chpter 4: Prt II Generl Vector Spces 163 คณ ตศาสตร ว ศวกรรม 3 สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา 1/2555 163 คณตศาสตรวศวกรรม 3 สาขาวชาวศวกรรมคอมพวเตอร
More informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
More 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 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 informationDEFINITION The inner product of two functions f 1 and f 2 on an interval [a, b] is the number. ( f 1, f 2 ) b DEFINITION 11.1.
398 CHAPTER 11 ORTHOGONAL FUNCTIONS AND FOURIER SERIES 11.1 ORTHOGONAL FUNCTIONS REVIEW MATERIAL The notions of generlized vectors nd vector spces cn e found in ny liner lger text. INTRODUCTION The concepts
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 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 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 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 informationMATH 174A: PROBLEM SET 5. Suggested Solution
MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous relvlued function on I), nd let L 1 (I) denote the completion
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 informationInterpolation. Gaussian Quadrature. September 25, 2011
Gussin Qudrture September 25, 2011 Approximtion of integrls Approximtion of integrls by qudrture Mny definite integrls cnnot be computed in closed form, nd must be pproximted numericlly. Bsic building
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 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 informationp(x) = 3x 3 + x n 3 k=0 so the right hand side of the equality we have to show is obtained for r = b 0, s = b 1 and 2n 3 b k x k, q 2n 3 (x) =
Norwegin University of Science nd Technology Deprtment of Mthemticl Sciences Pge 1 of 5 Contct during the exm: Elen Celledoni, tlf. 73593541, cell phone 48238584 PLESE NOTE: this solution is for the students
More informationLecture 3. Limits of Functions and Continuity
Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
More information(4.1) D r v(t) ω(t, v(t))
1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution
More informationpadic Egyptian Fractions
padic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Setup 3 4 pgreedy Algorithm 5 5 pegyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction
More informationPresentation Problems 5
Presenttion Problems 5 21355 A For these problems, ssume ll sets re subsets of R unless otherwise specified. 1. Let P nd Q be prtitions of [, b] such tht P Q. Then U(f, P ) U(f, Q) nd L(f, P ) L(f, Q).
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 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 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 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 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 informationConvex Sets and Functions
B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line
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 informationLecture 3 ( ) (translated and slightly adapted from lecture notes by Martin Klazar)
Lecture 3 (5.3.2018) (trnslted nd slightly dpted from lecture notes by Mrtin Klzr) Riemnn integrl Now we define precisely the concept of the re, in prticulr, the re of figure U(, b, f) under the grph of
More informationLecture Notes: Orthogonal Polynomials, Gaussian Quadrature, and Integral Equations
18330 Lecture Notes: Orthogonl Polynomils, Gussin Qudrture, nd Integrl Equtions Homer Reid My 1, 2014 In the previous set of notes we rrived t the definition of Chebyshev polynomils T n (x) vi the following
More informationReview of Riemann Integral
1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.
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 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 informationMATRICES AND VECTORS SPACE
MATRICES AND VECTORS SPACE MATRICES AND MATRIX OPERATIONS SYSTEM OF LINEAR EQUATIONS DETERMINANTS VECTORS IN SPACE AND SPACE GENERAL VECTOR SPACES INNER PRODUCT SPACES EIGENVALUES, EIGENVECTORS LINEAR
More informationInnerproduct spaces
Innerproduct spces Definition: Let V be rel or complex liner spce over F (here R or C). An inner product is n opertion between two elements of V which results in sclr. It is denoted by u, v nd stisfies:
More informationPhil Wertheimer UMD Math Qualifying Exam Solutions Analysis  January, 2015
Problem 1 Let m denote the Lebesgue mesure restricted to the compct intervl [, b]. () Prove tht function f defined on the compct intervl [, b] is Lipschitz if nd only if there is constct c nd function
More informationLECTURE 3. Orthogonal Functions. n X. It should be noted, however, that the vectors f i need not be orthogonal nor need they have unit length for
ECTURE 3 Orthogonl Functions 1. Orthogonl Bses The pproprite setting for our iscussion of orthogonl functions is tht of liner lgebr. So let me recll some relevnt fcts bout nite imensionl vector spces.
More informationHilbert Spaces. Chapter Inner product spaces
Chpter 4 Hilbert Spces 4.1 Inner product spces In the following we will discuss both complex nd rel vector spces. With L denoting either R or C we recll tht vector spce over L is set E equipped with ddition,
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 informationMTH 5102 Linear Algebra Practice Exam 1  Solutions Feb. 9, 2016
Nme (Lst nme, First nme): MTH 502 Liner Algebr Prctice Exm  Solutions Feb 9, 206 Exm Instructions: You hve hour & 0 minutes to complete the exm There re totl of 6 problems You must show your work Prtil
More informationMath 270A: Numerical Linear Algebra
Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner
More information1. On some properties of definite integrals. We prove
This short collection of notes is intended to complement the textbook Anlisi Mtemtic 2 by Crl Mdern, published by Città Studi Editore, [M]. We refer to [M] for nottion nd the logicl stremline of the rguments.
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 information1.3 The Lemma of DuBoisReymond
28 CHAPTER 1. INDIRECT METHODS 1.3 The Lemm of DuBoisReymond We needed extr regulrity to integrte by prts nd obtin the Euler Lgrnge eqution. The following result shows tht, t lest sometimes, the extr
More informationChapter 3. Vector Spaces
3.4 Liner Trnsformtions 1 Chpter 3. Vector Spces 3.4 Liner Trnsformtions Note. We hve lredy studied liner trnsformtions from R n into R m. Now we look t liner trnsformtions from one generl vector spce
More informationf(x)dx . Show that there 1, 0 < x 1 does not exist a differentiable function g : [ 1, 1] R such that g (x) = f(x) for all
3 Definite Integrl 3.1 Introduction In school one comes cross the definition of the integrl of rel vlued function defined on closed nd bounded intervl [, b] between the limits nd b, i.e., f(x)dx s the
More informationAdvanced Calculus: MATH 410 Uniform Convergence of Functions Professor David Levermore 11 December 2015
Advnced Clculus: MATH 410 Uniform Convergence of Functions Professor Dvid Levermore 11 December 2015 12. Sequences of Functions We now explore two notions of wht it mens for sequence of functions {f n
More informationThe RiemannLebesgue Lemma
Physics 215 Winter 218 The RiemnnLebesgue Lemm The Riemnn Lebesgue Lemm is one of the most importnt results of Fourier nlysis nd symptotic nlysis. It hs mny physics pplictions, especilly in studies of
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 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 informationPart IB Numerical Analysis
Prt IB Numericl Anlysis Theorems with proof Bsed on lectures by G. Moore Notes tken by Dexter Chu Lent 2016 These notes re not endorsed by the lecturers, nd I hve modified them (often significntly) fter
More informationSOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL (1 + µ(f n )) f(x) =. But we don t need the exact bound.) Set
SOLUTIONS FOR ANALYSIS QUALIFYING EXAM, FALL 28 Nottion: N {, 2, 3,...}. (Tht is, N.. Let (X, M be mesurble spce with σfinite positive mesure µ. Prove tht there is finite positive mesure ν on (X, M such
More informationg i fφdx dx = x i i=1 is a Hilbert space. We shall, henceforth, abuse notation and write g i f(x) = f
1. Appliction of functionl nlysis to PEs 1.1. Introduction. In this section we give little introduction to prtil differentil equtions. In prticulr we consider the problem u(x) = f(x) x, u(x) = x (1) where
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 informationHERMITEHADAMARDTYPE INEQUALITIES FOR GENERALIZED CONVEX FUNCTIONS
HERMITEHADAMARDTYPE INEQUALITIES FOR GENERALIZED CONVEX FUNCTIONS MIHÁLY BESSENYEI Institute of Mthemtics University of Debrecen H4010 Debrecen, Pf. 12, Hungry EMil: besse@mth.klte.hu HermiteHdmrdtype
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 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 informationQUADRATURE is an oldfashioned 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 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 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 informationARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More informationFarey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University
U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions
More informationIntroduction to Numerical Analysis
Introduction to Numericl Anlysis Doron Levy Deprtment of Mthemtics nd Center for Scientific Computtion nd Mthemticl Modeling (CSCAMM) University of Mrylnd June 14, 2012 D. Levy CONTENTS Contents 1 Introduction
More 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 informationIntroduction to Some Convergence theorems
Lecture Introduction to Some Convergence theorems Fridy 4, 005 Lecturer: Nti Linil Notes: Mukund Nrsimhn nd Chris Ré. Recp Recll tht for f : T C, we hd defined ˆf(r) = π T f(t)e irt dt nd we were trying
More informationSTUDY GUIDE FOR BASIC EXAM
STUDY GUIDE FOR BASIC EXAM BRYON ARAGAM This is prtil list of theorems tht frequently show up on the bsic exm. In mny cses, you my be sked to directly prove one of these theorems or these vrints. There
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 informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils
More informationHandout: Natural deduction for first order logic
MATH 457 Introduction to Mthemticl Logic Spring 2016 Dr Json Rute Hndout: Nturl deduction for first order logic We will extend our nturl deduction rules for sententil logic to first order logic These notes
More informationMatrices and Determinants
Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd GussJordn elimintion to solve systems of liner
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 information