Quadrature Methods for Numerical Integration
|
|
- Gabriella Craig
- 6 years ago
- Views:
Transcription
1 Qudrture Methods for Numericl Itegrtio Toy Sd Istitute for Cle d Secure Eergy Uiversity of Uth April 11, The Need for Numericl Itegrtio Nuemricl itegrtio ims t pproximtig defiite itegrls usig umericl techiques. There re my situtios where umericl itegrtio is eeded. For exmple, severl well defied fuctios do ot hve ti-derivtive, i.e. their ti-derivtive cot be expressed i terms of primitive fuctio. A populr exmple is the fuctio e x2 whose ti-derivtive does ot exist. This fuctio rises i vriety of pplictios such s those relted to propbility d sttistics lyses. Furthermore, my pplictios i sciece d egieerig re represeted by itegrodifferetil equtios tht require specil tretmet for the itegrl terms (e.g. expsio, lierliztio, closure...). Therefore, umericl itegrtio does ot oly provide mes for evlutig itegrls umericlly, but lso grts us the bility to pproximte specil fuctios tht re defied i terms of itegrls. Without loss of geerlity, there re two clsses of problems where umericl itegrtio is eeded. I the first clss, oe wishes to evlute the itegrl of well defied fuctio. I this cse, the itegrd c be evluted vrious poits becuse d umericl itegrtio techiques help defie the optimum umber of these poits s well s their loctios. The secod clss of problems for pplyig umericl itegrtio is foud i differetil equtios the most commo of which re those tht express coservtio priciples. For exmple, the popultio blce equtio, well kow prtil differetil equtio ecoutered i process modelig d biologicl systems, exhibits source terms tht re represeted s itegrls of the solutio vrible (e.g. the umber desity fuctio). The most commo techique for umericl itegrtio is clled qudrture. The recipe for qudrture cosists of three steps 1. Approximte the itegrd by iterpoltig polyomil usig specified umber of poits or odes 2. Substitute the iterpoltig polyomil ito the itegrl 3. Itegrte The resultig qudrture pproximtes the itegrl s summtio of the form f (x)dx = Postdoctorl Fellow. web: emil:sdtoy@gmil.com. w i f (x i ). (1) 1
2 Furthermore, if the odes re selected i specific mer usig orthogol polyomils, the ccurcy of the qudrture formul is substtilly improved s will be show i subsequet sectios. I will begi our study of qudrture methods by reviewig the theory of iterpoltig polyomils. The, I will itroduce qudrture pproximtios for eqully spced odes. This is followed by discussio of the theory of orthogol polyomils. Filly, I will show you how orthogol polyomils c help i improvig the degree of exctess of qudrture pproximtios. 2 Iterpoltig Polyomils Iterpoltig polyomils re used to pproximte rel vlued fuctio f (x) o rel itervl [,b] by polyomil p(x) tht ccurtely represets the fuctio over tht itervl. 2.1 Lgrge Iterpoltio The most commoly used iterpoltig polyomils re kow s Lgrge polyomils. Cosider rel, cotiuous fuctio f (x) o rel itervl [,b]. Also, ssume tht the vlues of this fuctio re kow t fiite umber of poits x i [,b], the, the Lgrge iterpoltig polyomil p (x) of order is the lowest degree polyomil such tht p (x i ) = f (x i ). Assumig tht the itervl is subdivided ito ( + 1) poits (x 0 < x 1 < < x [,b]), the p (x) is give by p (x) = i=0 where L i (x) is the i-th Lgrge iterpoltig polyomil defied s The first three Lgrge polyomils re give by f (x i )L i (x), (2) L i (x) = j=0, j i (x x j) j=0, j i (x i x j ). (3) L 0 (x) = x x 1 x 0 x 1 x x 2 x 0 x 2 x x 3 x 0 x 3 L 1 (x) = x x 0 x 1 x 0 x x 2 x 1 x 2 x x 3 x 1 x 3 (4) L 2 (x) = x x 0 x 2 x 0 x x 1 x 2 x 1 x x 3 x 2 x Exmple Fid the iterpoltig polyomil for the followig set of poits The fudmetl lgrgi polyomils re x 1.5 f (x 0 ) = x 1 = 0.75 f (x 1 ) = x 2 = 0 f (x 2 ) = 0 x 3 = 0.75 f (x 3 ) = x 4 = 1.5 f (x 4 ) = (5) 2
3 L 0 (x) = x x 1 x x 2 x x 3 x x 4 = 1 x(2x 3)(4x 3)(4x + 3), (6) x 0 x 1 x 0 x 2 x 0 x 3 x 0 x L 1 (x) = x x 0 x x 2 x x 3 x x 4 = 8 x(2x 3)(2x + 3)(4x 3), (7) x 1 x 0 x 1 x 2 x 1 x 3 x 1 x L 2 (x) = x x 0 x x 1 x x 3 x x 4 = 3 (2x + 3)(4x + 3)(4x 3)(2x 3), (8) x 2 x 0 x 2 x 1 x 2 x 3 x 2 x Iterpoltio Error. (9) The remider or iterpoltio error for usig the Lgrge iterpoltig polyomils is give by where ξ (,b). R (x) = f (x) P (x) = f (ξ ) ( + 1)! j=0 (x x j ); ξ ξ (x), (10) 3 Qudrture Numericl itegrtio is bsed o the ide of first pproximtig the itegrd usig iterpoltig polyomil d the itegrtig the resultig polyomil. Assume tht we wish to clculte the followig itegrl Let f (x)dx. (11) f (x) p 1 (x) = f (x i )L i (x), (12) deote the Lgrge iterpoltig polyomil for f (x). This is polyomil of order ( 1) give the ode pproximtio. Note tht L i (x) is give by We ow substitute the iterpoltig polyomil ito the itegrl where f (x)dx L i (x) = j=1, j i (x x j) j=1, j i (x i x j ). (13) f (x i )L i (x)dx = w i f (x i ) L i (x)dx = w i f (x i ), (14) L i (x)dx. (15) This formul is kow s qudrture pproximtio for itegrl. The poits x i re referred to s the bscisse or odes while w i re clled the weights. 3
4 3.1 Degree of Exctess A qudrture pproximtio is sid to hve degree of exctess m if it is exct whe f (x) is polyomil of degree less th or equl to m, while it is ot exct for polyomil of order m + 1. As rule of thumb, y iterpoltory qudrture formul tht uses distict odes hs degree of exctess of t lest Gussi Qudrture Gussi qudrture ims t improvig the degree of exctess of the qudrture pproximtio by crefully selectig the bscisse of the qudrture formul. It lso geerlizes the cocept of qudrture to itegrls of the form f (x)w(x)dx, (16) where w(x) is weight fuctio. A weight fuctio w(x) is positive mesurble fuctio o domi Ω such tht It lso hs the followig property Usig poit qudrture rule for Eq. (16), we hve f (x)dx = w i f (x i ); w(x) : Ω R +. (17) x w(x)dx < ; = 0,1,2,... (18) w i L i (x)w(x)dx; x 1 < x 2 < < x b. (19) Regrdless of how we choose the bscisse, this qudrture pproximtio hs degree of exctess t lest equl to ( 1). With Gussi qudrture, oe c chieve degree of exctess of more th twice! Before we see how how this is possible, we ll hve to go through some spects of the theory of orthogol polyomils. 4 Orthogolity Two rel fuctio f (x) d g(x) re sid to be orthogol if their ier product is zero. The ier product of two fuctios, o itervl [,b], is defied by the followig itegrl The, f (x) d g(x)re orthogol if The bove covolutio is lso kow s ier product. f,g f (x)g(x)dx. (20) f,g = 0. (21) 4
5 4.1 Orthogol Polyomils The ides of orthogol fuctios c be used to costruct set of polyomils tht c be used s bsis spig spce of rel fuctios. As result, every fuctio i tht spce c be writte s lier combitio of the orthogol bsis. But we will ot be cocered with group theory t this poit, d we c proceed to developig orthogol polyomils. Cosider sequece of polyomils p k (x) such tht For exmple, p k (x) = k i=0 α k,i x i ; α k,k = 1, k = 0,1,2, (22) p 0 (x) = α 0, 1, (23) p 1 (x) = α 1,0 + α 1,1 x = x + α 1,0, (24) p 2 (x) = x 2 + α 2,1 x + α 2,0. (25) By settig α k,k = 1, the polyomils re sid to be moic, i.e. the coefficiet of the term with highest order is oe. A sequece of polyomils P = {p m (x); m = 0,1,..., } is sid to be orthogol if { p, p m = 0 m p, p m = 0 = m, (26) or, i compct form p, p m = δ m M, (27) where δ m is the Kroecker delt d M = p, p. If, i dditio, p, p = 1, the the polyomils re sid to be orthoorml. Therefore, orthoorml set of polyomils is ormlized set of orthogol polyomils. Orthoorml polyomils re defied usig the followig compct ottio p, p m = δ m. (28) Oe c defie sequece of orthoorml polyomils q k (x) by ormlizig the orthogol oes s You c immeditely verify tht q k (x) = p k(x) pk, p k. (29) p (x) q,q m = p, p, p m (x) pm, p m = 1 p, p p m, p m p, p m = δ m. (30) hece, q k (x) re orthoorml. 5
6 4.2 Coctructig Orthogol Polyomils O c costruct sequece of orthogol polyomils usig the followig three-term recurrece reltio (TTRR) p 1 (x) = 0, p 0 (x) = 1, (31) p +1 (x) = (x α )p (x) β p 1 (x). The coefficiets c be clculted by usig orthogolity. First, for α, we multiply Eq. (31) by p p p +1 = (x α )p p β p p 1. (32) Next, we itegrte over [,b] p p +1 dx = But, by virtue of orthogolity, we set (x α )p p dx β p p 1 dx. (33) or filly (x α )p p dx 0, (34) xp p dx α p p dx, (35) α = For β, we multiply Eq. (31) by p 1 Agi, by itegrtig over [,b], we hve xp p dx p p dx = xp, p p, p. (36) p 1 p +1 = (x α )p 1 p β p 1 p 1. (37) p 1 p +1 dx = Usig orthogolity, we write (x α )p 1 p dx β p 1 p 1 dx. (38) filly xp 1 p dx β p 1 p 1 dx, (39) β = xp 1p dx p 1p 1 dx = xp 1, p p 1, p 1. (40) 6
7 At this poit, it would be esier to fid simpler form for the term xp 1 tht ppers i the umertor of Eq. (40). First, we observe tht xp 1 is polyomil of order. Also, becuse the polyomils re moic, we hve 1 p xp 1 = i=0 d,i p i (x) q(x) P 1. (41) I other wods, the differece p xp 1 is polyomil of order ( 1) d c be writte s lier combitio of ll the lower order orthogol polyomils. I fct, oe c determie the coefficiets of this lier combitio very esily by usig orthogolity. For exmple, for m 1, we form the followig ier products or 1 p, p m xp 1, p m = i=0 d,i p i, p m, (42) 0 xp 1, p m = d,m p m, p m, (43) so tht At the outset, we c write the followig the, by tkig the ier product, we recover d,m = xp 1, p m p m, p m. (44) xp 1 = p + q(x); q(x) P 1, (45) xp 1, p = p, p + q, p = p, p. (46) By substitutig Eq. (46) ito Eq. (31), the formul for clcultig β is t hd 4.3 Geerliztio β = p, p p 1, p 1. (47) Orthogol polyomils c lso be defied with respect to weight fuctio w(x). Two polyomils re sid to be orthogol with respect to weight fuctio w(x) if ˆ { b 0 m p (x)p m (x)w(x)dx = δ m M = M = m. (48) I similr fshio, the orthogol polyomils c be determied usig the TTRR give i Eq. (31). To clculte the coefficiets α d β, we impose orthogolity with respect to w(x). Strtig with α, we multiply Eq. (31) by p (x) Itegrtig over [,b], we hve p p +1 w = (x α )p p w β p p 1 w. (49) 7
8 p p +1 wdx = d, by virtue of orthogolity, we recover (x α )p p wdx β p p 1 wdx, (50) or filly (x α )p p wdx 0, (51) xp p wdx α p p wdx, (52) α = For β, we multiply Eq. (31) by p 1 xp p wdx p p wdx = xwp, p wp, p. (53) wp 1 p +1 = (x α )wp 1 p β wp 1 p 1, (54) or the filly wp 1 p +1 dx = w(x α )p 1 p dx β wp 1 p 1 dx, (55) xwp 1 p dx β wp 1 p 1 dx, (56) β = xwp 1p dx wp 1p 1 dx = xwp 1, p wp 1, p 1. (57) As we did for the o-weighted cse, we c simplify the umerictor for β by writig We ow form the ier product xw(x)p 1 (x) = w(x)xp 1 (x) = w(x)[p + q]; q P 1. (58) xwp 1, p = wp, p + wp 1, p = wp, p. (59) Upo substitutio ito Eq. (57), we recover the formul for clcultig β s β = wp, p wp 1, p 1. (60) Orthogol fuctios hve my other properties tht re outside the scope of this review. I ll get bck to those t lter occsio, but for ow, we hve eough iformtio to go hed d discuss how orthogol polyomils c be used to improve the degree of exctess of the qudrture pproximtio. 8
9 5 Gussi Qudrture Bck to Gussi qudrture, we sid tht it ims t improvig the degree of exctess of the qudrture pproximtio by crefully selectig the odes of the qudrture formul. For poit qudrture, if the bscisse re selected such tht they coicide with the roots of the correspodig orthogol polyomil p 1 (x), the the qudrture pproximtio hs degree of exctess of (2 1). Let us see how this is possible. Suppose tht f (x) is polyomil of degree m 2 1. The, oe c write f (x) = p (x)q(x) + r(x), (61) where q d r re polyomils of degree 1. By virtue of orthogolity, we hve f (x)w(x)dx = p (x)q(x)w(x)dx + Now the qudrture formul is f (x)w(x)dx = r(x)w(x)dx = r(x)w(x)dx. (62) w i f (x i ). (63) Now, suppose tht the odes re selected such tht they coicide with the roots of p (x), i.e. p (x i ) = 0, i = 1,2,...,. The, d our iterpoltory rule becomes f (x i ) = p (x i )q(x i ) + r(x i ) = 0 + r(x i ) (64) f (x)w(x)dx = w i f (x i ) = w i r(x i ) (65) d thus the pproximtio is exct becuse r(x) is polyomil of degree 1 (remember, -poit qudrture rule is exct for polyomil of degree ( 1)). As you c see, by usig oly odes, d by specificlly choosig those odes s the roots of the th order orthogol polyomil, the the qudrture pproximtio hs degree of exctess of (2 1). 9
Closed Newton-Cotes Integration
Closed Newto-Cotes Itegrtio Jmes Keeslig This documet will discuss Newto-Cotes Itegrtio. Other methods of umericl itegrtio will be discussed i other posts. The other methods will iclude the Trpezoidl Rule,
More informationlecture 16: Introduction to Least Squares Approximation
97 lecture 16: Itroductio to Lest Squres Approximtio.4 Lest squres pproximtio The miimx criterio is ituitive objective for pproximtig fuctio. However, i my cses it is more ppelig (for both computtio d
More informationlecture 24: Gaussian quadrature rules: fundamentals
133 lecture 24: Gussi qudrture rules: fudmetls 3.4 Gussi qudrture It is cler tht the trpezoid rule, b 2 f ()+ f (b), exctly itegrtes lier polyomils, but ot ll qudrtics. I fct, oe c show tht o qudrture
More information0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.
. Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric
More informationThe Definite Integral
The Defiite Itegrl A Riem sum R S (f) is pproximtio to the re uder fuctio f. The true re uder the fuctio is obtied by tkig the it of better d better pproximtios to the re uder f. Here is the forml defiitio,
More information1.3 Continuous Functions and Riemann Sums
mth riem sums, prt 0 Cotiuous Fuctios d Riem Sums I Exmple we sw tht lim Lower() = lim Upper() for the fuctio!! f (x) = + x o [0, ] This is o ccidet It is exmple of the followig theorem THEOREM Let f be
More informationApproximations of Definite Integrals
Approximtios of Defiite Itegrls So fr we hve relied o tiderivtives to evlute res uder curves, work doe by vrible force, volumes of revolutio, etc. More precisely, wheever we hve hd to evlute defiite itegrl
More informationFOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To -periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the
More informationWe will begin by supplying the proof to (a).
(The solutios of problem re mostly from Jeffrey Mudrock s HWs) Problem 1. There re three sttemet from Exmple 5.4 i the textbook for which we will supply proofs. The sttemets re the followig: () The spce
More informationEVALUATING DEFINITE INTEGRALS
Chpter 4 EVALUATING DEFINITE INTEGRALS If the defiite itegrl represets re betwee curve d the x-xis, d if you c fid the re by recogizig the shpe of the regio, the you c evlute the defiite itegrl. Those
More informationReview of the Riemann Integral
Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of
More informationDefinite Integral. The Left and Right Sums
Clculus Li Vs Defiite Itegrl. The Left d Right Sums The defiite itegrl rises from the questio of fidig the re betwee give curve d x-xis o itervl. The re uder curve c be esily clculted if the curve is give
More informationNumerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials
Numericl Solutios of Fredholm Itegrl Equtios Usig erstei Polyomils A. Shiri, M. S. Islm Istitute of Nturl Scieces, Uited Itertiol Uiversity, Dhk-, gldesh Deprtmet of Mthemtics, Uiversity of Dhk, Dhk-,
More informationNumerical Integration - (4.3)
Numericl Itegrtio - (.). Te Degree of Accurcy of Qudrture Formul: Te degree of ccurcy of qudrture formul Qf is te lrgest positive iteger suc tt x k dx Qx k, k,,,...,. Exmple fxdx 9 f f,,. Fid te degree
More information1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2
Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit
More informationChapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
More informationA GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD
Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method
More informationWeek 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:
Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the
More informationTHE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING
OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING MATHEMATICS MA008 Clculus d Lier
More informationLimit of a function:
- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive
More informationINFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1
Appedix A.. Itroductio As discussed i the Chpter 9 o Sequeces d Series, sequece,,...,,... hvig ifiite umber of terms is clled ifiite sequece d its idicted sum, i.e., + + +... + +... is clled ifite series
More informationInfinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex:
Ifiite Series Sequeces: A sequece i defied s fuctio whose domi is the set of positive itegers. Usully it s esier to deote sequece i subscript form rther th fuctio ottio.,, 3, re the terms of the sequece
More informationRiemann Integration. Chapter 1
Mesure, Itegrtio & Rel Alysis. Prelimiry editio. 8 July 2018. 2018 Sheldo Axler 1 Chpter 1 Riem Itegrtio This chpter reviews Riem itegrtio. Riem itegrtio uses rectgles to pproximte res uder grphs. This
More informationThe Weierstrass Approximation Theorem
The Weierstrss Approximtio Theorem Jmes K. Peterso Deprtmet of Biologicl Scieces d Deprtmet of Mthemticl Scieces Clemso Uiversity Februry 26, 2018 Outlie The Wierstrss Approximtio Theorem MtLb Implemettio
More informationB. Examples 1. Finite Sums finite sums are an example of Riemann Sums in which each subinterval has the same length and the same x i
Mth 06 Clculus Sec. 5.: The Defiite Itegrl I. Riem Sums A. Def : Give y=f(x):. Let f e defied o closed itervl[,].. Prtitio [,] ito suitervls[x (i-),x i ] of legth Δx i = x i -x (i-). Let P deote the prtitio
More informationReview of Sections
Review of Sectios.-.6 Mrch 24, 204 Abstrct This is the set of otes tht reviews the mi ides from Chpter coverig sequeces d series. The specific sectios tht we covered re s follows:.: Sequces..2: Series,
More informationNotes 17 Sturm-Liouville Theory
ECE 638 Fll 017 Dvid R. Jckso Notes 17 Sturm-Liouville Theory Notes re from D. R. Wilto, Dept. of ECE 1 Secod-Order Lier Differetil Equtios (SOLDE) A SOLDE hs the form d y dy 0 1 p ( x) + p ( x) + p (
More informationMA123, Chapter 9: Computing some integrals (pp )
MA13, Chpter 9: Computig some itegrls (pp. 189-05) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how
More informationGeneral properties of definite integrals
Roerto s Notes o Itegrl Clculus Chpter 4: Defiite itegrls d the FTC Sectio Geerl properties of defiite itegrls Wht you eed to kow lredy: Wht defiite Riem itegrl is. Wht you c ler here: Some key properties
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 0 FURTHER CALCULUS II. Sequeces d series. Rolle s theorem d me vlue theorems 3. Tlor s d Mcluri s theorems 4. L Hopitl
More informationChapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures
Chpter 5 The Riem Itegrl 5.1 The Riem itegrl Note: 1.5 lectures We ow get to the fudmetl cocept of itegrtio. There is ofte cofusio mog studets of clculus betwee itegrl d tiderivtive. The itegrl is (iformlly)
More informationTaylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best
Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =
More informationNotes on Dirichlet L-functions
Notes o Dirichlet L-fuctios Joth Siegel Mrch 29, 24 Cotets Beroulli Numbers d Beroulli Polyomils 2 L-fuctios 5 2. Chrcters............................... 5 2.2 Diriclet Series.............................
More informationINTEGRATION IN THEORY
CHATER 5 INTEGRATION IN THEORY 5.1 AREA AROXIMATION 5.1.1 SUMMATION NOTATION Fibocci Sequece First, exmple of fmous sequece of umbers. This is commoly ttributed to the mthemtici Fibocci of is, lthough
More informationn 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1
Ifiite Series Some Tests for Divergece d Covergece Divergece Test: If lim u or if the limit does ot exist, the series diverget. + 3 + 4 + 3 EXAMPLE: Show tht the series diverges. = u = + 3 + 4 + 3 + 3
More informationNumerical Integration by using Straight Line Interpolation Formula
Glol Jourl of Pure d Applied Mthemtics. ISSN 0973-1768 Volume 13, Numer 6 (2017), pp. 2123-2132 Reserch Idi Pulictios http://www.ripulictio.com Numericl Itegrtio y usig Stright Lie Iterpoltio Formul Mhesh
More informationRemarks: (a) The Dirac delta is the function zero on the domain R {0}.
Sectio Objective(s): The Dirc s Delt. Mi Properties. Applictios. The Impulse Respose Fuctio. 4.4.. The Dirc Delt. 4.4. Geerlized Sources Defiitio 4.4.. The Dirc delt geerlized fuctio is the limit δ(t)
More informationSome New Iterative Methods Based on Composite Trapezoidal Rule for Solving Nonlinear Equations
Itertiol Jourl of Mthemtics d Sttistics Ivetio (IJMSI) E-ISSN: 31 767 P-ISSN: 31-759 Volume Issue 8 August. 01 PP-01-06 Some New Itertive Methods Bsed o Composite Trpezoidl Rule for Solvig Nolier Equtios
More informationMath 3B Midterm Review
Mth 3B Midterm Review Writte by Victori Kl vtkl@mth.ucsb.edu SH 643u Office Hours: R 11:00 m - 1:00 pm Lst updted /15/015 Here re some short otes o Sectios 7.1-7.8 i your ebook. The best idictio of wht
More information[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.
[ 0 ]. Iequlity eists oly betwee two rel umbers (ot comple umbers).. If be y rel umber the oe d oly oe of there hold.. If, b 0 the b 0, b 0.. (i) b if b 0 (ii) (iii) (iv) b if b b if either b or b b if
More informationConvergence rates of approximate sums of Riemann integrals
Jourl of Approximtio Theory 6 (9 477 49 www.elsevier.com/locte/jt Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsukub, Tsukub Ibrki
More informationVectors. Vectors in Plane ( 2
Vectors Vectors i Ple ( ) The ide bout vector is to represet directiol force Tht mes tht every vector should hve two compoets directio (directiol slope) d mgitude (the legth) I the ple we preset vector
More informationParticle in a Box. and the state function is. In this case, the Hermitian operator. The b.c. restrict us to 0 x a. x A sin for 0 x a, and 0 otherwise
Prticle i Box We must hve me where = 1,,3 Solvig for E, π h E = = where = 1,,3, m 8m d the stte fuctio is x A si for 0 x, d 0 otherwise x ˆ d KE V. m dx I this cse, the Hermiti opertor 0iside the box The
More informationSequence and Series of Functions
6 Sequece d Series of Fuctios 6. Sequece of Fuctios 6.. Poitwise Covergece d Uiform Covergece Let J be itervl i R. Defiitio 6. For ech N, suppose fuctio f : J R is give. The we sy tht sequece (f ) of fuctios
More informationThe Reimann Integral is a formal limit definition of a definite integral
MATH 136 The Reim Itegrl The Reim Itegrl is forml limit defiitio of defiite itegrl cotiuous fuctio f. The costructio is s follows: f ( x) dx for Reim Itegrl: Prtitio [, ] ito suitervls ech hvig the equl
More informationChapter System of Equations
hpter 4.5 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee
More informationMATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n
MATH 04 FINAL SOLUTIONS. ( poits ech) Mrk ech of the followig s True or Flse. No justifictio is required. ) A ubouded sequece c hve o Cuchy subsequece. Flse b) A ifiite uio of Dedekid cuts is Dedekid cut.
More informationOptions: Calculus. O C.1 PG #2, 3b, 4, 5ace O C.2 PG.24 #1 O D PG.28 #2, 3, 4, 5, 7 O E PG #1, 3, 4, 5 O F PG.
O C. PG.-3 #, 3b, 4, 5ce O C. PG.4 # Optios: Clculus O D PG.8 #, 3, 4, 5, 7 O E PG.3-33 #, 3, 4, 5 O F PG.36-37 #, 3 O G. PG.4 #c, 3c O G. PG.43 #, O H PG.49 #, 4, 5, 6, 7, 8, 9, 0 O I. PG.53-54 #5, 8
More information( ) dx ; f ( x ) is height and Δx is
Mth : 6.3 Defiite Itegrls from Riem Sums We just sw tht the exct re ouded y cotiuous fuctio f d the x xis o the itervl x, ws give s A = lim A exct RAM, where is the umer of rectgles i the Rectgulr Approximtio
More information( ) k ( ) 1 T n 1 x = xk. Geometric series obtained directly from the definition. = 1 1 x. See also Scalars 9.1 ADV-1: lim n.
Sclrs-9.0-ADV- Algebric Tricks d Where Tylor Polyomils Come From 207.04.07 A.docx Pge of Algebric tricks ivolvig x. You c use lgebric tricks to simplify workig with the Tylor polyomils of certi fuctios..
More informationCertain sufficient conditions on N, p n, q n k summability of orthogonal series
Avilble olie t www.tjs.com J. Nolier Sci. Appl. 7 014, 7 77 Reserch Article Certi sufficiet coditios o N, p, k summbility of orthogol series Xhevt Z. Krsiqi Deprtmet of Mthemtics d Iformtics, Fculty of
More informationOrthogonal functions - Function Approximation
Orthogol uctios - Fuctio Approimtio - he Problem - Fourier Series - Chebyshev Polyomils he Problem we re tryig to pproimte uctio by other uctio g which cosists o sum over orthogol uctios Φ weighted by
More informationBC Calculus Path to a Five Problems
BC Clculus Pth to Five Problems # Topic Completed U -Substitutio Rule Itegrtio by Prts 3 Prtil Frctios 4 Improper Itegrls 5 Arc Legth 6 Euler s Method 7 Logistic Growth 8 Vectors & Prmetrics 9 Polr Grphig
More informationConvergence rates of approximate sums of Riemann integrals
Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsuku Tsuku Irki 5-857 Jp tski@mth.tsuku.c.jp Keywords : covergece rte; Riem sum; Riem
More informationChapter 2 Infinite Series Page 1 of 9
Chpter Ifiite eries Pge of 9 Chpter : Ifiite eries ectio A Itroductio to Ifiite eries By the ed of this sectio you will be ble to uderstd wht is met by covergece d divergece of ifiite series recogise geometric
More informationSection 6.3: Geometric Sequences
40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul cost-of-livig icrese to keep slries cosistet with ifltio. Suppose, for exmple, recet college grdute fids positio s sles mger erig ul slry of $6,000.
More information2.1.1 Definition The Z-transform of a sequence x [n] is simply defined as (2.1) X re x k re x k r
Z-Trsforms. INTRODUCTION TO Z-TRANSFORM The Z-trsform is coveiet d vluble tool for represetig, lyig d desigig discrete-time sigls d systems. It plys similr role i discrete-time systems to tht which Lplce
More informationPOWER SERIES R. E. SHOWALTER
POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise
More informationMAS221 Analysis, Semester 2 Exercises
MAS22 Alysis, Semester 2 Exercises Srh Whitehouse (Exercises lbelled * my be more demdig.) Chpter Problems: Revisio Questio () Stte the defiitio of covergece of sequece of rel umbers, ( ), to limit. (b)
More information2. Infinite Series 3. Power Series 4. Taylor Series 5. Review Questions and Exercises 6. Sequences and Series with Maple
Chpter II CALCULUS II.4 Sequeces d Series II.4 SEQUENCES AND SERIES Objectives: After the completio of this sectio the studet - should recll the defiitios of the covergece of sequece, d some limits; -
More information n. A Very Interesting Example + + = d. + x3. + 5x4. math 131 power series, part ii 7. One of the first power series we examined was. 2!
mth power series, prt ii 7 A Very Iterestig Emple Oe of the first power series we emied ws! + +! + + +!! + I Emple 58 we used the rtio test to show tht the itervl of covergece ws (, ) Sice the series coverges
More information10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form
0.5 Power Series I the lst three sectios, we ve spet most of tht time tlkig bout how to determie if series is coverget or ot. Now it is time to strt lookig t some specific kids of series d we will evetully
More informationTest Info. Test may change slightly.
9. 9.6 Test Ifo Test my chge slightly. Short swer (0 questios 6 poits ech) o Must choose your ow test o Tests my oly be used oce o Tests/types you re resposible for: Geometric (kow sum) Telescopig (kow
More informationSimpson s 1/3 rd Rule of Integration
Simpso s 1/3 rd Rule o Itegrtio Mjor: All Egieerig Mjors Authors: Autr Kw, Chrlie Brker Trsormig Numericl Methods Eductio or STEM Udergrdutes 1/10/010 1 Simpso s 1/3 rd Rule o Itegrtio Wht is Itegrtio?
More informationMTH 146 Class 16 Notes
MTH 46 Clss 6 Notes 0.4- Cotiued Motivtio: We ow cosider the rc legth of polr curve. Suppose we wish to fid the legth of polr curve curve i terms of prmetric equtios s: r f where b. We c view the cos si
More informationMATRIX ALGEBRA, Systems Linear Equations
MATRIX ALGEBRA, Systes Lier Equtios Now we chge to the LINEAR ALGEBRA perspective o vectors d trices to reforulte systes of lier equtios. If you fid the discussio i ters of geerl d gets lost i geerlity,
More informationSolutions to Problem Set 7
8.0 Clculus Jso Strr Due by :00pm shrp Fll 005 Fridy, Dec., 005 Solutios to Problem Set 7 Lte homework policy. Lte work will be ccepted oly with medicl ote or for other Istitute pproved reso. Coopertio
More informationVariational Iteration Method for Solving Volterra and Fredholm Integral Equations of the Second Kind
Ge. Mth. Notes, Vol. 2, No. 1, Jury 211, pp. 143-148 ISSN 2219-7184; Copyright ICSRS Publictio, 211 www.i-csrs.org Avilble free olie t http://www.gem.i Vritiol Itertio Method for Solvig Volterr d Fredholm
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probbility d Stochstic Processes: A Friedly Itroductio for Electricl d Computer Egieers Roy D. Ytes d Dvid J. Goodm Problem Solutios : Ytes d Goodm,4..4 4..4 4..7 4.4. 4.4. 4..6 4.6.8 4.6.9 4.7.4 4.7.
More information3.7 The Lebesgue integral
3 Mesure d Itegrtio The f is simple fuctio d positive wheever f is positive (the ltter follows from the fct tht i this cse f 1 [B,k ] = for ll k, ). Moreover, f (x) f (x). Ideed, if x, the there exists
More informationProbability for mathematicians INDEPENDENCE TAU
Probbility for mthemticis INDEPENDENCE TAU 2013 21 Cotets 2 Cetrl limit theorem 21 2 Itroductio............................ 21 2b Covolutio............................ 22 2c The iitil distributio does
More informationTrapezoidal Rule of Integration
Trpezoidl Rule o Itegrtio Civil Egieerig Mjors Authors: Autr Kw, Chrlie Brker http://umericlmethods.eg.us.edu Trsormig Numericl Methods Eductio or STEM Udergrdutes /0/00 http://umericlmethods.eg.us.edu
More informationSchrödinger Equation Via Laplace-Beltrami Operator
IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume 3, Issue 6 Ver. III (Nov. - Dec. 7), PP 9-95 www.iosrjourls.org Schrödiger Equtio Vi Lplce-Beltrmi Opertor Esi İ Eskitşçioğlu,
More informationApproximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
More informationInterpolation. 1. What is interpolation?
Iterpoltio. Wht is iterpoltio? A uctio is ote give ol t discrete poits such s:.... How does oe id the vlue o t other vlue o? Well cotiuous uctio m e used to represet the + dt vlues with pssig through the
More informationUNIVERSITY OF BRISTOL. Examination for the Degrees of B.Sc. and M.Sci. (Level C/4) ANALYSIS 1B, SOLUTIONS MATH (Paper Code MATH-10006)
UNIVERSITY OF BRISTOL Exmitio for the Degrees of B.Sc. d M.Sci. (Level C/4) ANALYSIS B, SOLUTIONS MATH 6 (Pper Code MATH-6) My/Jue 25, hours 3 miutes This pper cotis two sectios, A d B. Plese use seprte
More informationGeometric Sequences. Geometric Sequence. Geometric sequences have a common ratio.
s A geometric sequece is sequece such tht ech successive term is obtied from the previous term by multiplyig by fixed umber clled commo rtio. Exmples, 6, 8,, 6,..., 0, 0, 0, 80,... Geometric sequeces hve
More informationCalculus II Homework: The Integral Test and Estimation of Sums Page 1
Clculus II Homework: The Itegrl Test d Estimtio of Sums Pge Questios Emple (The p series) Get upper d lower bouds o the sum for the p series i= /ip with p = 2 if the th prtil sum is used to estimte the
More information10.5 Test Info. Test may change slightly.
0.5 Test Ifo Test my chge slightly. Short swer (0 questios 6 poits ech) o Must choose your ow test o Tests my oly be used oce o Tests/types you re resposible for: Geometric (kow sum) Telescopig (kow sum)
More informationINTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
Mthemtics Revisio Guides Itegrtig Trig, Log d Ep Fuctios Pge of MK HOME TUITION Mthemtics Revisio Guides Level: AS / A Level AQA : C Edecel: C OCR: C OCR MEI: C INTEGRATION TECHNIQUES (TRIG, LOG, EXP FUNCTIONS)
More informationThe Basic Properties of the Integral
The Bsic Properties of the Itegrl Whe we compute the derivtive of complicted fuctio, like x + six, we usully use differetitio rules, like d [f(x)+g(x)] d f(x)+ d g(x), to reduce the computtio dx dx dx
More informationStudent Success Center Elementary Algebra Study Guide for the ACCUPLACER (CPT)
Studet Success Ceter Elemetry Algebr Study Guide for the ACCUPLACER (CPT) The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetry Algebr test. Reviewig these smples
More informationFrequency-domain Characteristics of Discrete-time LTI Systems
requecy-domi Chrcteristics of Discrete-time LTI Systems Prof. Siripog Potisuk LTI System descriptio Previous bsis fuctio: uit smple or DT impulse The iput sequece is represeted s lier combitio of shifted
More informationPROGRESSIONS AND SERIES
PROGRESSIONS AND SERIES A sequece is lso clled progressio. We ow study three importt types of sequeces: () The Arithmetic Progressio, () The Geometric Progressio, () The Hrmoic Progressio. Arithmetic Progressio.
More informationDouble Sums of Binomial Coefficients
Itertiol Mthemticl Forum, 3, 008, o. 3, 50-5 Double Sums of Biomil Coefficiets Athoy Sofo School of Computer Sciece d Mthemtics Victori Uiversity, PO Box 448 Melboure, VIC 800, Austrli thoy.sofo@vu.edu.u
More informationSupplemental Handout #1. Orthogonal Functions & Expansions
UIUC Physics 435 EM Fields & Sources I Fll Semester, 27 Supp HO # 1 Prof. Steve Errede Supplemetl Hdout #1 Orthogol Fuctios & Epsios Cosider fuctio f ( ) which is defied o the itervl. The fuctio f ( )
More information( a n ) converges or diverges.
Chpter Ifiite Series Pge of Sectio E Rtio Test Chpter : Ifiite Series By the ed of this sectio you will be ble to uderstd the proof of the rtio test test series for covergece by pplyig the rtio test pprecite
More informationy udv uv y v du 7.1 INTEGRATION BY PARTS
7. INTEGRATION BY PARTS Ever differetitio rule hs correspodig itegrtio rule. For istce, the Substitutio Rule for itegrtio correspods to the Chi Rule for differetitio. The rule tht correspods to the Product
More informationAPPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES
Scietific Reserch of the Istitute of Mthetics d Coputer Sciece 3() 0, 5-0 APPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES Jolt Borows, Le Łcińs, Jowit Rychlews Istitute of Mthetics,
More informationis an ordered list of numbers. Each number in a sequence is a term of a sequence. n-1 term
Mthemticl Ptters. Arithmetic Sequeces. Arithmetic Series. To idetify mthemticl ptters foud sequece. To use formul to fid the th term of sequece. To defie, idetify, d pply rithmetic sequeces. To defie rithmetic
More informationNATIONAL OPEN UNIVERSITY OF NIGERIA
NATIONAL OPEN UNIVERSITY OF NIGERIA SCHOOL OF SCIENCE AND TECHNOLOGY COURSE CODE: MTH COURSE TITLE: NUMERICAL ANALYSIS Course Code MTH Course Title NUMERICAL ANALYSIS Course Developer Dr. Ajibol S. O.
More informationdenominator, think trig! Memorize the following two formulas; you will use them often!
7. Bsic Itegrtio Rules Some itegrls re esier to evlute th others. The three problems give i Emple, for istce, hve very similr itegrds. I fct, they oly differ by the power of i the umertor. Eve smll chges
More informationTrapezoidal Rule of Integration
Trpezoidl Rule o Itegrtio Mjor: All Egieerig Mjors Authors: Autr Kw, Chrlie Brker Trsormig Numericl Methods Eductio or STEM Udergrdutes /0/200 Trpezoidl Rule o Itegrtio Wht is Itegrtio Itegrtio: The process
More informationCrushed Notes on MATH132: Calculus
Mth 13, Fll 011 Siyg Yg s Outlie Crushed Notes o MATH13: Clculus The otes elow re crushed d my ot e ect This is oly my ow cocise overview of the clss mterils The otes I put elow should ot e used to justify
More informationLinear Programming. Preliminaries
Lier Progrmmig Prelimiries Optimiztio ethods: 3L Objectives To itroduce lier progrmmig problems (LPP To discuss the stdrd d coicl form of LPP To discuss elemetry opertio for lier set of equtios Optimiztio
More informationBRAIN TEASURES INDEFINITE INTEGRATION+DEFINITE INTEGRATION EXERCISE I
EXERCISE I t Q. d Q. 6 6 cos si Q. Q.6 d d Q. d Q. Itegrte cos t d by the substitutio z = + e d e Q.7 cos. l cos si d d Q. cos si si si si b cos Q.9 d Q. si b cos Q. si( ) si( ) d ( ) Q. d cot d d Q. (si
More information4 The Integral. 4.0 Area. (base) (height). If the units for each side of the rectangle are meters,
4 The Itegrl Previous chpters delt with differetil clculus. They strted with the simple geometricl ide of the slope of tget lie to curve, developed it ito combitio of theory bout derivtives d their properties,
More information,... are the terms of the sequence. If the domain consists of the first n positive integers only, the sequence is a finite sequence.
Chpter 9 & 0 FITZGERALD MAT 50/5 SECTION 9. Sequece Defiitio A ifiite sequece is fuctio whose domi is the set of positive itegers. The fuctio vlues,,, 4,...,,... re the terms of the sequece. If the domi
More informationBC Calculus Review Sheet
BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl). This is wht you thik of doig.... Fid the re
More informationM3P14 EXAMPLE SHEET 1 SOLUTIONS
M3P14 EXAMPLE SHEET 1 SOLUTIONS 1. Show tht for, b, d itegers, we hve (d, db) = d(, b). Sice (, b) divides both d b, d(, b) divides both d d db, d hece divides (d, db). O the other hd, there exist m d
More information