麻省理工学院：《偏微分方程式数字方法》（英文版）Lecture 21 notes

Numerical methods for pdes Integral Equation Methods, Lecture 2 Numerical Quadrature Notes by suvranu De and J. white April 28, 2003

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1 Outl Easy technique for computing integrals Piecewise constant approach sian Quadra Convergence pI ssential role of orthogonal polynomials Multidimensional Integra Techniques for singular kernels Adapt ation and variable transformation Singular quadrature 2 3D Laplace's equation 2.1.1 Centroid collocation SLIDE 1 Put collocation points at panel centroids (x)= 甲(x) Note 1 In the last lecture we were introduced to integral equations and several different techniques for discretizing them were described. It was pointed out that one of the most popular means of obtaining a discrete set of equations is to use a piecewise constant centroid collocation scheme. We consider a simple problem of solving Laplace's equation in 3D. The potential, y is prescribed on the sur- face of the cube and we need to compute the charge distribution o. In order to do that we break the surface of the cube up into n panels and assume a constant charge distribution on each panel. Mathematically, this corresponds to assuming piecewise const ant basis functions, each basis function, Pi being compactly supported on the ith panel. The resulting semi-discrete equation is a function of the spatial variable z. In order to obt ain a discrete set of equations we assume that this semi-discrete equation is satisfied exactly at the centroids of the panels. This gives rise to the set of n equations corresponding to the n panels. Mathematically, this process of collocation corresponds to setting the residual orthogonal to a set of delta functions located at the panel centroids

￾ ✁✄✂✆☎✞✝✠✟✠✡☞☛ ✌✎✍✑✏✓✒✕✔✗✖✙✘✛✚✞✜✣✢✥✤✧✦✣✖✩★✫✪✭✬✆✘✮✪✰✯✲✱✳✦✴✔✵✢✥✜✣✶✷✢✸✜✹✔✗✖✙✶✺✬✻✍✑✼✸✏ ✽✿✾❁❀✗❂✓❀✓❃❄✾❆❅❇❀❈❂✠❉❋❊✺❅❇●■❍❏❊❑●▲❍❏▼✺▼✑◆❖❉❑❍✮❂◗P ❘✍✑✦✣✏✗✏✵✢✥✍✺✜❚❙❯✦✞✍✺❱✴✬✻✍❲✔✵✦✴✬✻✖ ❳❉✮❊✙❨❋❀✓◆■❩✮❀✓❊✭❂✠❀❬▼✺◆❖❉❋▼✭❀✗◆❇●■✾❭❀✵❅ ❪❫❅■❅❇❀✗❊❑●❖✾❆❍❏❴✧◆■❉✮❴❁❀✎❉✮❵✞❉❋◆❇●■P✑❉✮❩❋❉✮❊✺❍✮❴✭▼✰❉✮❴❁❛✙❊✑❉✮❜❝✾❆❍❏❴❆❅ ❞❢❡✑❴❭●❖✾❆❣❲✾❁❜❝❀✓❊✺❅❖✾❭❉❋❊✺❍❏❴✧❤✐❊❑●■❀✓❩✮◆◗❍❏❴❆❅ ❥✖❑✘✛✚✣✜✞✢✸✤✧✦✞✖✙✏❦★✫✪✭✬✆✏✗✢✸✜✞✶✰✦✣✼✥✍❲✬❦❧✰✖❋✬✻✜✣✖✙✼✥✏ ♠❣✑❍✮▼❲●■❍❏●❖✾❁❉✮❊♥❍❏❊✺❣♦❨✻❍❏◆■✾❁❍✮♣✑❴❭❀✎●■◆■❍✮❊✺❅q❵r❉❋◆❖❜❦❍❏●❖✾❁❉✮❊ s✾❁❊✑❩✮❡✺❴❁❍✮◆✉t❑❡✺❍❋❣❲◆◗❍✻●❖❡✺◆❖❀❋✈ ✇ ①③② ④⑥⑤❈⑦♥✝✠⑤③⑧⑨☛③⑩q❶❸❷❺❹❝✂♦⑤❻☎✣✟✠❼③✡ ❽✉❾❇❿ ➀⑥➁➃➂✙➄❇➂➆➅▲➇➉➈❬➊✰➋✑➄q➌➍➈➏➎➑➐➉➐❬➒✑➌❫➁✳➊✭➓ ➔➣→✥↔✰→✥↔ ↕✖✙✜✹✔✓✬✻✪✰✢✥❱ ↕✪✭✼✸✼✥✪➣✘❋✍❲✔✗✢✸✪✭✜ ➙✰➛❑➜➞➝✴➟✕➠ ✽✿❡❲●❬❂✓❉✮❴❁❴❭❉❲❂✗❍✻●❖✾❁❉✮❊☞▼✰❉✮✾❁❊❋●◗❅❄❍✻●▲▼✺❍✮❊✑❀✓❴➣❂✓❀✓❊❑●❖◆■❉✮✾❆❣✑❅ SMA-HPC ©1999 MIT Laplace’s Equation in 3-D Basis Function Approach ( ) 1 , 1 i i n c j j c i j panel j x dS x x A α = Ψ = ′ − ′ " ! "## $###% ( ) ( ) 1,1 1, 1 1 ,1 , n c n n n n n c A A x A A x α α #Ψ $ # $# $ % & % &% & % & % & = % & % & % & % & % & %Ψ & ' (' ( ' ( & & ' ( ' ' ' ' ( ' ' ' & & Put collocation points at panel centroids i c x Collocation point ➡➤➢✧➥➧➦➩➨ ❤✐❊➫●❖P✑❀❬❴❁❍❋❅q●✿❴❁❀✗❂✠●❖❡✑◆■❀❄❃➍❀❄❃⑨❀✓◆■❀❄✾❁❊❋●■◆❖❉❲❣❲❡✭❂✠❀✗❣➭●■❉❈✾❭❊❑●❖❀✗❩✮◆◗❍❏❴❲❀✵t❋❡✭❍✻●❖✾❁❉✮❊✭❅➃❍❏❊✺❣❝❅❖❀✓❨✮❀✗◆■❍✮❴✑❣❲✾❭➯✰❀✗◆❖❀✗❊❋● ●❖❀✵❂◗P✑❊✑✾❆t❑❡✑❀✗❅③❵r❉✮◆➭❣✑✾❁❅■❂✠◆■❀✠●■✾❭➲✗✾❭❊✑❩♥●■P✑❀✓❜➳❃⑨❀✓◆■❀❯❣❲❀✗❅■❂✠◆■✾❁♣✭❀✵❣✧✈❢❤✫●➭❃✉❍✮❅❈▼✰❉✮✾❁❊❑●❖❀✵❣✲❉✮❡❲●➵●❖P✺❍❏●➵❉✮❊✑❀ ❉❏❵❫●❖P✑❀❝❜❝❉❋❅❇●✎▼✰❉✮▼✺❡✑❴❁❍✮◆❻❜➫❀✵❍❏❊✺❅✎❉✮❵➍❉✮♣✑●■❍❏✾❁❊✑✾❁❊✑❩♥❍♦❣❲✾❆❅■❂✠◆■❀✠●❖❀❝❅❖❀✠●❻❉❏❵➍❀✵t❑❡✺❍✻●■✾❭❉❋❊✺❅❬✾❆❅✎●❖❉☞❡✺❅❖❀➫❍ ▼✑✾❁❀✗❂✓❀✓❃❄✾❆❅❇❀➵❂✓❉✮❊✺❅❇●■❍✮❊❑●➉❂✠❀✗❊❋●■◆❖❉❋✾❁❣❢❂✓❉✮❴❁❴❭❉❲❂✗❍✻●❖✾❁❉✮❊✩❅■❂◗P✑❀✗❜➫❀❋✈▲➸➤❀➭❂✠❉❋❊✺❅❖✾❁❣❲❀✗◆➉❍♦❅❇✾❁❜❝▼✑❴❭❀➭▼✑◆■❉✮♣✑❴❁❀✓❜ ❉❏❵✿❅❖❉✮❴❁❨✙✾❭❊✑❩❯➺✞❍❏▼✑❴❆❍✮❂✓❀✮➻ ❅❄❀✗t❑❡✺❍❏●❖✾❁❉✮❊♥✾❁❊➩➼✮➽➭✈✭➾✉P✑❀➵▼✭❉✮●❖❀✓❊❑●■✾❁❍✮❴✸➚✧➪➶✾❆❅▲▼✺◆❖❀✵❅❖❂✓◆❖✾❁♣✭❀✵❣♥❉✮❊☞●❖P✺❀➭❅❖❡✑◆❇➹ ❵✥❍✮❂✓❀❝❉❏❵⑨●❖P✑❀✆❂✓❡✑♣✰❀❯❍❏❊✺❣➤❃⑨❀❦❊✑❀✓❀✵❣➩●■❉✩❂✠❉❋❜➫▼✺❡❲●❖❀❝●■P✑❀❯❂◗P✺❍❏◆■❩✮❀❦❣❲✾❆❅❇●❖◆■✾❭♣✑❡✑●❖✾❁❉✮❊✲➘✳✈❯❤✐❊⑥❉✮◆◗❣❲❀✓◆ ●❖❉✲❣❲❉✷●❖P✭❍✻●❯❃⑨❀♦♣✑◆■❀✗❍✮➴⑥●❖P✑❀✩❅❖❡✑◆❖❵✥❍✮❂✓❀♦❉✮❵❬●❖P✺❀♥❂✓❡✑♣✰❀❢❡✑▼✕✾❁❊❑●❖❉⑥➷➬▼✭❍❏❊✑❀✗❴❁❅❝❍❏❊✭❣✕❍❋❅❖❅❖❡✑❜❝❀☞❍ ❂✠❉❋❊✺❅❇●■❍❏❊❑●③❂◗P✺❍✮◆❖❩❋❀❝❣✑✾❁❅❇●❖◆■✾❭♣✺❡❲●❖✾❁❉✮❊⑥❉✮❊⑥❀✵❍✮❂◗P⑥▼✺❍❏❊✺❀✓❴➮✈♦❞✩❍❏●❖P✑❀✗❜❝❍❏●❖✾❆❂✓❍✮❴❭❴❁❛✮➚✴●❖P✑✾❆❅➵❂✓❉✮◆■◆❖❀✵❅❇▼✰❉✮❊✺❣✺❅ ●❖❉➤❍✮❅■❅❇❡✺❜➫✾❁❊✑❩✩▼✺✾❭❀✵❂✠❀✓❃❄✾❆❅❖❀✆❂✠❉❋❊✺❅q●◗❍❏❊❑●➭♣✭❍✮❅❖✾❁❅③❵r❡✑❊✺❂➧●■✾❭❉❋❊✺❅✗➚➃❀✗❍✮❂◗P✲♣✭❍✮❅❖✾❁❅③❵r❡✑❊✺❂➧●■✾❭❉❋❊➣➚❫➱✳✃➉♣✭❀✗✾❭❊✑❩ ❂✠❉❋❜❝▼✺❍✮❂✠●❖❴❁❛➭❅❖❡✑▼✑▼✰❉✮◆❖●❖❀✗❣❯❉✮❊❝●❖P✺❀❬❐✫❒✥❮❈▼✺❍✮❊✑❀✓❴➮✈➃➾✉P✑❀▲◆■❀✗❅❖❡✑❴❭●❖✾❁❊✑❩➵❅❖❀✓❜❝✾❰➹✐❣❲✾❆❅❖❂✓◆❖❀✓●❖❀❄❀✵t❋❡✭❍✻●❖✾❁❉✮❊❦✾❆❅❫❍ ❵r❡✑❊✺❂✠●❖✾❁❉✮❊❯❉✮❵➣●❖P✺❀➉❅❖▼✺❍✻●■✾❁❍✮❴✭❨✻❍❏◆■✾❆❍❏♣✑❴❁❀▲Ï✴✈➃❤✐❊✆❉✮◆◗❣❲❀✓◆❫●❖❉➵❉❋♣❲●■❍✮✾❭❊♦❍➵❣❲✾❆❅❖❂✓◆❖❀✓●❖❀❬❅❖❀✠●✉❉❏❵➣❀✵t❑❡✺❍✻●■✾❭❉❋❊✺❅✓➚ ❃⑨❀➵❍✮❅■❅❇❡✺❜➫❀③●❖P✺❍❏●▲●❖P✑✾❆❅✎❅❇❀✗❜➫✾❭➹✐❣❲✾❁❅■❂✠◆■❀✠●■❀③❀✗t❑❡✺❍❏●❖✾❁❉✮❊❢✾❆❅❬❅■❍✻●■✾❁❅❇Ð✺❀✗❣♥❀✓Ñ✑❍✮❂➧●■❴❭❛♥❍✻●❬●❖P✑❀➫❂✠❀✓❊❑●■◆❖❉❋✾❁❣✑❅ ❉❏❵➃●■P✑❀➫▼✺❍❏❊✑❀✗❴❁❅✗✈❻➾✉P✑✾❁❅➉❩✮✾❁❨✮❀✵❅❬◆■✾❆❅❇❀❈●■❉✆●■P✑❀❝❅❇❀✓●➉❉❏❵➍➷✲❀✗t❑❡✺❍✻●■✾❭❉❋❊✺❅✎❂✠❉✮◆■◆■❀✗❅❖▼✭❉❋❊✺❣❲✾❁❊✑❩❝●❖❉✆●❖P✑❀➫➷ ▼✺❍✮❊✑❀✓❴❆❅✓✈❯❞➩❍✻●❖P✺❀✓❜❦❍✻●■✾❁❂✗❍❏❴❁❴❭❛❋➚✧●❖P✑✾❆❅❈▼✑◆■❉❲❂✠❀✵❅❖❅❻❉✮❵✉❂✓❉✮❴❁❴❭❉❲❂✓❍❏●❖✾❁❉✮❊➤❂✠❉✮◆■◆■❀✗❅❖▼✭❉❋❊✺❣✑❅➉●❖❉✩❅❖❀✠●❖●❖✾❁❊✑❩♥●❖P✑❀ ◆■❀✗❅❖✾❁❣❲❡✭❍❏❴✹❉✮◆❖●❖P✑❉❋❩✮❉❋❊✺❍❏❴✰●❖❉❦❍❝❅❖❀✠●▲❉❏❵✳❣❲❀✓❴❭●■❍➫❵r❡✺❊✺❂➧●■✾❭❉❋❊✺❅❄❴❁❉✙❂✗❍✻●■❀✗❣☞❍✻●✉●❖P✑❀❈▼✭❍❏❊✑❀✗❴✴❂✠❀✗❊❋●■◆❖❉❋✾❁❣✺❅✓✈ Ò

T2iMprdi u MuaouMa Mo i, 2 a M, df n invar axt u\taiz u Aaa, idnM,d Myou. T2i Aaan, i, iuMdf in, arum/ vant, 2uidwydia, idn pdin, o ait2, Mhi M, ddas mui, aru wo iyi dni am, ra, t dn d wainint, 2a am, riuMdf, 2wn-WS-n u a, rix ne do n in, 2u Tida a dou I, 2u iun, did i drydi a, idn asi2niAau,, 2u u a, rix a, riuminod you at raydf 2u Granm'Mfani, idn doar, 2u panu E2S/i aws, 2u, uru Ai df iMI a, rix idrrulodndM d, 2u pd, n, iaya,, 2u im, did df, 2 u i th panuydau, d ani, i2artu dumlifs dimriva, idn dn, 2uj panzy 2.1.2 Caldhlatisg Matrix allr Ists SLIDE 2 On ourS ipy o as df idu pa, int 2u in, tray in Aij, fdr a panty j o 2ii2 iM far rai doad frdu panty i, iM, d liu pyS rupyaiu, 2u in, tray WS, 2u in, atrand aaaa, ud a,, 2u ium, rdid df 2u panty ]. Of idard,, 2iMim, dd liu pvn fdr panu Mo 2ii2 in, rai, "hrant" i, 2, 2u panyi. Far, 2uh panty o 2i 2 aru Mr wso uu as and a fdar-pdin, in, atra, idn 12/2nu u. I dar u indO u mpi 2u panuyap in, d fdar u Aaay Aavanuyand o ri, w, 2u in, tray dor, 2u m, ira panty j aM, 2u Au df fdar in, utraydn, 2uht fd ar laviana T2m add Mu u, riik aMWifdru, rupaiint uai 2 df, 2uh fdar in, utrayMvs, 2u prddai, df in, ut rand, uoaaa, ud a,, 2u ium, rdid df uai 2 df, 2 uht nvanumand, 2u arua df 2u Nanay Frdu in, a, idn o u Wiuou, 2a,, 2iM272anu u iMt dint d tiou a Ma dru aii ara, u anM ur. T2u AauMidn, 2do uour, iMo 2u, 2ur, 2iMiM, 2u wimo as d td dr aru, 2ru w, ar, an 2ni Aau Nan o u dd, 2u Mu u kind df in, utra, idn dr panyi? Td W awi, d anM ur, uhi AaulyidnMo u o iy, aku a weiaf ydk a, 2de nau wii ayin, ut ra, idn iMpurfdru ad, a fiund df Mads kndo n aM Aaadra, aru

➾✉P✑✾❆❅❬▼✺◆❖❉❲❂✠❀✵❅❖❅➉❴❁❀✗❍✛❨✮❀✵❅▲❡✺❅➉❃❄✾❭●❖P✷❍♦❅❇❀✓●✎❉✮❵✿➷➆❴❁✾❭❊✑❀✵❍❏◆❻❍❏❴❁❩✮❀✗♣✑◆■❍✮✾❁❂❻❀✵t❑❡✺❍✻●■✾❭❉❋❊✺❅▲●■❉♦❅❖❉✮❴❁❨✮❀❋✈✎➾✉P✑❀ t❑❡✺❍❏❊❑●■✾❰●■✾❭❀✵❅✳❉❏❵✹✾❭❊❑●❖❀✗◆❖❀✵❅q●➃♣✰❀✓✾❁❊✑❩❻●❖P✺❀❬❂✠❉❋❴❭❴❁❉❲❂✓❍✻●■✾❭❉❋❊➵▼✰❉✮✾❁❊❑●✿❃➍❀✗✾❭❩❋P❋●◗❅✁￾✄✂✮➻ ❅✗✈✞❤✐❊➫●❖❉❲❣✑❍✛❛✹➻ ❅✳❴❭❀✵❂➧●■❡✑◆❖❀ ❃⑨❀✎❃❄✾❁❴❁❴✞❂✠❉❋❊✺❂✠❀✗❊❑●❖◆◗❍✻●❖❀✎❉✮❊♥❉❋♣❲●■❍✮✾❭❊✑✾❁❊✑❩➫●❖P✺❀③❀✓❊❑●❖◆■✾❁❀✗❅✉❉❏❵✣●❖P✺❀❈➷✹➹➮♣✙❛❑➹➮➷✷❜❦❍✻●❖◆■✾❭Ñ♦❅❇P✺❉✻❃❄❊♦✾❁❊♥●❖P✑❀ ❅❖❴❭✾❆❣❲❀♦❍✮♣✭❉✻❨❋❀✮✈✷❤✐❊➆●❖P✺❀♦❂✠❀✗❊❑●❖◆■❉✮✾❆❣➆❂✠❉❋❴❭❴❁❉❲❂✓❍✻●■✾❭❉❋❊⑥●❖❀✵❂◗P✑❊✑✾❆t❋❡✺❀✮➚✳●❖P✺❀♦❜❦❍✻●■◆❖✾❭Ñ✲❀✓❊❑●❖◆■✾❁❀✗❅➭✾❭❊✙❨✮❉❋❴❭❨❋❀ ✾❁❊❋●■❀✓❩❋◆■❍✮❴❁❅❻❉✮❵⑨●■P✑❀✆☎✎◆■❀✓❀✗❊➣➻ ❅❈❵r❡✑❊✺❂➧●■✾❭❉❋❊➆❉✻❨✮❀✗◆❻●❖P✑❀✆▼✺❍❏❊✑❀✗❴❁❅✗✈☞✽✿P✙❛✙❅❖✾❆❂✓❍❏❴❁❴❁❛✮➚✞●❖P✑❀✆●❖❀✓◆■❜✞✝✃✟✂ ❉✮❵ ●❖P✺✾❁❅❄❜❦❍❏●❖◆■✾❰Ñ♦❂✓❉✮◆■◆❖❀✵❅❇▼✰❉✮❊✭❣✑❅➍●❖❉❝●❖P✺❀❈▼✭❉✮●❖❀✓❊❑●■✾❁❍✮❴➣❍✻●❄●❖P✺❀③❂✠❀✓❊❑●■◆❖❉❋✾❁❣♦❉❏❵✣●❖P✑❀❈❐✫❒✥❮❝▼✭❍❏❊✑❀✗❴➣❣❲❡✑❀❻●❖❉ ❡✑❊✑✾❭●❬❂◗P✺❍✮◆❖❩❋❀✎❣❲❀✗❊✺❅❖✾❰●q❛✆❣✑✾❁❅❇●❖◆■✾❭♣✺❡❲●❖✾❁❉✮❊♥❉✮❊♦●❖P✺❀✡✠✮❒✥❮❦▼✺❍❏❊✑❀✗❴✸✈ ➔➣→✥↔✰→❆➔ ↕✍✑✼✸✘✮✦✣✼✥✍❲✔✵✢✥✜✣✶☞☛➶✍❲✔✓✬✻✢✍✌✕✌✎✼✥✖✙✯➆✖✙✜✹✔✗✏ ➙✰➛❑➜➞➝✴➟✏✎ 3-D Laplace’s Equation Basis Function Approach Panel j i c x Collocation point , 1 i i j panel j c x x A dS′ − ′ = ! , i j c centr j id i o Panel Area x x A − ≈ One point quadrature Approximation x y z t 4 , 1 in 0.25* i j c o i j j p Ar a x x A e = − ≈ " Four point quadrature Approximation ➡➤➢✧➥➧➦✒✑ ✓❊✑❀❝❨✮❀✗◆❖❛✷❅❇✾❁❜❝▼✑❴❭❀✆❃⑨❍✛❛❢❉✮❵✉❂✓❉✮❜❝▼✑❡❲●■✾❭❊✑❩♥●■P✑❀❯✾❁❊❑●❖❀✓❩❋◆■❍✮❴✳✾❭❊✔✝➉✃✟✂❋➚✴❵r❉✮◆➵❍☞▼✺❍✮❊✑❀✓❴✕✠❢❃❄P✑✾❆❂◗P⑥✾❁❅ ❵✥❍❏◆❦◆■❀✓❜❝❉✻❨❋❀✗❣✲❵r◆■❉✮❜ ▼✺❍❏❊✑❀✗❴❄❐◗➚➍✾❆❅➫●❖❉✲❅❖✾❭❜❝▼✑❴❁❛ ◆■❀✓▼✑❴❆❍✮❂✓❀✆●■P✑❀❢✾❁❊❋●■❀✓❩❋◆■❍✮❴⑨♣✙❛✲●■P✑❀❢✾❁❊❋●■❀✓❩❋◆■❍✮❊✺❣ ❀✓❨✻❍✮❴❭❡✺❍❏●❖❀✵❣✲❍✻●➵●■P✑❀♦❂✓❀✓❊❑●❖◆■❉✮✾❆❣➤❉❏❵❄●■P✑❀✆▼✺❍✮❊✑❀✓❴✁✠✭✈ ✓❵❬❂✠❉❋❡✑◆◗❅❇❀❋➚✴●■P✑✾❁❅➭✾❆❅❈●❖❉✙❉➤❅❇✾❁❜❝▼✑❴❭✾❆❅❇●❖✾❆❂❦❵r❉✮◆ ▼✺❍✮❊✑❀✓❴❆❅✎❃❄P✑✾❁❂◗P✷✾❭❊❑●■❀✓◆◗❍✮❂➧●✗✖✠❅❇●❖◆■❉✮❊✺❩✮❴❁❛✄✖❈❃❄✾❰●■P➩●■P✑❀❝▼✺❍❏❊✑❀✗❴➃❐◗✈✙✘✑❉❋◆➉●■P✑❀✗❅❖❀❝▼✺❍❏❊✺❀✓❴❆❅➉❃❄P✺✾❁❂◗P➤❍❏◆■❀ ❂✠❴❁❉❋❅❖❀✓◆❻♣✙❛✮➚✣❃⑨❀➫❜❦❍✛❛✷❡✺❅❇❀❦❍☞❵r❉❋❡✑◆❇➹✫▼✰❉✮✾❁❊❋●❈✾❁❊❑●❖❀✓❩❋◆■❍❏●❖✾❁❉✮❊⑥❅❖❂◗P✺❀✓❜❝❀✮✈❦❤✐❊✲❉❋❡✑◆❈❜❝✾❭❊✺❣✺❅❈❃➍❀❯❅❖▼✑❴❁✾❰● ●❖P✺❀✆▼✺❍✮❊✑❀✓❴⑨❡✑▼➆✾❁❊❑●❖❉✩❵r❉❋❡✑◆➵❀✵t❑❡✺❍❏❴⑨❅❇❡✑♣✺▼✺❍❏❊✑❀✗❴❁❅➫❍❏❊✭❣✲❃❄◆❖✾❭●❖❀❯●❖P✑❀☞✾❭❊❑●■❀✓❩✮◆◗❍❏❴❫❉✻❨✮❀✓◆③●❖P✑❀☞❀✓❊❑●❖✾❁◆■❀ ▼✺❍✮❊✑❀✓❴✚✠ ❍✮❅❝●■P✑❀✩❅❇❡✺❜ ❉❏❵✎❵r❉✮❡✺◆❦✾❭❊❑●❖❀✗❩✮◆◗❍❏❴❆❅❝❉✮❊✕●❖P✑❀✵❅❇❀♥❵r❉❋❡✑◆✆❅❖❡✑♣✑▼✺❍✮❊✑❀✓❴❆❅✗✈❺➾✉P✑❀✓❊➑❡✺❅❖❀♥●❖P✑❀ ❅■❍❏❜❝❀✉●❖◆■✾❁❂◗➴❝❍✮❅✿♣✭❀✓❵r❉✮◆■❀✮➚✮◆■❀✓▼✺❴❁❍❋❂✠✾❁❊✑❩❈❀✗❍❋❂◗P➫❉✮❵✹●❖P✑❀✵❅❇❀❄❵r❉❋❡✑◆❫✾❁❊❑●❖❀✗❩✮◆◗❍❏❴❆❅➃♣✙❛③●❖P✺❀❬▼✑◆■❉❲❣❲❡✺❂➧●➍❉❏❵✹●❖P✑❀ ✾❁❊❋●■❀✓❩❋◆■❍✮❊✺❣✧➚✑❀✗❨✛❍✮❴❭❡✭❍✻●❖❀✵❣♥❍✻●❄●❖P✺❀➭❂✠❀✗❊❑●❖◆■❉✮✾❆❣✆❉✮❵✳❀✵❍✮❂◗P♥❉❏❵✳●■P✑❀✗❅❖❀③❅❇❡✺♣✑▼✺❍❏❊✺❀✓❴❆❅▲❍❏❊✺❣☞●■P✑❀➵❍✮◆❖❀✵❍➭❉✮❵ ●❖P✺❀❝❅❖❡✑♣✑▼✭❍❏❊✑❀✗❴✸✈✙✘✺◆❖❉❋❜ ✾❭❊❑●■❡❲●❖✾❁❉✮❊✷❃➍❀➫♣✭❀✗❴❭✾❁❀✓❨❋❀➫●❖P✺❍❏●➉●■P✑✾❁❅❈❅■❂◗P✑❀✗❜➫❀➫✾❆❅➉❩❋❉✮✾❁❊✑❩❯●■❉♥❩✮✾❁❨✮❀➵❡✭❅✎❍ ❜❝❉✮◆■❀➵❍✮❂✗❂✠❡✑◆◗❍✻●■❀➭❍✮❊✺❅❇❃⑨❀✓◆✵✈➉➾✉P✺❀➭t❑❡✑❀✗❅❇●❖✾❁❉✮❊✴➚✰P✺❉✻❃➍❀✗❨✮❀✓◆✵➚✺✾❆❅✎❃❄P✑❀✠●■P✑❀✓◆➉●❖P✺✾❁❅✎✾❆❅▲●■P✑❀➭♣✰❀✗❅❇●❻❃⑨❍✛❛ ●❖❉❈❩❋❉❈❉✮◆❫❍✮◆❖❀⑨●■P✑❀✓◆■❀▲♣✰❀✠●❇●■❀✓◆❫●❖❀✗❂◗P✺❊✑✾❁t❑❡✑❀✵❅✜✛ ❳❍✮❊➫❃⑨❀▲❣❲❉❈●■P✑❀❬❅■❍❏❜❝❀❄➴✙✾❭❊✭❣❝❉❏❵✹✾❭❊❑●❖❀✗❩✮◆◗❍✻●■✾❭❉❋❊➭❉❋❊ ▼✺❍✮❊✑❀✓❴✞❐✢✛⑥➾✞❉❯♣✰❀➭❍✮♣✑❴❭❀③●❖❉✆❍✮❊✺❅❇❃⑨❀✓◆▲●❖P✑❀✵❅❇❀➵t❑❡✑❀✵❅q●■✾❭❉❋❊✺❅❄❃⑨❀③❃❄✾❁❴❭❴✞●■❍✮➴✮❀③❍❦♣✑◆■✾❁❀✠❵➃❴❭❉✙❉❋➴☞❍❏●❬P✑❉✻❃ ❊✙❡✑❜❝❀✓◆■✾❁❂✗❍❏❴✹✾❁❊❋●■❀✓❩❋◆■❍❏●❖✾❁❉✮❊☞✾❁❅✉▼✰❀✓◆❖❵r❉✮◆■❜❝❀✗❣✧➚✑❍➭Ð✺❀✗❴❁❣☞❉❏❵➃❅❇●❖❡✺❣❲❛✆➴❑❊✺❉✻❃❄❊☞❍❋❅✣✖✠t❑❡✺❍✮❣✑◆■❍❏●❖❡✑◆■❀✤✖❏✈ ✥

3 Normalized 1d problem 3.1 Basis Function Approach ()=/g(,x)(x)ds′x∈,1 Centroid collocated piecewise constant scheme 今+++++4 olaya)ds Note 3 Lets take a simple example in 1D. The domain is the segment [0, 1] of the real scheme, we divide the domain into n segments [==/.9 de on this le. We want to solve the integral equation shown at the top of the shi do ain. The t reen' s function is denoted by g(a, a. In o=0 and En=1. The charge density o is assumed to be piecewise constant on each of these intervals. The potential, y is then evaluated at the centroids ci This results in n equations in n variables, the collocation weights li, i=l,., n which can be written in matrix form. Our task is to first evaluate the entries entry of the matrix ss'quently solve the set of equations. Note that the i>th of the matrix and an integral of the treen's function, evaluated at the collocation point ci, over the interval [-ucull, which is the interval over which the basis function uis nonzero(recall that we have chosen a piecewise constant approximation). If, however, we decided to choose a different set of function and [0, 1.D Sral would be nonzero only on the support of the basis basis functions this inte ise, on the intersection of the support of the basis

① ￾❼✂✁☎✄⑤③✝✠✟✝✆⑨☛✟✞ ￾▲② ✠✡✁✣❼✂☛♥✝✠☛☞✄ ✌✉❾❇❿ ➀⑥➁➃➂✙➄❇➂➆➅▲➇➉➈❬➊✰➋✑➄q➌➍➈➏➎➑➐➉➐❬➒✑➌❫➁✳➊✭➓ ✍✴→✥↔✰→✥↔ ↕✪✰✼✥✼✥✪✴✘✮✍❲✔✵✢✥✪✰✜✏✎☞✢✥✏✵✘❏✬✻✖❑✔✗✢✒✑✮✍❲✔✵✢✥✪✰✜❚✪✭★ ↔ ✎➳✌➉✤✧✦✣✍❲✔✵✢✥✪✰✜ ➙✰➛❑➜➞➝✴➟✔✓ ➪✖✕✥Ï✘✗✚✙✜✛✏✢ ✣✥✤ ✕rÏ✧✦❇Ï✘★✩✗❇➘☎✕rÏ✘★✪✗✬✫✮✭☎★ Ï✰✯✔✱✲✳✦✵✴✝✶ ❳❀✓❊❑●■◆❖❉❋✾❁❣♦❂✠❉✮❴❁❴❁❉✙❂✗❍✻●■❀✗❣♦▼✑✾❭❀✵❂✠❀✗❃❄✾❁❅❖❀❈❂✠❉✮❊✭❅q●◗❍❏❊❑●▲❅❖❂◗P✑❀✗❜❝❀ Normalized 1-D Problem Basis Function Approach Collocation Discretization of 1-D Equation ( ) ( ) ( ) 1 0 Ψ = x g x x, ′ ′ σ x dS′ " x∈[0,1] x0 = 0 xn =1 1 x n 1 x x2 − σ1 σ n 1c x 2c x nc x ( ) ( ) 1 1 , to be evaluated j i j x j x n c j x S σ g x x d − = Ψ = ! ′ ′ " !""#""$ ➪✖✕rÏ✘✷✹✸✺✗✚✙✼✻✾✽✂✺✿ ✢ ➘✂❀✛❂❁✵❃ ❁ ❃✬❄❆❅ ✤ ✕✥Ï✘✷✹✸❇✦❖Ï★ ✗✬✫✮✭★ ❈ ❉✝❊ ❋ ❒❍●✟■✹❏❑❏▼▲✝◆❇❖◗P❘◆◗❒❍❏❚❙ ➡➤➢✧➥➧➦❱❯ ➺➣❀✓●■❅✉●◗❍❏➴❋❀➵❍❦❅❇✾❁❜❝▼✑❴❁❀③❀✠Ñ✑❍❏❜❝▼✑❴❁❀❈✾❁❊ Ò➽➭✈✺➾✉P✑❀➵❣✑❉✮❜❦❍❏✾❁❊❢✾❁❅▲●❖P✑❀➭❅❇❀✗❩✮❜❝❀✓❊❑●❲✱✲✳✦❳✴❳✶✴❉❏❵✳●■P✑❀③◆■❀✗❍❏❴ ❴❁✾❭❊✑❀❋✈✞➸➤❀❄❃✉❍❏❊❑●✞●■❉③❅❇❉❋❴❭❨❋❀❫●■P✑❀❄✾❁❊❋●■❀✓❩❋◆■❍✮❴❑❀✵t❑❡✺❍✻●■✾❭❉❋❊➭❅❖P✑❉✻❃❄❊➫❍❏●✳●❖P✑❀✉●■❉✮▼➫❉❏❵✰●❖P✑❀❄❅❖❴❭✾❆❣❲❀❄❉✮❊➭●❖P✺✾❁❅ ❣❲❉❋❜❝❍✮✾❭❊✴✈✉➾✉P✺❀ ☎✎◆❖❀✗❀✓❊➣➻ ❅✉❵r❡✑❊✭❂➧●❖✾❁❉✮❊➩✾❁❅➉❣❲❀✓❊✑❉✮●❖❀✵❣♥♣❑❛ ✤ ✕rÏ❨✦❖Ï★ ✗➧✈✉❤✐❊✩❍❯❂✓❀✓❊❑●❖◆■❉✮✾❆❣♥❂✠❉❋❴❭❴❁❉❲❂✓❍✻●■✾❭❉❋❊ ❅■❂◗P✑❀✓❜❝❀✮➚✣❃⑨❀❯❣❲✾❭❨✙✾❆❣❲❀❦●■P✑❀✆❣❲❉❋❜❝❍✮✾❭❊⑥✾❁❊❑●❖❉✩➷❚❅❖❀✓❩✮❜❝❀✗❊❋●◗❅❩✱Ï✂ ✦❇Ï✂✺❬ ✢ ✶ ✠❭✙❪✲✳✦❳❫✵❫❳❫✝✦❖➷➃➚✳❃❄✾❰●■P Ï ✣ ✙✥✲➵❍✮❊✺❣❦Ï✽❵❴ Ò ✈✳➾✉P✑❀➉❂◗P✭❍❏◆■❩✮❀▲❣❲❀✗❊✺❅❇✾❭●q❛❦➘❢✾❆❅➍❍❋❅❖❅❖❡✑❜❝❀✗❣❝●■❉➵♣✰❀❬▼✑✾❁❀✗❂✓❀✓❃❄✾❆❅❇❀✎❂✠❉❋❊✺❅q●◗❍❏❊❑●❫❉❋❊ ❀✗❍❋❂◗P✆❉✮❵✞●■P✑❀✗❅❖❀❻✾❭❊❑●❖❀✗◆❖❨✻❍✮❴❁❅✗✈➃➾✉P✑❀❻▼✭❉✮●❖❀✓❊❑●■✾❁❍✮❴✸➚✭➪ ✾❁❅✉●■P✑❀✓❊☞❀✓❨✻❍✮❴❭❡✺❍❏●❖❀✵❣♦❍❏●⑨●■P✑❀③❂✠❀✗❊❑●❖◆■❉✮✾❆❣✑❅⑨Ï✘✷✹✸➧✈ ➾✉P✑✾❆❅✣◆❖❀✵❅❇❡✑❴❭●■❅✣✾❁❊➭➷✆❀✗t❑❡✺❍✻●■✾❭❉❋❊✺❅✣✾❭❊➫➷❯❨✻❍❏◆■✾❆❍❏♣✑❴❁❀✗❅✗➚✛●❖P✑❀✉❂✠❉❋❴❭❴❁❉❲❂✓❍❏●❖✾❁❉✮❊➵❃⑨❀✓✾❁❩✮P❑●◗❅ ￾✃ ✦❖❐❀✙❛✴❜✦❳❫✵❫❳❫✝✦❖➷ ❃❄P✑✾❆❂◗P✷❂✗❍❏❊➩♣✰❀❝❃❄◆❖✾❭●❇●■❀✓❊➤✾❭❊➤❜❦❍✻●■◆❖✾❭Ñ❢❵r❉✮◆■❜♥✈ ✓❡✑◆➉●■❍❋❅❇➴❢✾❆❅✎●❖❉♦Ð✺◆◗❅q●❈❀✓❨✻❍✮❴❭❡✺❍❏●❖❀➭●❖P✑❀❝❀✗❊❋●■◆❖✾❁❀✗❅ ❉❏❵➍●❖P✑❀❦❜❦❍❏●❖◆■✾❰Ñ➩❍✮❊✺❣✷❅❖❡✑♣✺❅❖❀✗t❑❡✑❀✗❊❑●❖❴❁❛➩❅❇❉❋❴❭❨❋❀➵●■P✑❀❯❅❖❀✠●❈❉❏❵⑨❀✗t❑❡✺❍✻●■✾❭❉❋❊✺❅✗✈✖❝❬❉❏●❖❀➫●■P✺❍✻●❈●■P✑❀❝❐✠ ❒✥❮ ❀✓❊❑●■◆❖❛ ❉❏❵❈●■P✑❀➩❜❦❍✻●■◆❖✾❭Ñ❚✾❁❅☞❍❏❊❚✾❁❊❑●❖❀✗❩✮◆◗❍❏❴❬❉✮❵❈●❖P✑❀ ☎✎◆❖❀✗❀✓❊➣➻ ❅✆❵r❡✑❊✺❂➧●■✾❭❉❋❊➣➚▲❀✗❨✛❍✮❴❭❡✭❍✻●❖❀✵❣❺❍✻●♦●❖P✑❀ ❂✠❉❋❴❭❴❁❉❲❂✓❍❏●❖✾❁❉✮❊❚▼✰❉✮✾❁❊❋●❢Ï ✷ ✸◗➚➉❉✻❨❋❀✓◆❯●■P✑❀➤✾❭❊❑●❖❀✗◆❖❨✻❍✮❴❞✱Ï✂❜✦❖Ï✂✺❬ ✢ ✶✫➚❬❃❄P✑✾❆❂◗P➬✾❆❅♦●❖P✑❀➤✾❁❊❋●■❀✓◆■❨✻❍❏❴❻❉✻❨✮❀✗◆ ❃❄P✑✾❆❂◗P❢●❖P✺❀➵♣✺❍❋❅❇✾❆❅❄❵r❡✑❊✭❂➧●❖✾❁❉✮❊➤➱✂➵✾❁❅➉❊✑❉✮❊✺➲✓❀✓◆■❉✰❡r◆■❀✗❂✗❍❏❴❁❴➣●■P✺❍✻●❻❃➍❀➵P✺❍✛❨✮❀➵❂◗P✺❉❋❅❖❀✓❊✩❍❯▼✺✾❭❀✵❂✠❀✓❃❄✾❆❅❖❀ ❂✠❉❋❊✺❅❇●■❍❏❊❑●➵❍✮▼✑▼✑◆■❉✛Ñ✙✾❁❜❦❍✻●■✾❭❉❋❊❵❢➧✈♦❤✫❵q➚✞P✑❉✻❃⑨❀✓❨❋❀✓◆✵➚✴❃⑨❀❯❣❲❀✗❂✓✾❁❣✑❀✗❣➤●❖❉✷❂◗P✑❉✙❉❋❅❖❀❝❍❢❣✑✾❰➯✹❀✓◆■❀✓❊❑●➭❅❖❀✠●➵❉✮❵ ♣✺❍❋❅❇✾❆❅➍❵r❡✑❊✺❂➧●■✾❭❉❋❊✺❅✗➚✙●❖P✑✾❆❅✉✾❭❊❑●■❀✓❩✮◆◗❍❏❴✹❃⑨❉✮❡✑❴❆❣❯♣✰❀❻❊✑❉✮❊✺➲✓❀✓◆■❉➵❉❋❊✑❴❁❛❯❉✮❊✆●❖P✑❀❈❅❖❡✑▼✑▼✰❉✮◆❖●✉❉✮❵✴●■P✑❀❻♣✺❍✮❅❖✾❁❅ ❵r❡✑❊✺❂✠●❖✾❁❉✮❊✺❅❣❡✥❉✮◆✵➚❄●❖❉ ♣✭❀➤▼✑◆■❀✗❂✓✾❁❅❖❀✮➚▲❉❋❊❚●❖P✺❀✷✾❁❊❑●❖❀✓◆◗❅❖❀✗❂➧●■✾❭❉❋❊❺❉❏❵➵●❖P✑❀⑥❅❇❡✺▼✑▼✭❉❋◆❇●☞❉❏❵③●■P✑❀➤♣✺❍✮❅❖✾❁❅ ❵r❡✑❊✺❂✠●❖✾❁❉✮❊❢❍❏❊✭❣❭✱✲✳✦✵✴✝✶✪❢➧✈ ➼

3.2 Simple Quadrature Scheme 4 Area under the approximated by a rectangle x Note 4 Fdr, 2u, iu u wint yu, Mhidni an, ra, dn, 2u, dpi df duoarydpint a tddd nau aiiay yaa, Int, 2u in, urtraydf a far ddu ain [0, 1 aMau u, 2a,, 2u in, strand iMa T dd, 2"fani, idn,, 2dat2 o u o iau inu, 2iM aMnau p, idn ar. FirM o u aru tdint d duouydp a naion apprdai 2 fdr d wainint a tddd apprdxiu a, idn df, 2u in, utraydf, 2iMfani, idn dn, 2iMin, aoav o 2142 o u i ary, 2u lihu pyu Aaadra, ara h2wu a' u iu pyuma, 2int o u i an dd im, d rupa u, 2u in, utrayo 1,2,2 2u, 2u prddai df, 2u in, atrand, uoayaa, aNd a, a pdin, in /idu, 2u in, uroay and, 2u yant, 2 df, 2u 21i 2 in, 2iMi af imani, S Ifo u 22ddml, 2 df iun, rdid df, win, uroaw i u a=0.5, o uav, 2u 122n u"u idpdin, Aaadra, aru d uidpdin, Aaadra, aru M2 u u rupyaiuM, 2u arua andu, 2u i arou f(a)ws a ran, ant yo 2dM2ute, iM2u fani, idn f (a)upaya, d a, 2=0.5. T2w M2u uiM acaai, o 2m f(a)iMa idnMan,. Hdo uour, o 2a, iMu/Md oida MiM 2a,, 2u m2u u iMicxai, o 2m f(a) iMa invar fani, idn df a aM uy T2u u dM d woida Mo as df Mint, 2iMiMS ruajizint, 2a, o 2am f(a)iMa Trait 2, ynu,, 2u arua andur i, iM T2iM,rapuzdid 2a y,2 aM. 2a in,tray(ian Sd La, M,rS, d dario, 2iMin a ite, s diffar an, o as IMad df, 2u in, aroay wint [0,1] o u idn /idar an in, array[o, h],h>0, davit u dru tumuray Wu u aS apnd∫(x)aa,2uim, rid df,2iMn,moay=盘 )=f(2)+△( ∫ or some∈[0, runi aindar. Lu, Min, atra, 1, 2iMhxcapan Didn doar, 2 u in, ar cay[o, h] h d2f(e) f(e)dr=h/(2)+24 dr2 Hami u, 2u uardr in, 2uuidpdin, Aaadra, aru apprdxiu a, idn iM f() h df(e)

✌✉❾✸❽ ￾✉➄✂✁➐☎✄✝✆✟✞➇▲➁✡✠▲➒✑➁✣➋✑➇▲➒☛✆☞￾❄➊✺➓✌✆✍✁✎✆ ➙✰➛❑➜➞➝✴➟✑✏ ✛✢ ✣ ✓✒ ✕rÏ ✗✬✫❋Ï✕✔ ✒ ✖ ✴✗✙✘ Normalized 1-D Problem Simple Quadrature Scheme f x( ) x 0 1 1 2 Area under the curve is approximated by a rectangle ➡➤➢✧➥➧➦✕✚ ✘✑❉❋◆✿●■P✑❀▲●■✾❭❜❝❀✎♣✭❀✗✾❭❊✑❩➭❴❭❀✓●■❅⑨❂✠❉❋❊✺❂✠❀✗❊❋●■◆■❍❏●❖❀▲❉❋❊❝●❖P✑❀➉●❖❉❋▼✑✾❆❂❄❉❏❵✴❣❲❀✗❨✮❀✗❴❭❉❋▼✑✾❭❊✺❩➵❍③❩❋❉❑❉❲❣❝❊✙❡✑❜❝❀✓◆■✾❆❂✓❍❏❴ ●❖❀✵❂◗P✑❊✑✾❆t❑❡✑❀▲❵r❉❋◆➍❀✗❨✛❍✮❴❭❡✭❍✻●❖✾❁❊✑❩❈●■P✑❀✎✾❭❊❑●❖❀✗❩✮◆◗❍❏❴✭❉❏❵✴❍❈❵r❡✺❊✺❂➧●■✾❭❉❋❊✒ ✕rÏ ✗❫❉❋❊❦●❖P✺❀✎❣❲❉❋❜❦❍❏✾❁❊✰✱✲ ✦❳✴✝✶✫✈✳➸✷❀ ❍✮❅■❅❖❡✑❜❝❀➉●■P✺❍✻●❬●❖P✑❀❈✾❁❊❑●❖❀✓❩❋◆■❍✮❊✺❣♦✾❁❅▲❍✜✛❖❅❖❜➫❉✙❉✮●❖P✣✢✎❵r❡✑❊✺❂➧●■✾❭❉❋❊➣➚✑●■P✑❉✮❡✺❩✮P♥❃➍❀❈❃❄✾❁❴❁❴➣❀✠Ñ✑❍❏❜❝✾❁❊✑❀❻●❖P✺✾❁❅ ❍✮❅■❅❖❡✑❜❝▼❲●❖✾❁❉✮❊☞❴❁❍❏●❖❀✗◆✗✈✁✘✣✾❭◆◗❅❇●▲❃➍❀❈❍❏◆■❀❻❩✮❉❋✾❭❊✑❩➭●❖❉❯❣✑❀✓❨✮❀✗❴❭❉❋▼☞❍➫❊✺❍✮✾❭❨❋❀❈❍❏▼✑▼✑◆■❉❋❍❋❂◗P❯❵r❉❋◆❄❉✮♣❲●◗❍❏✾❁❊✑✾❭❊✺❩ ❍➭❩✮❉✙❉❲❣❯❍❏▼✑▼✺◆❖❉✛Ñ❲✾❁❜❝❍❏●❖✾❁❉✮❊❯❉✮❵➣●❖P✺❀✎✾❁❊❑●❖❀✗❩✮◆◗❍❏❴✰❉❏❵✴●■P✑✾❆❅❫❵r❡✺❊✺❂➧●■✾❭❉❋❊♦❉❋❊❯●❖P✺✾❁❅✉✾❁❊❑●❖❀✓◆■❨✻❍❏❴➮➚✙❃❄P✑✾❁❂◗P♦❃➍❀ ❂✓❍✮❴❭❴✹●■P✑❀ ✖✠❅❖✾❁❜➫▼✺❴❭❀❈t❑❡✺❍✮❣✑◆■❍❏●❖❡✑◆■❀✎❅■❂◗P✑❀✗❜➫❀ ✖✻✈ ➾✉P✑❀✆❅❖✾❁❜➫▼✺❴❭❀✵❅q●③●■P✑✾❁❊✑❩➩❃⑨❀✆❂✓❍✮❊✲❣❲❉✩✾❆❅❈●❖❉✷◆❖❀✗▼✑❴❆❍✮❂✠❀❝●■P✑❀❯✾❁❊❑●❖❀✓❩❋◆■❍✮❴✿❃❄✾❭●❖P✲●■P✑❀❯●■P✑❀❯▼✑◆■❉❲❣❲❡✺❂➧● ❉❏❵✣●❖P✺❀❈✾❭❊❑●❖❀✗❩✮◆◗❍❏❊✺❣➣➚❲❀✓❨✻❍❏❴❁❡✺❍✻●■❀✗❣☞❍✻●❬❍➫▼✰❉✮✾❁❊❋●▲✾❁❊✺❅❖✾❁❣✑❀❻●❖P✑❀③✾❁❊❑●❖❀✗◆❖❨✻❍❏❴➮➚✑❍❏❊✭❣✆●■P✑❀❈❴❁❀✓❊✑❩✮●❖P♥❉❏❵✣●❖P✑❀ ✾❁❊❋●■❀✓◆■❨✻❍❏❴➮➚❲❃❄P✑✾❁❂◗P♥✾❁❊☞●❖P✑✾❆❅▲❂✓❍❋❅❇❀❻✾❆❅❄❡✑❊✑✾❭●q❛✮✈❫❤✫❵✣❃⑨❀③❂◗P✑❉✙❉❋❅❖❀❬●■P✑❀③▼✰❉✮✾❁❊❑●▲❉❏❵✞❀✗❨✻❍❏❴❁❡✺❍✻●■✾❭❉❋❊♥❍❋❅⑨●❖P✑❀ ❂✠❀✗❊❑●❖◆■❉✮✾❆❣➭❉✮❵✹●❖P✑❀▲✾❁❊❑●❖❀✗◆❖❨✻❍❏❴➮➚✮✾➮✈ ❀✮✈✳Ï ✙✾✲ ❫ ✤✑➚✮❃⑨❀▲❂✓❍✮❴❭❴✺●❖P✑❀❬❅■❂◗P✑❀✗❜➫❀✙✖✠❜❝✾❁❣❲▼✰❉✮✾❁❊❑●➍t❑❡✺❍✮❣✑◆■❍❏●❖❡✑◆■❀✤✖❏✈ ♠ ❜❝✾❁❣✑▼✭❉❋✾❭❊❑●♦t❑❡✺❍❋❣❲◆■❍❏●❖❡✑◆■❀♥❅❖❂◗P✑❀✗❜❝❀☞◆■❀✓▼✺❴❁❍❋❂✠❀✗❅➫●■P✑❀✩❍❏◆■❀✗❍➤❡✑❊✭❣❲❀✓◆❦●■P✑❀✩❂✠❡✺◆❖❨❋❀ ✒ ✕✥Ï✘✗❝♣✙❛ ❍ ◆■❀✗❂➧●◗❍❏❊✑❩❋❴❭❀❄❃❄P✺❉❋❅❖❀▲P✑❀✓✾❁❩✮P❑●❫✾❆❅➃●■P✑❀▲❵r❡✺❊✺❂➧●■✾❭❉❋❊ ✒ ✕rÏ ✗➃❀✓❨✻❍✮❴❭❡✺❍❏●❖❀✵❣❦❍✻●➍Ï ✙✥✲✳❫✥✤❲✈✳➾✉P✑❀❬❅■❂◗P✑❀✗❜➫❀❬✾❁❅ ❀✠Ñ✑❍❋❂➧●✿❃❄P✑❀✗❊ ✒ ✕rÏ ✗➃✾❆❅❫❍❈❂✠❉❋❊✺❅❇●■❍❏❊❑●✵✈✧✦❬❉✻❃➍❀✗❨✮❀✓◆✵➚❏❃❄P✺❍❏●❫✾❆❅✿❴❁❀✗❅■❅✿❉✮♣✙❨✙✾❁❉✮❡✺❅➃✾❆❅➃●■P✺❍✻●❫●❖P✑❀✎❅❖❂◗P✑❀✗❜❝❀ ✾❆❅➉❀✓Ñ✑❍✮❂➧●❻❃❄P✑❀✗❊ ✒ ✕rÏ ✗➉✾❁❅❻❍♦❴❭✾❁❊✑❀✗❍✮◆❬❵r❡✺❊✺❂➧●■✾❭❉❋❊✷❉✮❵❫Ï✲❍✮❅➉❃➍❀✗❴❭❴➮✈➵➾✉P✑❀➫❜❝❉❋❅❇●✎❉❋♣❑❨✙✾❁❉✮❡✺❅➉❃✉❍✛❛☞❉✮❵ ❅❖❀✓❀✓✾❁❊✑❩➫●❖P✺✾❁❅❄✾❆❅❄♣✙❛❯◆■❀✗❍✮❴❭✾❁➲✓✾❁❊✑❩➫●❖P✺❍❏●▲❃❄P✑❀✓❊ ✒ ✕✥Ï✘✗⑨✾❆❅▲❍❝❅q●■◆■❍✮✾❭❩❋P❋●❄❴❁✾❭❊✺❀✮➚❲●■P✑❀③❍❏◆■❀✗❍➭❡✑❊✺❣❲❀✗◆▲✾❭●▲✾❁❅ ❍♦●❖◆◗❍❏▼✰❀✓➲✗❉✮✾❆❣✧✈➫➾✉P✺✾❁❅✎●■◆■❍✮▼✭❀✗➲✓❉✮✾❆❣✩P✭❍✮❅❻❀✠Ñ✑❍❋❂➧●❖❴❁❛♥●❖P✺❀❯❅❖❍✮❜➫❀❝❍❏◆■❀✗❍☞❍✮❅✎●■P✑❀❝◆❖❀✵❂➧●◗❍❏❊✑❩❋❴❭❀➫❃❄P✑✾❆❂◗P ●❖P✺✾❁❅▲❅■❂◗P✑❀✗❜➫❀❻❡✺❅❖❀✗❅⑨●■❉❦❍❏▼✑▼✑◆■❉✛Ñ❲✾❭❜❦❍❏●❖❀➉●❖P✑❀❈✾❁❊❑●❖❀✓❩❋◆■❍✮❴☎❡✥❂✓❍✮❊♥❛✮❉✮❡☞❅❇❀✗❀❈❃❄P✙❛✛ ❢✠✈ ➺➣❀✓●■❅▲●❖◆■❛♦●■❉✆❣❲❀✗◆❖✾❁❨✮❀❈●■P✑✾❆❅❬✾❁❊➩❍❯❅❖❴❭✾❁❩✮P❑●■❴❭❛♥❣❲✾❭➯✰❀✗◆❖❀✗❊❋●✎❃⑨❍✛❛❋✈❄❤✐❊✺❅❇●❖❀✗❍❋❣♥❉❏❵➃●■P✑❀➵✾❁❊❑●❖❀✓◆■❨✻❍❏❴✞♣✭❀✗✾❭❊✑❩ ✱✲✳✦❳✴❳✶➍❃⑨❀♦❂✓❉✮❊✺❅❖✾❁❣✑❀✓◆➭❍✮❊➆✾❁❊❋●■❀✓◆■❨✻❍❏❴ ✱✲✳✦✩★✶✹✦✪★✬✫ ✲❢●■❉➩♣✰❀♦❍➩♣✑✾❰●❝❜❝❉✮◆■❀❯❩✮❀✗❊✑❀✓◆◗❍❏❴➮✈✩➸➤❀♦❜❦❍✛❛ ❀✠Ñ✑❍✮▼✑❊✺❣ ✒ ✕rÏ✘✗✉❍✮♣✭❉❋❡❲●❄●❖P✺❀③❂✠❀✓❊❑●■◆❖❉❋✾❁❣✆❉❏❵✣●❖P✑✾❆❅❄✾❁❊❋●■❀✓◆■❨✻❍❏❴➮➚✮✭Ï ✙ ❮✯ ✒ ✕rÏ ✗✚✙ ✒ ✕✰✭Ï✘✗✍✱✳✲ ✕rÏ ✗ ✫✒ ✕✴✭Ï ✗ ✫❋Ï ✱ ✲ ✕✥Ï✘✗ ✯ ✗✶✵ ✫ ✯ ✒ ✕✸✷❜✗ ✫❋Ï✯ ✒✺✹✼✻✾✽✿✹✼❀❂❁ ✷✖✯❣✱✲✳✦✩★✶ ❃❄P✑❀✗◆❖❀❃✲ ✕✥Ï✘✗❂✙ Ï❅❄❆✭Ï✞✈❸➾✉P✺❀✷❴❆❍✮❅❇●♦●■❀✓◆■❜ ✾❭❊ ●❖P✺❀➤❀✓Ñ❲▼✺❍❏❊✺❅❖✾❁❉✮❊❺✾❆❅♦●❖P✑❀✲➾✣❍✛❛❑❴❁❉✮◆☞❅❇❀✗◆❖✾❁❀✗❅ ◆■❀✓❜❦❍❏✾❁❊✺❣❲❀✗◆✗✈✳➺➣❀✓●■❅❄✾❁❊❑●❖❀✓❩❋◆■❍❏●❖❀➉●❖P✑✾❆❅❄❀✓Ñ❲❍✮▼✺❍❏❊✭❅❇✾❁❉✮❊♦❉✻❨✮❀✓◆⑨●■P✑❀❈✾❭❊❑●■❀✓◆■❨✛❍✮❴✚✱✲ ✦✪★✳✶ ✛ ❮ ✣ ✒ ✕rÏ ✗✬✫❋Ï ✙❇★✒ ✕✰✭Ï✘✗✍✱ ★☛❈ ✗❊❉ ✫ ✯ ✒ ✕❋✷❜✗ ✫❋Ï✯ ✦▲❀✗❊✺❂✠❀✎●■P✑❀❈❀✓◆■◆❖❉❋◆✉✾❭❊☞●❖P✺❀❈❜➫✾❆❣❲▼✰❉✮✾❁❊❑●❬t❋❡✭❍✮❣❲◆◗❍✻●■❡✑◆❖❀❻❍❏▼✺▼✑◆❖❉✛Ñ❲✾❁❜❦❍✻●❖✾❁❉✮❊✆✾❁❅ ● ✙ ✛ ❮ ✣ ✒ ✕rÏ✘✗❚✫✮Ï❍❄❃★✒ ✕✴✭Ï ✗ ✙ ★■❈ ✗❊❉ ✫ ✯ ✒ ✕❋✷❜✗ ✫❋Ï✯ ❏

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Key questions about the method ors decay with n Are there better methods? As you can see dividing the interval into two reduces the error and there is no reason to stop at just two subinter vals when we can have n subintevals and repeat our midpoint quadrature rule on each subinterval. We obtain the scheme f(x)dx=∑ 2(+) no doubt that we gain, but the key question is by how much? How does this gain scale with the number of subintervals used? And finally, are there clever ways of obtaining better accuracy with less effort? 3.2.3 Numerical Example SLIDE 7 E Note 6 Lets look at an example of integrating f(a)= sin(a) on our domain. We obtain progressively bet ter answers to the integral by increasing the number of subintervals n. The error in evaluating the integral is plotted as a function of the number of subintervals(n). The error appears to be going down as o(-) why From what we have just seen, the error inside the ith subinterval(of length sh=i,is adia for some Si E[=, i]. Hence, for the entire inter val [0, 1] d obtain the error at ion usin

￾❀✓❛♦t❋❡✺❀✗❅❇●❖✾❁❉✮❊✺❅❄❍✮♣✭❉❋❡❲●✉●❖P✑❀❈❜❝❀✠●■P✑❉❲❣✁￾ ✂❢✪☎✄ ★✫✍✑✏✗✔❝❱✣✪⑥✔✗✚✣✖✩✖❑✬✛✬✻✪✺✬✻✏❦❱✣✖❑✘❋✍✮✒✆✄❦✢r✔✵✚ ✜✞✝ ✆❯✬✻✖❢✔✵✚✞✖❑✬✻✖✠✟❫✖❑✔✗✔✗✖❑✬❯✯➆✖❑✔✗✚✣✪➣❱✣✏✡✝ ➡➤➢✧➥➧➦✝☛ ♠❅❬❛✮❉❋❡♥❂✗❍❏❊✩❅❖❀✓❀➵❣✑✾❭❨✙✾❆❣❲✾❭❊✺❩❦●❖P✑❀➵✾❭❊❑●❖❀✗◆❖❨✻❍✮❴✴✾❁❊❋●■❉❝●q❃➍❉✆◆❖❀✵❣❲❡✺❂✠❀✵❅✉●❖P✺❀➵❀✓◆■◆■❉✮◆▲❍❏❊✭❣☞●❖P✑❀✗◆❖❀❈✾❆❅▲❊✑❉ ◆■❀✗❍✮❅❖❉✮❊➆●❖❉✲❅q●■❉✮▼➑❍✻● ✏q❡✺❅❇●❦●q❃➍❉⑥❅❖❡✑♣✑✾❁❊❋●■❀✓◆■❨✻❍❏❴❆❅❝❃❄P✑❀✓❊ ❃➍❀❢❂✗❍❏❊ P✭❍✛❨✮❀☞➷ ❅❖❡✑♣✑✾❁❊❑●❖❀✓❨✻❍✮❴❁❅❝❍❏❊✭❣ ◆■❀✓▼✰❀✗❍✻●➃❉❋❡✑◆➃❜❝✾❁❣❲▼✰❉✮✾❁❊❑●➍t❑❡✺❍✮❣✑◆■❍❏●❖❡✑◆■❀➍◆■❡✑❴❁❀❄❉✮❊❝❀✗❍❋❂◗P➫❅❖❡✑♣✑✾❁❊❑●❖❀✗◆❖❨✻❍❏❴➮✈✴➸➤❀▲❉❋♣❲●■❍✮✾❭❊➭●❖P✑❀➉❅❖❂◗P✑❀✗❜❝❀ ✛✢ ✣ ✒ ✕rÏ ✗✬✫❋Ï ✙ ✫✽ ✃✿ ✢ ✴ ➷ ❈❉✝❊❋ ❅❖❡✑♣✑✾❁❊❑●❖❀✓◆■❨✻❍❏❴ ❴❁❀✓❊✑❩✮●❖P ✒ ✕✥Ï✷ ✸✬✗ ❃❄P✑❀✗◆❖❀❈●■P✑❀❝❂✠❀✓❊❑●■◆❖❉❋✾❁❣♥❉✮❵✿●❖P✑❀➭❐✫❒✥❮✆❅❖❡✑♣✑✾❁❊❑●❖❀✓◆■❨✻❍❏❴✣✾❁❅➉Ï✷ ✸✟✙ ✢✯ ✕ ✃☞☛ ✽ ✢ ✱ ✃✽ ✗☞✙ ✃✌☛ ❅✍ ✽ ✈❈➾✉P✑❀✗◆❖❀➵✾❁❅ ❊✑❉♥❣❲❉✮❡✺♣❲●❈●❖P✺❍❏●❈❃➍❀❝❩❋❍✮✾❭❊✴➚✧♣✑❡❲●❈●❖P✺❀➫➴❋❀✓❛➩t❑❡✑❀✵❅q●■✾❭❉❋❊➩✾❆❅❻♣✙❛✩P✑❉✻❃➶❜➭❡✺❂◗P✛ ✦▲❉✻❃ ❣❲❉✙❀✗❅✎●❖P✺✾❁❅ ❩❋❍✮✾❭❊✷❅❖❂✗❍❏❴❁❀➭❃❄✾❭●❖P✷●❖P✑❀➫❊✙❡✑❜➵♣✰❀✓◆❻❉✮❵➍❅❇❡✺♣✑✾❭❊❑●■❀✓◆■❨✛❍✮❴❁❅✎❡✺❅❖❀✗❣✛ ♠❊✭❣✩Ð✺❊✭❍❏❴❁❴❭❛❋➚➣❍❏◆■❀③●❖P✺❀✓◆■❀➫❂✓❴❭❀✗❨✮❀✓◆ ❃✉❍✛❛✙❅⑨❉✮❵✞❉✮♣✑●■❍❏✾❁❊✑✾❁❊✑❩❝♣✭❀✓●❇●■❀✓◆❬❍❋❂✓❂✠❡✺◆■❍❋❂✠❛❦❃❄✾❭●❖P♥❴❭❀✵❅❖❅✉❀✓➯✰❉❋◆❇● ✛ ✍✴→❆➔➣→ ✍ ✎✦✣✯➆✖❑✬✻✢✥✘✮✍✺✼❫✌✌➣✍✑✯✲✱✳✼✥✖ ➙✰➛❑➜➞➝✴➟✑✏ ✛✢ ✣ ✓✽ ❐➮➷✚✕✥Ï✘✗❚✫✮Ï✕✔ ✫✽ ✃✿ ✢ ✴ ➷ ✽ ❐✫➷ ✖ ❐✍❄ ✢✯ ➷ ✘ SMA-HPC ©1999 MIT Normalized 1-D Problem Simple Quadrature Scheme Numerical Example E r r o r n ➡➤➢✧➥➧➦✓✒ ➺➣❀✓●■❅♦❴❭❉✙❉❋➴➑❍❏●☞❍✮❊❺❀✠Ñ✑❍❏❜❝▼✑❴❁❀➩❉✮❵③✾❭❊❑●■❀✓❩✮◆◗❍✻●■✾❭❊✺❩ ✒ ✕rÏ ✗❭✙ ✽ ❐➮➷✚✕✥Ï✘✗✆❉✮❊❺❉❋❡✑◆♥❣❲❉✮❜❦❍❏✾❁❊➣✈ ➸✷❀ ❉✮♣✑●■❍❏✾❁❊☞▼✑◆❖❉❋❩✮◆■❀✗❅■❅❇✾❁❨✮❀✗❴❭❛➫♣✰❀✠●❇●■❀✓◆▲❍✮❊✺❅❇❃⑨❀✓◆◗❅➍●❖❉➫●❖P✺❀❈✾❭❊❑●❖❀✗❩✮◆◗❍❏❴✹♣✙❛❯✾❁❊✺❂✠◆■❀✗❍❋❅❇✾❁❊✑❩➭●❖P✑❀❻❊✙❡✑❜➵♣✰❀✓◆❄❉✮❵ ❅❖❡✑♣✑✾❁❊❋●■❀✓◆■❨✻❍❏❴❆❅❬➷➃✈❈➾✉P✑❀➫❀✓◆■◆❖❉❋◆❬✾❁❊➩❀✗❨✛❍✮❴❭❡✭❍✻●❖✾❁❊✑❩✆●❖P✑❀❝✾❁❊❋●■❀✓❩❋◆■❍✮❴✴✾❆❅➉▼✺❴❭❉✮●❇●❖❀✵❣➩❍❋❅➉❍✆❵r❡✑❊✺❂✠●❖✾❁❉✮❊➩❉✮❵ ●❖P✺❀➉❊✙❡✑❜➭♣✭❀✗◆⑨❉✮❵✴❅❇❡✺♣✑✾❭❊❑●■❀✓◆■❨✛❍✮❴❁❅ ❡r➷❨❢✠✈✿➾✉P✑❀✎❀✓◆■◆❖❉❋◆➍❍❏▼✺▼✭❀✵❍❏◆◗❅✿●■❉➭♣✰❀➉❩❋❉✮✾❁❊✑❩➭❣✑❉✻❃❄❊✆❍✮❅✕✔✗✖ ✽✢ ✍☎✘ ✈ ➺➣❀✓●■❅❄❅❖❀✓❀❻❃❄P✙❛✮✈ ✘✑◆■❉✮❜ ❃❄P✺❍✻●❦❃⑨❀☞P✭❍✛❨✮❀ ✏q❡✺❅q●✆❅❖❀✓❀✗❊➣➚➍●❖P✑❀❢❀✗◆❖◆■❉✮◆➫✾❭❊✭❅❇✾❆❣❲❀♥●❖P✺❀☞❐✫❒✥❮✲❅❖❡✑♣✑✾❁❊❑●❖❀✗◆❖❨✻❍❏❴✂❡✥❉❏❵❻❴❭❀✗❊✑❩❏●■P ✙ ★ ✙ ✽✢ ➻ ❢❄✾❆❅ ❮✛✚ ✯ ✎ ❙ ✍✢✜✤✣✦✥ ✸✌✧ ❙ ❁ ✍ ❵r❉❋◆❬❅❖❉✮❜❝❀✜✷✓✃ ✯ ✱ ✃✌☛ ✽ ✢ ✦ ✃✽ ✶➮✈ ✦❬❀✓❊✺❂✓❀✮➚✭❵r❉✮◆❄●■P✑❀➵❀✗❊❑●❖✾❁◆❖❀❈✾❁❊❑●❖❀✓◆■❨✻❍❏❴ ✱✲ ✦❳✴✝✶ ❃⑨❀➫❂✗❍❏❊⑥❅❇❡✑❜ ●❖P✑❀✵❅❇❀➫❀✓◆■◆❖❉❋◆■❅✎❍❏❊✭❣➩❉❋♣❲●■❍✮✾❭❊➩●❖P✑❀❝❀✓◆■◆■❉✮◆✵➚✺●✽ ❵r❉❋◆❻❍✮❊➤❍✮▼✑▼✑◆■❉✛Ñ❲✾❭❜❦❍✻●■✾❭❉❋❊❢❡✺❅❖✾❭❊✑❩ ★

subintervals as E d2f si) It is easy to see that if f(a)is a continuous function, 'M(being the mean)must be bounded by the maximum and the minumum of f (a) on the interval [0, 1 and hence, there must exist some S E [0, 1]such that M=d2f(s)/dx2. Hence we obtain the estimate E h3d2f(5)1d2f(5) 24n2d since h= 1/n. This error estimate tells us that the scheme is again exact for constants and linear functions on the domain (no higher order polynomials and, for a smooth function, the error decays algebraically 3.3 General Quadrature Scheme 3.3.1 General 1D Form SLIDE 8 ∫(x)d ∫(x1) Free to pick the evaluation points Free to pick the weights fe An n-point formula has 2n degrees of freedom! After all the hard work we did dividing the domian into subintervals, we realize that we cannot even integrate a parabola exactly on the domain. There must e son we can back and look at the general form of the quadrature approximation sche Al approximating an integral by a weighted sum of function evaluations as shown n this slide. So far we have been choosing these weights as the subinterval lengths. We have also been choosing all the evaluation points. The weights are alizing factors which ensure that the ximation is exact if f(a)=l and the equality of areas of trapezoids and rectangles that we discussed gives us the extra polynomial accuracy of being able to obtain the area under straing line exactly. So, what would happen if ere to choose both the integration points and the weights intelligently? For an n-point formula we have

➻➷➃➻✑❅❇❡✑♣✺✾❭❊❑●❖❀✗◆❖❨✻❍✮❴❁❅❄❍❋❅ ● ✽ ✙ ➷✍★■❈ ✗❊❉ ￾ ✴ ➷ ✫✽ ✃✿ ✢ ✫ ✯ ✒ ✕❋✷✗✃✒✗ ✫❋Ï✯✂✁ ❈ ❉✝❊ ❋ ✷ ◆❇❖ ❖✹❒✥❮ ✃☎✄✝✆✞✠✟ ❤✫●➃✾❁❅✿❀✗❍✮❅❖❛❻●❖❉❈❅❖❀✓❀⑨●❖P✭❍✻●✿✾❭❵ ✒ ✕✥Ï✘✗✣✾❆❅➃❍✎❂✓❉✮❊❑●❖✾❁❊✙❡✑❉✮❡✭❅✴❵r❡✺❊✺❂➧●■✾❭❉❋❊➣➚ ✙☛✡➻✳❡r♣✰❀✓✾❁❊✑❩❻●❖P✺❀❄❜➫❀✵❍❏❊❵❢✞❜➭❡✺❅q● ♣✰❀❦♣✭❉❋❡✑❊✺❣❲❀✵❣➩♣✙❛❢●■P✑❀❦❜❦❍✻Ñ❲✾❭❜➭❡✑❜ ❍✮❊✺❣✩●■P✑❀❦❜❝✾❭❊✙❡✑❜➭❡✑❜ ❉✮❵ ✒ ✕✥Ï✘✗✎❉✮❊✷●❖P✺❀❦✾❭❊❑●❖❀✗◆❖❨✻❍✮❴☞✱✲ ✦❳✴✝✶ ❍❏❊✭❣♥P✑❀✓❊✺❂✓❀✮➚✰●❖P✺❀✓◆■❀③❜➵❡✺❅❇●➉❀✓Ñ❲✾❁❅❇●✎❅❖❉✮❜❝❀✜✷ ✯✏✱✲✳✦❳✴❳✶✞❅❖❡✺❂◗P❢●❖P✭❍✻● ✡ ✙✡✫✯ ✒ ✕❋✷❜✗✌☞❘✫❋Ï✯ ✈✌✦▲❀✗❊✺❂✠❀ ❃⑨❀✎❉❋♣❲●■❍✮✾❭❊♦●❖P✺❀❈❀✗❅❇●❖✾❁❜❝❍❏●❖❀ ● ✽ ✙ ➷✍★☛❈ ✗❊❉ ✫ ✯ ✒ ✕❋✷❜✗ ✫❋Ï✯ ✙ ✴ ✗❊❉➷ ✯ ✫ ✯ ✒ ✕❋✷❜✗ ✫❋Ï✯ ❅❖✾❭❊✺❂✓❀ ★✔✙ ✴✍☞✻➷➃✈③➾✉P✑✾❁❅❻❀✗◆❖◆■❉✮◆➉❀✗❅❇●❖✾❁❜❦❍✻●❖❀➭●❖❀✗❴❭❴❆❅❻❡✺❅➉●❖P✭❍✻●❻●❖P✑❀❦❅■❂◗P✑❀✓❜❝❀➫✾❁❅❻❍✮❩❋❍❏✾❁❊➩❀✓Ñ✑❍✮❂➧●✎❵r❉✮◆ ❂✠❉❋❊✺❅❇●■❍❏❊❑●◗❅➭❍✮❊✺❣➆❴❁✾❭❊✺❀✗❍❏◆➭❵r❡✑❊✺❂✠●❖✾❁❉✮❊✺❅➫❉❋❊➆●❖P✺❀♥❣✑❉✮❜❦❍❏✾❁❊✥❡r❊✺❉➤P✺✾❭❩❋P✑❀✓◆➫❉✮◆◗❣❲❀✓◆➫▼✰❉✮❴❁❛❑❊✺❉✮❜❝✾❁❍✮❴❁❅✏✎ ❢ ❍❏❊✭❣✧➚✙❵r❉✮◆▲❍❝❅❖❜➫❉✙❉✮●❖P☞❵r❡✑❊✺❂✠●❖✾❁❉✮❊➣➚❲●■P✑❀❈❀✓◆■◆■❉✮◆✉❣❲❀✗❂✗❍✛❛✙❅❄✍✺✼✥✶✰✖✟✣✬✻✍✑✢✥✘❋✍✑✼✥✼✥✒ → ✌✉❾✒✌ ✑✆✴➈✆➣➒✺➁✮✄ ✞➇▲➁✡✠❬➒✑➁✣➋❲➇❬➒✆✎￾✉➊✭➓✆✍✁✎✆ ✍✴→ ✍✴→✥↔ ❘✖✙✜✣✖❋✬✻✍✑✼ ↔✎ ★✴✪✭✬✻✯ ➙✰➛❑➜➞➝✴➟✓✒ ✛✢ ✣ ✒ ✕rÏ ✗✬✫❋Ï❂✔ ✫✽ ✃✿ ✢✕✔ ✃ ❈❉✝❊❋ ✖ ❏ ✃✘✗ ❮✗❒ ✒ ✕✥Ï✭✃✹✗ ❈ ❉❊ ❋ ✙▲✝◆❇❖ P◆➧❒ ✃ ●✽✛✚● ✃✽❒ ✘✑◆■❀✓❀✎●■❉➫▼✺✾❁❂◗➴✆●❖P✑❀③✖✄✑✍✑✼✸✦✞✍❲✔✵✢✥✪✰✜ ✱❫✪✭✢✸✜✹✔✗✏❋✈ ✘✑◆■❀✓❀✎●■❉➫▼✺✾❁❂◗➴✆●❖P✑❀ ✄➵✖✙✢✸✶✭✚✹✔✵✏✎❵r❉✮◆❄❀✗❍❋❂◗P✆▼✰❉✮✾❁❊❑●✗✈ ✆✆✜❚✜✥✤❖✱➍✪✭✢✸✜✹✔➫★✫✪✭✬✛✯➤✦✣✼✥✍➩✚✣✍✑✏ ➔ ✜➑❱✞✖✙✶✭✬✛✖✙✖✙✏❝✪✭★❬★✸✬✻✖✙✖✙❱✞✪✰✯✢✜ ➡➤➢✧➥➧➦✤✣ ♠❵➞●■❀✓◆✉❍❏❴❁❴✺●■P✑❀❬P✭❍❏◆◗❣❝❃➍❉❋◆❖➴➫❃➍❀➉❣❲✾❁❣♦❣❲✾❁❨❑✾❆❣❲✾❁❊✑❩③●■P✑❀➉❣✑❉✮❜❝✾❁❍✮❊❦✾❭❊❑●■❉➭❅❖❡✑♣✑✾❁❊❋●■❀✓◆■❨✻❍❏❴❆❅✓➚❑❃➍❀❬◆❖❀✵❍❏❴❁✾❭➲✗❀ ●❖P✭❍✻●❻❃➍❀❝❂✓❍✮❊✑❊✑❉❏●❻❀✓❨❋❀✓❊✩✾❁❊❑●❖❀✗❩✮◆◗❍✻●❖❀➫❍✆▼✭❍❏◆◗❍❏♣✰❉✮❴❆❍❦❀✠Ñ✑❍✮❂✠●❖❴❁❛❢❉✮❊➩●❖P✑❀❝❣❲❉✮❜❦❍✮✾❭❊➣✈③➾✉P✺❀✓◆■❀➭❜➭❡✺❅q● ♣✰❀➵❅❇❉❋❜❝❀✠●❖P✺✾❭❊✑❩❯●❖P✺❍❏●❬❃➍❀➭❂✓❍❏❊✩❣✑❉❦●❖❉❯✾❁❜❝▼✑◆■❉✻❨✮❀✎●❖P✑✾❆❅✎❅❖❂◗P✑❀✗❜❝❀✮✈✉➸✷❀➵❩✮❉❯♣✺❍❋❂◗➴♦❍✮❊✺❣♥❴❭❉✙❉❋➴♦❍✻● ●❖P✺❀➵❩✮❀✗❊✑❀✓◆◗❍❏❴➣❵r❉✮◆■❜ ❉✮❵✣●■P✑❀➭t❑❡✺❍❋❣❲◆◗❍✻●❖❡✺◆❖❀③❍✮▼✑▼✑◆■❉✛Ñ✙✾❁❜❦❍✻●■✾❭❉❋❊☞❅■❂◗P✑❀✓❜❝❀❋✈ ♠❴❁❴✞❃⑨❀➵❍❏◆■❀③❣❲❉❋✾❭❊✑❩✆✾❁❅ ❍❏▼✺▼✑◆❖❉✛Ñ❲✾❁❜❦❍✻●❖✾❁❊✑❩❝❍❏❊♥✾❁❊❋●■❀✓❩❋◆■❍✮❴➣♣✙❛♦❍❝❃⑨❀✓✾❁❩✮P❑●❖❀✵❣☞❅❖❡✑❜ ❉❏❵✳❵r❡✑❊✺❂✠●❖✾❁❉✮❊❢❀✓❨✻❍✮❴❭❡✺❍❏●❖✾❁❉✮❊✺❅❄❍❋❅▲❅❖P✑❉✻❃❄❊ ✾❁❊ ●■P✑✾❁❅✆❅❇❴❁✾❁❣✑❀✮✈ s❉➤❵✥❍❏◆❦❃⑨❀☞P✭❍✛❨✮❀☞♣✭❀✗❀✓❊➑❂◗P✑❉✙❉❋❅❖✾❭❊✑❩✷●❖P✺❀✗❅❖❀☞❃⑨❀✓✾❁❩✮P❑●◗❅❦❍✮❅➭●❖P✑❀✩❅❖❡✑♣✑✾❁❊❑●❖❀✗◆❖❨✻❍❏❴ ❴❁❀✓❊✑❩✮●❖P✺❅✗✈✳➸✷❀❈P✺❍✛❨❋❀✎❍✮❴❁❅❖❉➫♣✭❀✗❀✓❊♥❂◗P✑❉✙❉❋❅❖✾❭❊✑❩❝❍❏❴❁❴✹●❖P✺❀❈❀✓❨✻❍❏❴❁❡✺❍✻●■✾❭❉❋❊✆▼✰❉✮✾❁❊❑●■❅✗✈✿➾✉P✑❀❻❃➍❀✗✾❭❩❋P❋●◗❅✉❍❏◆■❀ ✏q❡✺❅❇●③❅❇❉❋❜➫❀➫❊✑❉❋◆❖❜❦❍❏❴❁✾❁➲✓✾❁❊✑❩✆❵✥❍❋❂➧●■❉✮◆◗❅❬❃❄P✑✾❆❂◗P➤❀✓❊✺❅❖❡✑◆■❀➵●■P✺❍✻●❻●❖P✺❀❝❍✮▼✑▼✑◆■❉✛Ñ❲✾❭❜❦❍✻●■✾❭❉❋❊✩✾❆❅➉❀✓Ñ✑❍✮❂➧●❈✾❭❵ ✒ ✕rÏ ✗✚✙❛✴⑨❍❏❊✺❣③●❖P✑❀⑨❀✗t❑❡✺❍✮❴❭✾❭●q❛❻❉❏❵✺❍✮◆❖❀✵❍✮❅➣❉✮❵❲●❖◆◗❍❏▼✰❀✓➲✗❉✮✾❆❣✑❅✞❍✮❊✺❣❈◆■❀✗❂✠●■❍❏❊✺❩✮❴❁❀✗❅✴●❖P✺❍❏●✞❃⑨❀⑨❣❲✾❆❅■❂✠❡✺❅■❅❇❀✵❣ ❩✮✾❁❨✮❀✵❅▲❡✺❅➉●■P✑❀➭❀✓Ñ✙●❖◆◗❍❯▼✰❉✮❴❁❛❑❊✺❉✮❜❝✾❁❍✮❴✣❍❋❂✓❂✓❡✑◆■❍❋❂✠❛♦❉❏❵❫♣✰❀✓✾❁❊✑❩♥❍❏♣✑❴❁❀③●■❉♦❉❋♣❲●■❍✮✾❭❊❢●■P✑❀❝❍❏◆■❀✗❍❦❡✑❊✭❣❲❀✓◆ ❍❢❅❇●❖◆◗❍❏✾❁❊✑❩❢❴❭✾❁❊✑❀❯❀✓Ñ✑❍✮❂➧●■❴❭❛❋✈ s ❉✺➚✳❃❄P✺❍✻●③❃⑨❉✮❡✺❴❁❣⑥P✺❍✮▼✑▼✰❀✓❊✲✾❭❵✉❃⑨❀❯❃⑨❀✓◆■❀❝●❖❉➩❂◗P✺❉❑❉❑❅❇❀❝♣✰❉❏●■P✲●❖P✑❀ ✾❁❊❋●■❀✓❩❋◆■❍❏●❖✾❁❉✮❊③▼✰❉✮✾❁❊❑●■❅➃❍❏❊✭❣③●■P✑❀✉❃➍❀✗✾❭❩❋P❋●◗❅✣✾❭❊❑●❖❀✗❴❭❴❁✾❁❩✮❀✓❊❑●■❴❭❛✛✒✘✑❉❋◆✳❍✮❊➵➷✹➹✫▼✭❉❋✾❭❊❑●✳❵r❉✮◆■❜➵❡✺❴❁❍➉❃➍❀⑨P✺❍✛❨❋❀ ✥

weight ard'n'eval ativ pirt ts ch\\ e. That give/ 2n’ degree if freedom. Herce we m( t be able ti integrate a p lyrsmial -f degree at m i(2n-1)'. Thi imple idea give ri e ts the s called/Ga a( adrat(re cheme 2.2 Po nt-We]ht Select on Cr ter 3 SLIDE 9 Re(It h ld be exact if f(a)i a pylyesmial f(x)‘a+a1x+a2x2+ x‘p(x) Select a;’afdw;’( ch that p(a)dz:∑up(x1) frIANY ply灬mial(pt、( ard ircl( dig) 7 n Order W th 2n de] rees of freedoM, L' 2n-1 Note 8 Let pr(a) derate a plyesmial sf degree l ig the variable a(a/0. We wart ts elect the weight ard integrative pirt (ch that the form( la f(x)dx‘∑u;p(x;) exact frr all plyesmial f degree(pt\ard ircl( dig)L. Obvi( ly, with 2n degree >f freedom, the be t we ca ds i l 2n-1 .2. Why the Ex3ctness Cr ter 3? Cv ider the Taylor erie for f(a) ∫(x)f(0)+ af(0) 84f(0) x2+R+1 Rit 184+1f(Gi) where t∈[0,x Note 9 Of all f( h al The rea come frsm the tr( ct( re sf Taylsr' erie expa/ The Taylor expa i\

✙ ➷➃➻✧❃➍❀✗✾❭❩❋P❋●◗❅❈❍❏❊✺❣ ✙ ➷➃➻➣❀✓❨✻❍❏❴❁❡✺❍❏●❖✾❁❉✮❊➤▼✭❉❋✾❭❊❑●■❅✎●■❉❢❂◗P✑❉✙❉❋❅❖❀✮✈❦➾✉P✺❍✻●❈❩❋✾❭❨❋❀✗❅✎❡✺❅ ✙ ✗➷➃➻➣❣❲❀✗❩✮◆■❀✓❀✵❅➉❉✮❵ ❵r◆■❀✓❀✗❣✑❉✮❜♥✈ ✦▲❀✓❊✭❂✠❀❦❃⑨❀➫❜➭❡✺❅❇●③♣✭❀❯❍✮♣✑❴❭❀➫●■❉☞✾❁❊❑●❖❀✗❩✮◆◗❍✻●❖❀❦❍☞▼✰❉✮❴❁❛❑❊✺❉✮❜❝✾❁❍✮❴➃❉❏❵✉❣❲❀✓❩❋◆❖❀✗❀❝❍❏●❈❜➫❉❑❅q● ✙ ✕ ✗➷❃❄✡✴ ✗✠➻❄✈ ➾✉P✺✾❁❅❦❅❖✾❭❜❝▼✑❴❁❀♥✾❁❣✑❀✗❍✷❩✮✾❁❨✮❀✗❅➫◆■✾❁❅❖❀♦●❖❉➤●❖P✑❀❢❅❖❉➤❂✗❍❏❴❁❴❭❀✵❣ ✖ ☎❻❍❏❡✺❅■❅❝t❋❡✭❍✮❣❲◆◗❍✻●■❡✑◆❖❀ ✖ ❅■❂◗P✑❀✓❜❝❀✮✈ ✍✴→ ✍✴→❆➔ ✦➉✪✭✢✸✜✹✔ ✤✁￾✖❑✢✸✶✭✚✹✔✄✂✴✖✙✼✥✖✙✘❏✔✵✢✥✪✰✜ ↕✬✻✢r✔✵✖❋✬✻✢✸✍ ➙✰➛❑➜➞➝✴➟✆☎ ✝▲❀✗❅❖❡✑❴❰●❬❅❖P✑❉❋❡✑❴❁❣☞♣✰❀❈❀✠Ñ✑❍✮❂✠●❄✾❰❵ ✒ ✕rÏ ✗⑨✾❁❅❄❍❝▼✰❉✮❴❁❛✙❊✑❉✮❜❝✾❆❍❏❴ ✒ ✕rÏ ✗ ✙✟✞✣ ✱✠✞ ✢ Ï✾✱✠✞✯ Ï ✯ ✱☛✡☞✡☞✡✼✱✌✞❖ Ï❖ ✙✌✍❖ ✕✥Ï✘✗ s❀✗❴❭❀✵❂➧●▲Ï✭✃❖➻ ❅❄❍❏❊✺❣ ✔ ✃❖➻ ❅❄❅❖❡✺❂◗P✆●■P✺❍✻● ✛✢ ✣ ✍ ❖ ✕✥Ï✘✗❚✫✮Ï❱✙ ✫✽ ✃✿ ✢ ✔ ✃✎✍❖ ✕rÏ✭✃▼✗ ❵r❉✮◆ ♠❝✑✏➶▼✰❉✮❴❁❛✙❊✑❉✮❜❝✾❆❍❏❴✧❡✑▼❲●■❉✰❡✥❍✮❊✺❣✆✾❁❊✺❂✓❴❭❡✺❣✑✾❭❊✑❩✮❢✓✒r❒✥❮❝❉❋◆■❣✑❀✓◆ ￾✢r✔✵✚ ✗➷ ❱✣✖❑✶✭✬✻✖✙✖❑✏❝✪✰★▲★➮✬✻✖❑✖✙❱✣✪✭✯☛✔✕✒❨✙ ✗➷❍❄ ✴ ➡➤➢✧➥➧➦✗✖ ➺➣❀✓●✘✍❖ ✕rÏ✘✗➉❣❲❀✗❊✑❉❏●■❀➭❍❦▼✰❉✮❴❁❛✙❊✑❉✮❜❝✾❆❍❏❴✣❉❏❵✿❣❲❀✓❩❋◆❖❀✗❀✙✒✿✾❁❊❢●■P✑❀➵❨✻❍❏◆■✾❆❍❏♣✑❴❁❀③Ï ❡✚✞❖✜✛✙ ✲ ❢➧✈❬➸✷❀➫❃⑨❍✮❊❑● ●❖❉❦❅❖❀✓❴❁❀✗❂✠●❄●❖P✑❀❈❃⑨❀✓✾❁❩✮P❑●■❅✉❍✮❊✺❣☞✾❭❊❑●❖❀✗❩✮◆◗❍✻●■✾❭❉❋❊✆▼✰❉✮✾❁❊❋●◗❅❄❅❇❡✭❂◗P✆●■P✺❍✻●❄●■P✑❀❻❵r❉✮◆■❜➵❡✑❴❆❍ ✛✢ ✣ ✒ ✕rÏ ✗✬✫❋Ï ✙ ✫✽ ✃✿ ✢ ✔ ✃✍ ❖ ✕rÏ✃ ✗ ✾❆❅❻❀✠Ñ✑❍✮❂✠●✎❵r❉❋◆③❍❏❴❁❴✿▼✰❉✮❴❁❛❑❊✺❉✮❜❝✾❁❍✮❴❁❅✎❉✮❵✉❣❲❀✗❩✮◆■❀✓❀➫❡✑▼❲●■❉❂❡✥❍❏❊✭❣✷✾❁❊✺❂✓❴❭❡✺❣✑✾❭❊✑❩✮❢✢✒q✈ ✓♣✙❨✙✾❭❉❋❡✺❅❇❴❁❛✮➚✴❃❄✾❰●■P ✗➷➤❣❲❀✓❩❋◆❖❀✗❀✗❅⑨❉❏❵✞❵r◆❖❀✗❀✗❣❲❉❋❜♥➚❲●❖P✑❀❈♣✰❀✗❅❇●▲❃⑨❀❻❂✗❍❏❊♥❣❲❉❝✾❁❅✣✒ ✙ ✗➷✕❄ ✴✮✈ ✍✴→ ✍✴→ ✍ ￾✚✹✒ ✔✗✚✣✖❢✌✡✌➣✍✑✘✮✔✗✜✣✖✙✏✗✏ ↕✬✻✢r✔✵✖❋✬✻✢✸✍✝ ➙✰➛❑➜➞➝✴➟✕➠✥✤ ❳❉❋❊✺❅❖✾❁❣❲❀✗◆✉●❖P✑❀③➾✣❍✛❛✙❴❭❉❋◆❄❅❇❀✗◆❖✾❁❀✗❅➍❵r❉✮◆ ✒ ✕✥Ï✘✗ ✒ ✕rÏ✘✗ ✙ ✒ ✕✒✲ ✗✙✱✧✦✒ ✕✒✲ ✗ ✦ Ï Ï ✱★✡✩✡☞✡✿✱ ✴ ✒ ✵ ✦ ❖ ✒ ✕❍✲ ✗ ✦ Ï❖ Ï❖ ✱✠✪❖❬ ✢ ✪ ❖❬ ✢ ✾❆❅⑨●❖P✑❀❈✬✻✖✙✯➆✍✑✢✸✜✞❱✣✖❑✬ ✪ ❖❬ ✢ ✙ ✴ ✕✫✒☛✱ ✴ ✗ ✵ ✦ ❖❬ ✢ ✒ ✕☞✬Ï✘✗ ✦ Ï❖ ❬ ✢ Ï❖❬ ✢ ❃❄P✑❀✗◆❖❀✭✬Ï❱✯ ✱✲ ✦❇Ï✶ ➡➤➢✧➥➧➦✯✮ ✓❵➃❍❏❴❁❴✹❵r❡✑❊✺❂➧●■✾❭❉❋❊✺❅✗➚✑❃❄P✙❛♦❍❏◆■❀✎❃➍❀❈✾❁❊❑●❖❀✓◆■❀✗❅❇●❖❀✵❣✆✾❁❊♥✾❭❊❑●■❀✓❩✮◆◗❍✻●■✾❭❊✺❩➭▼✰❉✮❴❁❛✙❊✑❉✮❜❝✾❆❍❏❴❆❅✜✛❢➾✉P✺❀❈◆❖❀✵❍✮❅❖❉✮❊ ❂✠❉❋❜❝❀✗❅❈❵r◆■❉✮❜ ●❖P✺❀♦❅❇●❖◆■❡✺❂➧●■❡✑◆❖❀✆❉❏❵▲➾✞❍✛❛✙❴❁❉✮◆✵➻ ❅➵❅❇❀✗◆❖✾❁❀✗❅❈❀✓Ñ❲▼✺❍❏❊✺❅❖✾❁❉✮❊➣✈✩➾✉P✑❀♦➾✞❍✛❛✙❴❁❉✮◆③❀✓Ñ✙▼✭❍❏❊✺❅❖✾❭❉❋❊ ✰

of a function in a local neighborhood of a point (here this point is chosen as 0 without loss of generality) is nothing but a power series expansion! The higher he order of polynomials that our scheme can integrate the higher the order of he remainder term in the expansion. The integral of the remainer over the domain is precisely the error in numerical integration 3.3.4 Estimating the Error Using the Taylor series results and the exactness criteria ∫(x)dr-∑1tf(x) 1/3t y(2(a)2/+Idz assuming deriv (+1)! Remainder ives of f(a) are bounded on [0, 1 ()d-∑m()57+ Assume that our scheme is exact upto a polynomial order l. That means we can integrate the first (L+1) terms in this Taylor series expansion exactly. The E=/f(x)dx-∑mf(x) +1f(2(x)x1+1d +1 3.3.5 Meeting the Exactness Criteria SLIDE 12 P(x)d=/(a+a1x+a2x2+…+ad)dx=∑p(x1) Equivalently do (x) This slide needs little clarification. Our exactness criterion is ()d=/(+0+2+…+)h=∑mn()

❉❏❵✿❍❝❵r❡✺❊✺❂➧●■✾❭❉❋❊✩✾❭❊✩❍❯❴❭❉❲❂✓❍✮❴✴❊✑❀✗✾❭❩❋P✙♣✭❉❋◆❖P✑❉✙❉❲❣☞❉❏❵❫❍❝▼✰❉✮✾❁❊❑●✖❡rP✺❀✓◆■❀❈●❖P✑✾❆❅➉▼✭❉❋✾❭❊❑●❬✾❆❅❬❂◗P✑❉❑❅❇❀✗❊✩❍✮❅✁￾ ❃❄✾❭●❖P✑❉❋❡❲●▲❴❭❉❑❅❖❅⑨❉✮❵✣❩❋❀✓❊✑❀✗◆■❍✮❴❭✾❭●q❛✳❢✿✾❆❅✉❊✑❉❏●■P✑✾❁❊✑❩❦♣✑❡❲●❬❍➫▼✰❉✻❃➍❀✗◆✉❅❖❀✓◆■✾❁❀✗❅✉❀✠Ñ❲▼✺❍✮❊✺❅❖✾❭❉❋❊ ✎✣➾✉P✺❀❈P✑✾❭❩❋P✑❀✓◆ ●❖P✺❀③❉✮◆◗❣❲❀✓◆❄❉✮❵✣▼✰❉✮❴❁❛✙❊✑❉✮❜❝✾❆❍❏❴❆❅✉●❖P✭❍✻●▲❉✮❡✺◆❬❅❖❂◗P✺❀✓❜❝❀③❂✓❍✮❊☞✾❁❊❑●❖❀✗❩✮◆◗❍✻●❖❀❻●■P✑❀❈P✑✾❁❩✮P✑❀✗◆❄●❖P✺❀③❉✮◆◗❣❲❀✓◆❄❉✮❵ ●❖P✺❀♦◆■❀✓❜❦❍❏✾❁❊✺❣❲❀✗◆➫●❖❀✓◆■❜ ✾❁❊ ●■P✑❀♦❀✓Ñ❲▼✺❍❏❊✺❅❖✾❁❉✮❊➣✈ ➾✉P✺❀♦✾❁❊❑●❖❀✓❩❋◆■❍✮❴➍❉✮❵▲●❖P✺❀☞◆■❀✓❜❦❍✮✾❭❊✑❀✗◆➭❉✻❨❋❀✓◆➭●❖P✑❀ ❣❲❉❋❜❝❍✮✾❭❊☞✾❆❅✉▼✑◆❖❀✵❂✠✾❆❅❇❀✗❴❭❛❯●❖P✑❀❈❀✗◆❖◆■❉✮◆⑨✾❭❊♥❊✙❡✑❜❝❀✓◆■✾❆❂✓❍❏❴✧✾❁❊❑●❖❀✗❩✮◆◗❍✻●❖✾❁❉✮❊✴✈ ✍✴→ ✍✴→✄✂ ✌✎✏✓✔✵✢✥✯➆✍✑✔✗✢✥✜✣✶✷✔✗✚✣✖❢✌❬✬✛✬✻✪✺✬ ➙✰➛❑➜➞➝✴➟✕➠✑➠ ☎❬❅❖✾❁❊✑❩➫●❖P✑❀③➾✣❍✛❛✙❴❭❉❋◆✉❅❖❀✓◆■✾❁❀✗❅⑨◆❖❀✵❅❇❡✺❴❰●◗❅❄❍❏❊✺❣♦●❖P✺❀❈❀✠Ñ✑❍✮❂✠●❖❊✑❀✵❅❖❅✉❂✠◆■✾❭●❖❀✓◆■✾❆❍ ✆ ✣ ✢ ✒ ✕✥Ï✘✗✬✫❋Ï✕❄ ✻✾✽✃✿ ✢ ✔ ✃ ✒ ✕✥Ï✭✃✹✗ ✙ ✴ ✕✫✒☛✱ ✴ ✗ ✵ ✛✏✢ ✣ ✦ ❖❬ ✢ ✒ ✕☞✬Ï ✕rÏ✘✗✺✗ ✦ Ï❖❬ ✢ Ï❖❬ ✢ ✫✮Ï ❈ ❉❊ ❋ ✝❏✟✞✟◆ ✃✽❙❏✟✠ ♠❅■❅❇❡✺❜➫✾❁❊✑❩✆❣✑❀✓◆■✾❭❨✻❍✻➹ ●❖✾❁❨✮❀✵❅✉❉❏❵ ✒ ✕rÏ✘✗▲❍❏◆■❀➉♣✰❉✮❡✑❊✭❣❲❀✗❣☞❉✮❊❣✱✲✳✦✵✴✝✶ ✡ ✡ ✡ ✡ ✡ ✛✢ ✣ ✓✒ ✕rÏ✘✗❚✫✮Ï❍❄ ✫✽ ✃✿ ✢ ✔ ✃ ✒ ✕rÏ✃ ✗ ✡ ✡ ✡ ✡ ✡☞☛ ✌ ✕✚✒☛✱✾✴ ✗ ✵ ❳❉✮❊✙❨❋❀✓◆■❩✮❀✓❊✭❂✠❀❬✾❆❅ ✄✭✖❑✬✛✒⑥❵✥❍✮❅❇●✏✎☎✎ ➡➤➢✧➥➧➦➩➨✎✍ ♠❅■❅❖❡✑❜❝❀③●❖P✭❍✻●❻❉✮❡✑◆❻❅❖❂◗P✺❀✓❜❝❀➵✾❆❅❬❀✓Ñ✑❍✮❂➧●❻❡✑▼❲●■❉♦❍✆▼✭❉❋❴❭❛✙❊✑❉❋❜➫✾❆❍❏❴✣❉✮◆◗❣❲❀✗◆ ✙ ✒q➻❁✈❻➾✉P✭❍✻●➉❜❝❀✵❍❏❊✺❅➉❃➍❀ ❂✓❍✮❊❯✾❁❊❋●■❀✓❩❋◆■❍❏●❖❀▲●■P✑❀➉Ð✺◆■❅❇●✂✕✚✒ ✱✏✴ ✗✳●■❀✓◆■❜❝❅⑨✾❁❊❯●■P✑✾❁❅✉➾✣❍✛❛✙❴❭❉❋◆⑨❅❖❀✓◆■✾❭❀✵❅✿❀✓Ñ✙▼✭❍❏❊✺❅❖✾❭❉❋❊❯❀✠Ñ✑❍❋❂➧●❖❴❁❛✮✈➃➾✉P✑❀ ❀✓◆■◆■❉✮◆⑨✾❭❊♥❊✙❡✑❜❝❀✓◆■✾❁❂✗❍❏❴✧✾❁❊❑●❖❀✓❩❋◆■❍❏●❖✾❁❉✮❊ ● ✙ ✛✢ ✣ ✒ ✕rÏ ✗✬✫❋Ï ❄ ✫✽ ✃✿ ✢ ✔ ✃ ✒ ✕rÏ✃ ✗✚✙ ✴ ✕✚✒☛✱✾✴ ✗ ✵ ✛✢ ✣ ✦ ❖❬ ✢ ✒ ✕✩✬Ï✧✕rÏ ✗❚✗ ✦ Ï❖❬ ✢ Ï❖❬ ✢ ✫❋Ï❨❫ ✍✴→ ✍✴→✑✏ ☛➶✖✙✖❑✔✗✢✸✜✞✶➤✔✵✚✞✖❢✌✌➣✍✑✘❏✔✵✜✞✖✙✏✵✏ ↕✬✛✢✥✔✗✖❑✬✻✢✥✍ ➙✰➛❑➜➞➝✴➟✕➠ ✎ ❪➃Ñ✑❍✮❂✠●❖❊✑❀✵❅❖❅❄❂✓❉✮❊✺❣✑✾❰●■✾❭❉❋❊♦◆■❀✗t❑❡✑✾❁◆❖❀✵❅ ✛✢ ✣ ✍ ❖ ✕rÏ ✗✬✫❋Ï ✙ ✛✢ ✣ ✕✫✞✣ ✱✌✞ ✢ Ï✾✱✌✞✯ Ï ✯ ✱☛✡☞✡✩✡✼✱✠✞❖ Ï❖ ✗✬✫❋Ï ✙ ✫✽ ✃✿ ✢ ✔ ✃✍ ❖ ✕✥Ï✃ ✗ ❵r❉✮◆▲❍✮❊✙❛✆❅❇❀✓●❄❉❏❵ ✒☛✱ ✴❈❂✠❉✙❀✓✒❦❂✓✾❭❀✗❊❑●■❅✑✞✣ ✦ ✞ ✢ ✦❳❫✵❫❳❫❳✦ ✞ ❖ ✌✎✤✧✦✣✢✄✑✍✑✼✸✖❑✜✹✔✵✼r✒ ✛✢ ✣ ✞ ✣ ✫✮Ï✾✱ ✛✢ ✣ ✞ ✢ Ï❵✫❋Ï ✱ ✛✢ ✣ ✞✯ Ï ✯ ✫❋Ï✾✱★✡✩✡☞✡✿✱ ✛✢ ✣ ✞ ❖ Ï❖ ✫❋Ï ✙ ✫✽ ✃✿ ✢ ✔ ✃✍ ❖ ✕✥Ï✃ ✗ ➡➤➢✧➥➧➦➩➨✰➨ ➾✉P✑✾❆❅❄❅❇❴❁✾❆❣❲❀❈❊✑❀✓❀✵❣✑❅✉❴❁✾❰●❖●❖❴❁❀③❂✠❴❆❍❏◆■✾❰Ð✭❂✗❍✻●■✾❭❉❋❊➣✈ ✓❡✑◆❄❀✠Ñ✑❍✮❂✠●❖❊✑❀✵❅❖❅✉❂✓◆❖✾❭●❖❀✗◆❖✾❁❉✮❊☞✾❁❅ ✛✢ ✣ ✍ ❖ ✕rÏ ✗✬✫❋Ï ✙ ✛✢ ✣ ✕✫✞✣ ✱✌✞ ✢ Ï✾✱✌✞✯ Ï ✯ ✱☛✡☞✡✩✡✼✱✠✞❖ Ï❖ ✗✬✫❋Ï ✙ ✫✽ ✃✿ ✢ ✔ ✃✍ ❖ ✕✥Ï✃ ✗ ✔