Discrete Leastsquares Approximations


 Tracey Shepherd
 1 years ago
 Views:
Transcription
1
2 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 tht is considered to be the fit best for the dt, in some sense Severl types of fits cn be considered But the one tht is used most in pplictions is the lestsqures fit Mthemticlly, the problem is the following: Discrete LestSqures Approximtion Problem Given set of dt points (x k, y i ), i =,, m, find n lgebric polynomil P n (x) = + x + + n x n (n < m) such tht the error in the lestsqures m sense in minimized; tht is, E = (y i x i n x n i ) is minimum For E to be minimum, we must hve i= E j =, j =,, n Now, E = E = m (y i x i n x n i ) i= m x i (y i x i n x n i ) i= E n = m x n i (y i x i n x n i ) i= Setting these equtions to be zero we hve + m i= m i= x i + x i + m i= m i= x i + + n x i + + n m i= m i= x n i = x n+ i = m i= y i m x i y i i= m i= x n i + m i= x n+ i + + n m i= x n i = m x n i y i i=
3 Set now m x k i = s k, k =,,, n, nd denoting the right hnd side entries s b,, b n, i= the bove eqution cn be written s: s + s + + s n n = b (Note tht m i= x i = s = m) s + s + + s n+ n = b s n + s n+ + s n n = b n This is system of (n + ) equtions in (n + ) unknowns,,, n These equtions re clled Norml Equtions This system now cn be solved to obtin these (n + ) unknowns, provided solution to the system exists We will not show tht this system hs unique solution if x i s re distinct The system cn be written in the following mtrix form: or where Define s = s s s n s s s n+ s n s n+ s n s s s n s s s n+ s n s n+ s n Then the bove system hs the form: s = b n, = = n x x x n x x x n V = x 3 x 3 x n 3 x m x m x n m V T V = b b b b n, b = The mtrix V is known s the Vndermonde mtrix, nd it cn be shown [Exercise] tht it hs full rnk if x i s re distinct In this cse the mtrix S = V T V is symmetric nd positive definite [Exercise] nd is therefore nonsingulr Thus, if x i s re distinct, the eqution S = b hs unique solution b b b n
4 Theorem (Existence nd uniqueness of Discrete LestSqures Solutions) Let (x, y ), (x, y ),, (x n, y n ) be n distinct points Then the discrete lestsqure pproximtion problem hs unique solution LestSqures Approximtion of Function We hve described lestsqures pproximtion to fit set of discrete dt Here we describe continuous lestsqure pproximtions of function f(x) by using polynomils The problem cn be stted s follows: LestSqure Approximtions of Function Using Stndrd Polynomils Given function f(x), continuous on [, b], find polynomil P n (x) of degree t most n: P n (x) = + x + x + + n x n such tht the integrl of the squre of the error is minimized Tht is, is minimized E = [f(x) P n (x)] dx The polynomil P n (x) is clled the LestSqures Polynomil,,, n, we denote this by E(,,, n ) For minimiztion, we must hve E i =, i =,,, n Since E is function of As before, these conditions will give rise to norml system of (n + ) equtions in (n + ) unknowns:,,, n Solution of these equtions will yield the unknowns Setting up the Norml Equtions Since We hve E = [f(x) ( + x + x + + n x n )] dx 3
5 E = E = E n = so, E = [f(x) x x n x n ]dx x[f(x) x x n x n ]dx x n [f(x) x x n x n ]dx Similrly, E = i,, 3,, n dx + x dx + x dx + n x n dx = So, (n + ) norml equtions in this cse re: x i dx+ x i+ dx+ x i+ dx + n x i+n dx = f(x)dx x i dx, i = i = : i = : dx + x dx + x dx + + n x n dx = x dx + x dx + 3 x 3 dx + + n x n dx = f(x) xf(x)dx i = n : Denote x n dx + x n+ dx + x n+ dx + + n x n dx = x n f(x)dx x i dx = s i, i =,,, 3,, n, nd b i = Then the bove (n + ) equtions cn be written s x i f(x)dx, i =,,, n s + s + s + + n s n = b s + s + s n+ s n+ = b s n + s n+ + s n+ + + n s n = b n 4
6 or in mtrix nottion s s s s n s s s 3 s n+ s n s n+ s n n = b b b n Denote S = (s ii ), = n, b = b b b n Then we hve the liner system: S = b The solution of these equtions will yield the coefficients,,, n of the lestsqures polynomil P n (x) A Specil Cse: Let the intervl be [, ] Then s i = x i dx = i +, Thus, in this cse the mtrix of the norml equtions i =,,, n n S = 3 n + n n+ n + n + n which is Hilbert Mtrix It is wellknown to be illconditioned Algorithm: LestSqures Approximtion using Polynomils Inputs: (i) f(x)  A continuous function on [, b] (ii) n  The degree of the desired lestsqure polynomil 5
7 Output: The coefficients,,, n of the desired lestsqures polynomil: P n (x) = + x + + n x n Step Compute s, s,, s n For i =,, n do End s i = x i dx Step Compute b, b,, b n : For i =,,, n do b i = End x i f(x)dx Step 3 Form the mtrix S nd the vector b s s s n s s s n+ S = s n s n+ s n b = b b b n Step 4 Solve the (n + ) (n + ) system of equtions: S = b, where = n Exmple Find Liner nd Qudrtic lestsqures pproximtions to f(x) = e x on [, ] 6
8 Liner Approximtion: n = ; P (x) = + x s = dx = [ ] x s = xdx = = ( ) = [ ] x s = x 3 dx = = ( ) 3 3 = 3 3 ( ) ( ) s s Thus, S = = s s 3 b = b = e x dx = e e 354 e x xdx = e 7358 The norml system of equtions is: ( 3 ) ( ) = ( b b ) This gives = 75, = 37 The liner lestsqures polynomil P (x) = x Check Accurcy: P (5) = 77 e 5 = 6487 Reltive Error = 453 7
9 Qudrtic Fitting: n = P (x) = + x + x s =, s =, s = 3 s 3 = [ ] x 4 x3 dx = = 4 s 4 = [ ] x 5 x4 dx = = 5 5 b = b = b = The system of norml equtions is: e x dx = e e 354 xe x dx = e 7358 x e x dx = e 5 e 8789 = The solution is: = 9963, = 37, = The qudrtic lestsqures polynomil P (x) = x x 8
10 Check the ccurcy: P (5) = 6889 e 5 = 6487 Reltive error P (5) e 5 e 5 = = 4 Exmple Find liner nd Qudrtic lestsqures polynomil pproximtion to f(x) = x + 5x + 6 in [, ] Liner Fit: b = b = The norml equtions re: 3 P (x) = + x s = s = s = dx = xdx = x dx = 3 (x + 5x + 6)dx = = 53 6 x(x + 5x + 6)dx = ( = = 59 ) = (x 3 + 5x + 6x)dx ] = = 6 The liner lest squres polynomil P (x) = x Check Accurcy: f(5) = 875; P (5) =
11 Reltive error: = 95 Qudrtic LestSqure Approximtion: P (x) = + x + x b = 53 6, b = 59 b = x (x + 5x + 6)dx = S = (x 4 + 5x 3 + 6x )dx = = 69 The solution of the liner system is: = 6, = 5, = P (x) = 6 + 5x + x (Exct) Use of Orthogonl Polynomils in Lestsqures Approximtions The lestsqures pproximtion using polynomils, s described bove, is not numericlly effective; since the system mtrix S of norml equtions is very often illconditioned For exmple, when the intervl is [,], we hve seen tht S is Hilbert mtrix, which is notoriously illconditioned for even modest vlues of n When n = 5, the condition number of this mtrix = cond(s) = O( 5 ) Such computtions cn, however, be mde computtionlly effective by using specil type of polynomils, clled orthogonl polynomils Definition The set of functions φ, φ,, φ n is clled set of orthogonl functions, with respect to weight function w(x), if { if i j w(x)φ j (x)φ i (x)dx = if i = j where C j is rel positive number Furthermore, if C j =, j =,,, n, then the orthogonl set is clled n orthonorml set Using this interesting property, lestsqures computtions cn be more numericlly effective, s shown below Without ny loss of generlity, let s ssume tht w(x) = Ide: The ide is to find n pproximtion of f(x) on [, b] by mens of polynomil of the form P n (x) = φ (x) + φ (x) + + n φ n (x), where {φ n } n k= is set of orthogonl polynomils Tht is, the bsis for generting P n(x) in this cse is set of orthogonl polynomils C j
12 Lestsqures Approximtion of Function Using Orthogonl Polynomils Given f(x), continuous on [, b], find,,, n using polynomil of the form: P n (x) = φ (x) + φ (x) + + n φ n (x), where {φ k (x)} n k= is given set of orthogonl polynomils on [, b], such tht the error function: is minimized As before, we set E(,,, n ) = [f(x) ( φ (x) + n φ n (x))] dx Now E i =, i =,,, n E = Setting this equl to zero, we get φ (x)[f(x) φ (x) φ (x) n φ n (x)]dx φ (x)f(x)dx = Since, {φ k (x)} n k= is n orthogonl set, we hve, nd ( φ (x) + + n φ n (x))φ (x)dx φ (x) dx = C, φ (x)φ i (x) dx =, i Applying the bove orthogonl property, we see from bove tht Tht is, φ (x)f(x)dx = C
13 = φ (x)f(x)dx C Similrly, E = φ (x)[f(x) φ (x) φ (x) n φ n (x)]dx The orthogonl property of {φ j (x)} n j= implies tht so, setting E =, we get φ (x) = C nd φ (x)φ i (x) =, i, = φ (x)f(x)dx C In generl, k = φ k (x)f(x)dx, k =,,, n, C k where C k = φ k(x)dx Expresions for k with Weight Function w(x) If the weight function w(x) is included, we obtin k = w(x)f(x)φ k (x)dx, k =,, n C k
14 Algorithm: LestSqures Approximtion Using Orthogonl Polynomils Inputs: f(x)  A continuous function on [, b] w(x)  A weight function (n integrble function on [, b]) {φ k (x)} n k=  A set of n orthogonl functions on [, b] Output: The coefficients,,, n such tht is minimized w(x)[f(x) φ (x) φ (x) n φ n (x)] dx Step Compute C k, k =,,, n s follows: For k =,,,, n do End C k = w(x)φ k(x)dx Step Compute k, k =,, n s follows: For k =,,,, n do k = w(x)f(x)φ k (x)dx C k End LestSqures Approximtion Using Legendre s Polynomils Recll tht the Legendre Polynomils {φ k (x)} re given by φ (x) = φ (x) = x φ (x) = x 3 φ 3 (x) = x x etc re orthogonl polynomils on [, ], with respect to the weight function w(x) = If these polynomils re used for lestsqures pproximtion, then it is esy to see tht 3
15 C = C = C = φ (x)dx = φ (x)dx = φ (x)dx = dx = x dx = ( 3 x ) dx = nd so on Exmple: Find liner nd qudrtic lestsqures pproximtion to f(x) = e x using Legendre polynomils Liner Approximtion: P (x) = φ (x) + φ (x) φ (x) =, φ (x) = x C = φ (x)dx = = φ (x)e x dx C The liner lestsqures polynomil dx = [x] = So, = e x dx = [ex ] = [ x C = φ (x)dx = x 3 dx = 3 = 3 xe x dx = 3 [ ] = 3 e e ( e ) e ] P (x) = φ (x) + φ (x) = [ e ] + 3 e e x = = 3 Accurcy Check: P (5) = e 5 = 6487 [ e ] + 3 e e 5 = 77 Reltive error: = 475 4
16 Qudrtic Approximtion: P (x) = φ (x) + φ (x) + φ (x) = ( e ), = 3 e e C = φ (x)dx = = ( x x3 3 + x = e x φ (x)dx C = 45 8 (x 3 ) dx ) ( e x x ) dx 3 = 8 45 = e 7 e Qudrtic lestsqures polynomil: P (x) = ( e ) + 3e ( e x + e 7 ) ( x ) e 3 Accurcy check: Reltive error P n (5) = 5868 e 5 = = 375 Compre this reltive error with tht obtined erlier with n nonorthogonl polynomil of degree 5
17 Chebyshev polynomils: Another wonderful fmily of orthogonl polynomils Definition: The set of polynomils defined by T n (x) = cos[n rccos x], n on [, ] re clled the Chebyshev polynomils To see tht T n (x) is polynomil of degree n in our fmilir form, we derive recursive reltion by noting tht T (x) = (A polynomil of degree zero) T (x) = x (A polynomil of degree ) A Recursive Reltion for Generting Chebyshev Polynomils: Substitute θ = rc cos x Then, T n (x) = cos(nθ), θ π T n+ (x) = cos(n + )θ = cos nθ cos θ sin nθ sin θ T n (x) = cos(n )θ = cos nθ cos θ + sin nθ sin θ Adding the lst two equtions, we obtin T n+ (x) + T n (x) = cos nθ cos θ The right hnd side still does not look like polynomil in x But note tht cos θ = x So, or T n+ (x) = cos nθ cos θ T n (x) = x cos(n cos rc x) T n (x) = xt n (x) T n (x) T n+ (x) = xt n (x) T n (x), n Using this recursive reltion, the Chebyshev polynomils of the succesive degrees cn be generted n = : T (x) = xt (x) T (x) = x n = : T 3 (x) = xt (x) T (x) = x(x ) x = 4x 3 3x nd so on 6
18 The orthogonl property of the Chebyshev polynomils We now show tht Chebyshev polynomils re orthogonl with respect to the weight function w(x) =, in the intervl [, ] x To demonstrte the orthogonl property of these polynomils, consider = = T m (x)t n (x)dx, m n x π = = = cos(rccos x) cos(n rccos x) x dx cos mθ cos nθdθ ( By chnging the vrible from x to θ with substitution of rccosx = θ) π [ cos(m + n)θdθ + sin(m + n)θ (m + n) ] π π + cos(m n)θdθ [ sin(m n)θ (m n) ] π Similrly, it cn be shown [Exercise] tht Summrizing: T n(x)dx x = π for n Orthogonl Property of the Chebyshev Polynomils T m (x)t n (x) dx = x if m n π if m = n The LestSqure Approximtion using Chebyshev Polynomils 7
19 As before, the Chebyshev polynomils cn be used to find lestsqures pproximtions to function f(x) s stted below The lestsqures pproximting polynomil P n (x) of f(x) using Chebyshev polynomils is given by: P n (x) = C T (x) + C T (x) + + C n T n where nd C i = π f(x)t i (x)dx, i =,, n x C = π f(x)dx x Find liner lestsqures pproximtion of f(x) = e x using Chebyshev poly Exmple: nomils Here P (x) = φ (x) + φ i (x) = T (x) + T (x) = + x, where Thus, P (x) = x Check the ccurcy: = π = π e x dx x 66 xe x dx 33 x P (5) = 975; e 5 = 6487 Reltive error = 4 Monic Chebyshev Polynomils Note tht T k (x) is Chebyshev polynomil of degree k with the leding coefficient k, k Thus we cn generte set of monic Chebyshev polynomils from the polynomils T k (x) s follows: 8
20 The Monic Chebyshev Polynomils, T k (x), re then given by T (x) =, T k (x) = k T k(x), k The k zeros of T k (x) re esily clculted [Exercise]: ( ) j x j = cos k π, j =,,, k The mximum or minimum vlues of T k (x) occur t x j = cos T k ( x j ) = ()j, j =,,, k k ( ) jπ, nd k Polynomil Approximtions with Chebyshev the polynomils: As seen bove the Chebyshev polynomils cn, of course, be used to find lestsqures polynomil pproximtions However, these polynomils hve severl other wonderful polynomil pproximtion properties Some of them re stted below The mximum bsolute vlue of ny monic polynomil of degree n over [, ] is lwys greter thn or equl to tht of T n (x) over the sme intervl; which is, by the lst property, n Minimx Property of the Chebyshev Polynomils If P n (x) is ny monic polynomil of degree n, then = mx T n n (x) mx P n (x) x [,] x [,] Moreover, this hppens when P n (x) = T n (x) Proof: By contrdiction [Exercise] 9
21 Choosing the interpolting nodes with the Chebyshev Zeros Recll tht error in polynomil interpoltion by polynomil P n (x) of degree t most n is given by where Ψ(x) = (x x )(x x ) (x x n ) E = f(x) P (x) = f n+ (ξ) (n + )! Ψ(x), The question is: How to choose these (n + ) nodes x, x,, x n so tht Ψ(x) is minimized in [, ]? The nswer cn be given from the lstmentioned property of the monic Chebyshev polynomils Note tht Ψ(x) is monic polynomil of degree (n + ) So, by the minimx property mx T n+ (x) mx Ψ(x) x [,] x [,] Tht is, the mximum vlue of ψ(x) is smllest when x, x,, x n re chosen s the (n + ) zeros of T n+ (x) nd this mximum vlue is n Choosing the Nodes for Minimizing Polynomil Interpoltion error To minimize the polynomil interpoltion error, choose the nodes x, x,, x n s the (n+) zeros of the (n + )th degree monic Chebyshev polynomil Note (Working with n rbitrry intervl) If the intervl is [, b], different from [, ], then, the zeros of T n+ (x) need to be shifted by using the trnsformtion: x = [(b )x + ( + b)] Exmple Let the interpolting polynomil be of degree t most nd the intervl be [5, ] The three zeros of T 3 (x) in [, ] re given by x = cos π 6, x = cos π, nd x 3 = cos 5 6 π These zeros re to be shifted using trnsformtion: x new = [( 5) x i + ( + 5)]
22 Use of Chebyshev Polynomils to Economize Power Series Power Series Economiztion Let P n (x) = + x+ + n x n be polynomil of degree n obtined by truncting power series expnsion of continuous function on [, b] The problem is to find polynomil P r (x) of degree r (< n) such tht P n (x) P r (x) < ɛ, where ɛ is tolernce supplied by users The problem is esily solved by using the Minimx Property of the Chebyshev polynomils First note tht n P n (x) P n (x) is monic polynomil So, by the minimx property, we hve Thus, if we choose mx P n (x) P n (x) n mx T n (x) = n n P n (x) = P n (x) n Tn (x), then the minimum vlue of mx P n (x) P n (x) = n n n If this quntity,, plus error due to the trunction of the power series is within the n permissible tolernce ɛ, we cn then repet the process by constructing P n (x) from P n (x) s bove The process cn be continued until the ccumulted error exceeds the tolernce ɛ So, the process cn be summrized s follows: Power Series Economiztion Process by Chebyshev Polynomils Obtin P n (x) = + x n + + n x n by truncting the power series expnsion of f(x) Find the error of trunction E P T Compute P n (x): P n (x) = P n (x) n Tn (x) Check if the totl error ( E P T + n n ) is less thn ɛ If so, continue the process by decresing the degree of the polynomils successively until the ccumulted error becomes greter thn ɛ
Orthogonal 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 informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationLecture 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 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 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 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 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 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 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 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 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 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 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 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 informationWeek 10: Riemann integral and its properties
Clculus nd Liner Algebr for Biomedicl Engineering Week 10: Riemnn integrl nd its properties H. Führ, Lehrstuhl A für Mthemtik, RWTH Achen, WS 07 Motivtion: Computing flow from flow rtes 1 We observe the
More informationUNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE
UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence
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 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 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 informationEulerMaclaurin Summation Formula 1
Jnury 9, EulerMclurin Summtion Formul Suppose tht f nd its derivtive re continuous functions on the closed intervl [, b]. Let ψ(x) {x}, where {x} x [x] is the frctionl prt of x. Lemm : If < b nd, b Z,
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 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 informationCalculus and linear algebra for biomedical engineering Week 11: The Riemann integral and its properties
Clculus nd liner lgebr for biomedicl engineering Week 11: The Riemnn integrl nd its properties Hrtmut Führ fuehr@mth.rwthchen.de Lehrstuhl A für Mthemtik, RWTH Achen Jnury 9, 2009 Overview 1 Motivtion:
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 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 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 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 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 informationNumerical Integration
Numericl Integrtion Wouter J. Den Hn London School of Economics c 2011 by Wouter J. Den Hn June 3, 2011 Qudrture techniques I = f (x)dx n n w i f (x i ) = w i f i i=1 i=1 Nodes: x i Weights: w i Qudrture
More 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 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 informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
More informationChapter 3 MATRIX. In this chapter: 3.1 MATRIX NOTATION AND TERMINOLOGY
Chpter 3 MTRIX In this chpter: Definition nd terms Specil Mtrices Mtrix Opertion: Trnspose, Equlity, Sum, Difference, Sclr Multipliction, Mtrix Multipliction, Determinnt, Inverse ppliction of Mtrix in
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 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 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 informationQuantum Physics II (8.05) Fall 2013 Assignment 2
Quntum Physics II (8.05) Fll 2013 Assignment 2 Msschusetts Institute of Technology Physics Deprtment Due Fridy September 20, 2013 September 13, 2013 3:00 pm Suggested Reding Continued from lst week: 1.
More informationMath 113 Exam 2 Practice
Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This
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 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 informationWe know that if f is a continuous nonnegative function on the interval [a, b], then b
1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going
More information1 Ordinary Differential Equations
1 Ordinry Differentil Equtions 1. Mthemticl Bckground 1..1 Smoothness Definition 1.1 A function f defined on [, b] is continuous t x [, b] if lim x x f(x) = f(x ). Remrk Note tht this implies existence
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 informationSOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014
SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
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 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 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 informationLecture 14 Numerical integration: advanced topics
Lecture 14 Numericl integrtion: dvnced topics Weinn E 1,2 nd Tiejun Li 2 1 Deprtment of Mthemtics, Princeton University, weinn@princeton.edu 2 School of Mthemticl Sciences, Peking University, tieli@pku.edu.cn
More informationk and v = v 1 j + u 3 i + v 2
ORTHOGONAL FUNCTIONS AND FOURIER SERIES Orthogonl functions A function cn e considered to e generliztion of vector. Thus the vector concets like the inner roduct nd orthogonlity of vectors cn e extended
More informationFINALTERM EXAMINATION 2011 Calculus &. Analytical GeometryI
FINALTERM EXAMINATION 011 Clculus &. Anlyticl GeometryI Question No: 1 { Mrks: 1 )  Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive...
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 information0606 ADDITIONAL MATHEMATICS
CAMBRIDGE INTERNATIONAL EXAMINATIONS Interntionl Generl Certificte of Secondry Eduction MARK SCHEME for the October/November 0 series 00 ADDITIONAL MATHEMATICS 00/ Pper, mximum rw mrk 80 This mrk scheme
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More informationMatrix Eigenvalues and Eigenvectors September 13, 2017
Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues
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 information4037 ADDITIONAL MATHEMATICS
CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinry Level MARK SCHEME for the October/November 0 series 07 ADDITIONAL MATHEMATICS 07/ Pper, mximum rw mrk 80 This mrk scheme is published s n id to techers
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 informationMATH 573 FINAL EXAM. May 30, 2007
MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.
More informationQUADRATIC EQUATIONS OBJECTIVE PROBLEMS
QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.
More informationTopic 6b Finite Difference Approximations
/8/8 Course Instructor Dr. Rymond C. Rump Oice: A 7 Pone: (95) 747 6958 E Mil: rcrump@utep.edu Topic 6b Finite Dierence Approximtions EE 486/5 Computtionl Metods in EE Outline Wt re inite dierence pproximtions?
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 informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationCLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED NEWTONCOTES QUADRATURES
Filomt 27:4 (2013) 649 658 DOI 10.2298/FIL1304649M Published by Fculty of Sciences nd Mthemtics University of Niš Serbi Avilble t: http://www.pmf.ni.c.rs/filomt CLOSED EXPRESSIONS FOR COEFFICIENTS IN WEIGHTED
More informationREVIEW Chapter 1 The Real Number System
Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole
More informationSTURMLIOUVILLE THEORY, VARIATIONAL APPROACH
STURMLIOUVILLE THEORY, VARIATIONAL APPROACH XIAOBIAO LIN. Qudrtic functionl nd the EulerJcobi Eqution The purpose of this note is to study the SturmLiouville problem. We use the vritionl problem s
More informationa a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.
Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting
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 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 informationEnergy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon
Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,
More informationMidpoint Approximation
Midpoint Approximtion Sometimes, we need to pproximte n integrl of the form R b f (x)dx nd we cnnot find n ntiderivtive in order to evlute the integrl. Also we my need to evlute R b f (x)dx where we do
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 informationSection 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40
Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since
More informationIf deg(num) deg(denom), then we should use longdivision of polynomials to rewrite: p(x) = s(x) + r(x) q(x), q(x)
Mth 50 The method of prtil frction decomposition (PFD is used to integrte some rtionl functions of the form p(x, where p/q is in lowest terms nd deg(num < deg(denom. q(x If deg(num deg(denom, then we should
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 informationMA Handout 2: Notation and Background Concepts from Analysis
MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,
More informationHomework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.
Homework 4 (1) If f R[, b], show tht f 3 R[, b]. If f + (x) = mx{f(x), 0}, is f + R[, b]? Justify your nswer. (2) Let f be continuous function on [, b] tht is strictly positive except finitely mny points
More informationNumerical Methods I. Olof Widlund Transcribed by Ian Tobasco
Numericl Methods I Olof Widlund Trnscribed by In Tobsco Abstrct. This is prt one of two semester course on numericl methods. The course ws offered in Fll 011 t the Cournt Institute for Mthemticl Sciences,
More informationIntegrals along Curves.
Integrls long Curves. 1. Pth integrls. Let : [, b] R n be continuous function nd let be the imge ([, b]) of. We refer to both nd s curve. If we need to distinguish between the two we cll the function the
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 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 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More 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 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 information4.1. Probability Density Functions
STT 1 4.14. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile  vers  discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of
More informationIntroduction to Determinants. Remarks. Remarks. The determinant applies in the case of square matrices
Introduction to Determinnts Remrks The determinnt pplies in the cse of squre mtrices squre mtrix is nonsingulr if nd only if its determinnt not zero, hence the term determinnt Nonsingulr mtrices re sometimes
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More informationNumerical Analysis Lecture 1 1
Mthemticl Tripos Prt IB: Ester 006 Numericl Anlysis Lecture LU fctoriztion of mtrices. Definition nd pplictions Let A be rel n n mtrix. We sy tht the n n mtrices L nd U re n LU fctoriztion of A if () L
More information7.6 The Use of Definite Integrals in Physics and Engineering
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems
More informationCzechoslovak Mathematical Journal, 55 (130) (2005), , Abbotsford. 1. Introduction
Czechoslovk Mthemticl Journl, 55 (130) (2005), 933 940 ESTIMATES OF THE REMAINDER IN TAYLOR S THEOREM USING THE HENSTOCKKURZWEIL INTEGRAL, Abbotsford (Received Jnury 22, 2003) Abstrct. When relvlued
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 informationChapter 4. Additional Variational Concepts
Chpter 4 Additionl Vritionl Concepts 137 In the previous chpter we considered clculus o vrition problems which hd ixed boundry conditions. Tht is, in one dimension the end point conditions were speciied.
More informationDEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS
3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive
More informationA basic logarithmic inequality, and the logarithmic mean
Notes on Number Theory nd Discrete Mthemtics ISSN 30 532 Vol. 2, 205, No., 3 35 A bsic logrithmic inequlity, nd the logrithmic men József Sándor Deprtment of Mthemtics, BbeşBolyi University Str. Koglnicenu
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationMATH 222 Second Semester Calculus. Fall 2015
MATH Second Semester Clculus Fll 5 Typeset:August, 5 Mth nd Semester Clculus Lecture notes version. (Fll 5) This is self contined set of lecture notes for Mth. The notes were written by Sigurd Angenent,
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 informationVariational Techniques for SturmLiouville Eigenvalue Problems
Vritionl Techniques for SturmLiouville Eigenvlue Problems Vlerie Cormni Deprtment of Mthemtics nd Sttistics University of Nebrsk, Lincoln Lincoln, NE 68588 Emil: vcormni@mth.unl.edu Rolf Ryhm Deprtment
More information