Lecture Notes: Orthogonal Polynomials, Gaussian Quadrature, and Integral Equations

Transcription

1 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 logic: Given function f(x) on the intervl [ 1, 1], define g(θ) = f(cos θ) Being n even 2π-periodic function, g(θ) hs Fourier cosine series expnsion g(θ) = ã0 2 + ν ãν cos(νθ), whereupon our originl function f(x) hs the expnsion f(x) = ã0 2 + ν ν cos ( ν rccos x ) or f(x) = ã0 2 + ν ν T ν (x), T ν (x) = cos ( n rccos x ) which defines the function T ν (x) In these notes we will investigte the following lterntive chrcteriztion of the functions {T n }: The Chebyshev polynomils {T n (x)} re the unique polynomils normlized to T n (0) = 1 nd orthogonl with respect to the inner product Contents f, g = 1 1 f(x)g(x)dx 1 x 2 1 Orthogonl Sets of Polynomils 2 2 Roots of orthogonl polynomils 7 3 Gussin qudrture 9 4 Integrl equtions nd Nyström s method 12 1

2 18330 Lecture Notes 2 1 Orthogonl Sets of Polynomils An orthogonl set of polynomils is fully specified by three ingredients: 1 An intervl of the rel line [, b] over which we will be integrting 2 A weight function W (x) defined over [, b] 3 A normliztion convention, which just defines n overll multiplictive prefctor You cn think of items 1 nd 2 here s together specifying n inner product on the vector spce of rel-vlued functions on the intervl [, b]: f, g f(x)g(x)w (x)dx (1) An inner product is just rule for ssigning rel number to ny pir of functions f, g, nd different choices of [, b] nd W (x) yield different inner products [Note tht the inner product is liner, ie αf, g = α f, g nd f +g, h = f, h + g, h ] For our purposes, the most importnt fct bout the inner product is tht it doesn t vnish when you stick the sme function into both slots, ie f, f 0 1 Given n inner product nd normliztion convention, n orthogonl set of polynomils is simply collection of polynomils {Q n (x)} (where n indexes the degree of the polynomil, ie Q 0 is constnt, Q 1 (x) is liner function, Q 2 (x) is second-degree polynomil, etc) tht stisfy the normliztion convention nd tht re orthogonl with respect to the inner product, ie Exmples Q n, Q m = 0 for n m The following tble summrizes the ingredients tht define some of the commonlyused sets of orthogonl polynomils Nme Symbol Intervl Weight function Normliztion Legendre P n (x) [ 1, 1] 1 P n (1) = 1 Chebyshev T n (x) [ 1, 1] 1 1 x 2 T n (0) = 1 Lguerre L n (x) [0, ] e x L n (0) = 1 Hermite H n (x) [, ] e x2 H n, H n = π2 n n! 1 Note tht one convenient wy to define normliztion convention [item (3) bove] would be to scle ll functions f such tht f, f = 1, but this is not the convention tht is typiclly used

3 18330 Lecture Notes 3 Construction from inner product Given n inner product nd normliztion convention, there is simple constructive procedure for computing every element in the corresponding fmily of orthogonl polynomils It is the nlogue for polynomil vector spces of the usul Grm-Schmidt orthogonliztion process used to construct orthogonl bses in geometry, nd it goes like this: 1 First, choose Q 0 (x) to be the unique degree-zero polynomil (ie constnt) tht stisfies the normliztion convention (In most cses we simply hve Q 0 = 1) 2 Now construct Q 1 s the product of liner fctor times Q 0 : where 2 B 1 = Q 1 (x) = A 1 (x B 1 )Q 0 (2) xq0, Q 0 Q0, Q 0 (3) nd A 1 is chosen to ensure tht Q 1 stisfies the normliztion condition You cn esily verify tht Q 1 (x) s defined by (2) is orthogonl to Q 0 by construction 3 Now construct Q 2 s the product of liner fctor times Q 1 plus constnt fctor times Q 0 : [ ] Q 2 (x) = A 2 (x B 2 )Q 1 C 2 Q 0 (4) where B 2 = xq1, Q 1 Q1, Q 1, C 2 =, Q1, Q 1 A 1 Q0, Q 0 nd A 2 is chosen to ensure tht Q 2 stisfies the normliztion condition You cn esily verify tht Q 2 (x) s defined by (4) is orthogonl to both Q 1 nd Q 0 4 Now construct Q 3 s the product of liner fctor times Q 2 plus constnt fctor times Q 1 : [ ] Q 3 (x) = A 3 (x B 3 )Q 2 C 3 Q 1 (5) where xq2, Q 2 Q2, Q 2 B 3 =, C 3 = Q2, Q 2 A 2 Q1, Q 1 2 Just to clrify: The numertor of the following eqution is the inner product (1) with the function f(x) tken to be xq 0 (x) nd the function g(x) tken to be Q 0 (x)

4 18330 Lecture Notes 4 nd A 3 is chosen to ensure tht Q 3 stisfies the normliztion condition You cn esily verify, s before, tht Q 3 (x) s defined by (5) is orthogonl to both Q 2 nd Q 1 Wht is surprising is tht this Q 3 is lso orthogonl to Q 0 Indeed, more generlly 5 we construct the generl element Q n (x) s the product of liner fctor times Q n 1 (x) plus constnt fctor times Q n 2 : [ ] Q n (x) = A n (x B n )Q n 1 C n Q n 2 (6) where xqn 1, Q n 1 Qn 1, Q n 1 B n =, C n = (7) Qn 1, Q n 1 A n 1 Qn 2, Q n 2 nd A n is chosen to ensure tht Q n stisfies the normliztion condition Agin, wht is surprising here is tht the polynomil constructed in (6) is orthogonl not only to Q n 1 nd Q n 2 but indeed to ll previous members of the set, Q n 3, Q n 4,, Q 1, Q 0 In constructing (6) it seems like we re only ensuring orthogonlity ginst Q n 2 nd Q n 1 But the orthogonlity ginst the previous members of the set turns out to follow for free, utomticlly, from the wy the previous functions were defined This is not obvious Recurrence reltions By scutinizing the generl cse of the inductive procedure discussed bove, it is generlly possible to write down recurrence reltions tht relte the next element in set of orthogonl polynomils to previous elements For exmple, the Legendre polynomils stisfy the recurrence P n+1 (x) = ( 2n + 1 n + 1 ) xp n (x) ( n n + 1 The Chebyshev polynomils stisfy the recurrence ) P n 1 (x) T n+1 (x) = 2xT n (x) T n 1 (x) (8) The Lguerre polynomils stisfy the recurrence L n+1 (x) = (2n + 1) x L n (x) k (n + 1) k + 1 L n+1(x) (9) The Hermite polynomils stisfy the recurrence H n+1 (x) = 2xH n (x) 2nH n 1 (x) (10)

5 18330 Lecture Notes 5 Differentil equtions Mny sets of orthogonl polynomils rise s solutions to differentil equtions For exmple, the nth Legendre polynomil P n (x) stisfies (1 x 2 ) d2 P n dx 2 nd the nth Chebyshev polynomil stisfies (1 x 2 ) d2 T n dx 2 Generting functions 2xdP n dx + n(n + 1)P n(x) = 0 xdt n dx + n2 T n (x) = 0 It is curious, nd in some cses useful, to note tht mny functions of orthogonl polynomils hve generting function which encodes the properties of the entire set of functions nd from which individul functions cn be recovered by performing lgebric nd derivtive mnipultions For exmple, for the Legendre polynomils we hve d n P n (x) = 1 2 n n! dx n (x2 1) n For the Chebyshev polynomils, it turns out tht T n (x) rises s precisely the coefficient of y n in the expnsion of the quntity (1 xy)/(1 2xy + y 2 ) in powers of y : 1 xy 1 2xy + y 2 = T n (x)y n Differentiting ech side of this eqution n times nd setting y 0 then yields n expression for T n (x) n=0 Properties of orthogonl polynomils There re few common properties tht re common to ll sets of orthogonl polynomils 1 The first N elements in the set constitute bsis for the vector spce of ll polynomils of degree N Wht this mens is tht ny rbitrry degree-n polynomil F (x) my be represented exctly nd uniquely s liner combintion of the Q n functions: F (x) = c 0 Q 0 (x) + c 1 Q 1 (x) + + c N Q N (x) (11) 2 Only the constnt element in the set hs nonvnishing integrl over the intervl with respect to the weight function, ie Q n (x)w (x)dx = 0, n 1 (11b)

6 18330 Lecture Notes 6 This is ctully just consequence of orthogonlity: We must hve Q n, Q 0 = 0 for n 0, but Q 0 is just constnt nd my be pulled out of the integrl (1), leving behind (11b)

7 18330 Lecture Notes 7 2 Roots of orthogonl polynomils For mny pplictions, including Gussin qudrture s discussed in the following section, we need to compute the roots of the Nth element in some set of orthogonl polynomils, ie we need the N points x n tht stisfy Q N (x n ) = 0, n = 1, 2,, N (12) It turns out to be esy to compute the numbers x n using numericl eigenvlue techniques, nd indeed numericl eigenvlue techniques re the preferred wy to compute these roots, s other methods tend to be numericlly unstble The trick is to mke use of the recurrence reltion (6) to write xq n in terms of other Q functions: xq n = α n Q n 1 + β n Q n + γ n Q n+1 (13) where the α, β, γ coefficients my be written down in closed form nd tke different forms for vrious different sets of orthogonl polynomils; for exmple, in the cse of Legendre polynomils we hve α n = n 2n + 1, β n = 0, γ n = n + 1 2n + 1 If we now write out eqution (13) for n = 0, 1,, N 1, we obtin n N N liner system of equtions: β 0 γ Q 0 (x) α 1 β 1 γ Q 1 (x) 0 α 2 β 2 γ Q 2 (x) 0 0 α 3 β Q 3 (x) β N 3 γ N α N 2 β N 2 γ N α N 1 β N 1 Q 0 (x) 0 Q 1 (x) 0 Q 2 (x) 0 Q 3 (x) 0 = x Q N 3 (x) Q N 2 (x) Q N 1 (x) γ N 1 Q N (x) Q N 3 (x) Q N 2 (x) Q N 1 (x) Wht this eqution sys is tht x is lmost n eigenvlue of the mtrix on the LHS The only thing tht spoils the eigenvlue condition is the extr term in the lst slot of the second vector on the RHS However, this term vnishes whenever x is root of Q N! This mens tht the roots of Q N re precisely the eigenvlues of the tridigonl mtrix on the RHS Here s little juli code tht will compute nd return n N-dimensionl vector contining the roots of the Nth Legendre polynomil, P N (x):

8 18330 Lecture Notes 8 function LegendreRoots(N) A=zeros(N,N) A[1,2] = 1; for n=1:n-2 A[n+1,n] = n/(2*n+1); A[n+1,n+2] = (n+1)/(2*n+1); end A[N,N-1] = (N-1)/(2*N-1); (lmbd,u)=eig(a); lmbd end

9 18330 Lecture Notes 9 3 Gussin qudrture In this section we consider the evlution of integrls of the form f(x)w (x) dx (14) where W (x) is some weight function nd f(x) is n rbitrry function whose integrl (times W ) we re trying to compute We would like to construct n N-point qudrture rule consisting of N points nd weights {{x n }, {w n }) such tht N w n f(x n ) f(x)w (x)dx (15) n=1 Note tht the sum on the LHS here only involves smples of f, not W ; the weight function W (x) is bked in to the definition of the qudrture weights w n Let {Q n } be the set of orthogonl polynomils {Q n (x)} defined with respect to n inner product of the form (1) with intervl [, b] nd weight function W (x) mtching those of the integrl we re trying to compute in (14) [In the common cse in which W (x) = 1, these will be just the Legendre polynomils {P n (x)}] It s esy to construct n N-point qudrture rule tht exctly integrtes polynomils up to degree N 1 If you give me ny set of N points {x n } distributed throughout the intervl [, b], I cn find set of N weights {w n } such tht the qudrture rule [{x n }, {w n }] exctly integrtes ll polynomils of degree N 1 or less All I hve to do is to require my qudrture rule to be exct for the first N elements in the orthogonl set {Q n } Since ny polynomil of degree N 1 or lower cn be exctly represented s liner combintion of these elements, its integrl will be computed exctly by our qudrture rule The condition tht our qudrture rule be exct for the first N polynomils in the set {Q n } mounts to set of N simultneous liner equtions on the N qudrture weights {w n } Indeed, the requirement tht my qudrture rule be exct when I use it to integrte the function Q 0 gives me the condition w 1 Q 0 (x 1 ) + w 2 Q 0 (x 2 ) + + w N Q 0 (x N ) = The condition tht the rule be exct for Q 1 yields w 1 Q 1 (x 1 ) + w 2 Q 1 (x 2 ) + + w N Q 1 (x N ) = Q 0 (x)w (x)dx Q 1 (x)w (x)dx (16) (16b)

10 18330 Lecture Notes 10 Proceeding similrly, I obtin totl of N equtions, culminting in w 1 Q N 1 (x 1 ) + w 2 Q N 1 (x 2 ) + + w N Q N 1 (x N ) = Q N 1 (x)w (x)dx (16c) Equtions (16) together constitute n N N liner system for the qudrture weights w n Note lso tht the RHS of this system is simpler thn it looks: s we noted erlier, ll the RHS integrls vnish except for the one involving Q 0, so the RHS vector of our liner system hs only one nonzero entry But Guss discovered wy to construct n N-point qudrture rule tht exctly integrtes polynomils up to degree 2N 1 The proceeding development tells me tht, given ny choice of N points {x n }, I cn find set of N weights tht mkes the qudrture rule (15) exct for ll polynomils up to degree N 1 However, mong ll possible wys to choose the set of qudrture points {x n }, there is one choice tht is distinguished: It is the set of roots of the polynomil Q N (x) It is n stonishing fct tht the qudrture rule (15), computed with the {x n } tken s the roots of Q N nd the weights computed s discussed bove, is exct for ll polynomils up to degree 2N 1 This mssively expnds the spce of functions over which our qudrture rule is exct; the technique is known s Gussin qudrture The proof of this sttement is mzingly simple Let f(x) be polynomil of degree 2N 1 or less If we divide 3 f(x) by the polynomil Q N (x), we obtin some quotient p(x) nd some reminder r(x), nd becuse Q N hs degree N we re gurnteed tht tht p(x) nd r(x) both hve degree N 1 or less In other words, ny polynomil f of degree 2N 1 my be written exctly in the form f(x) = Q N (x)p(x) + r(x), deg p, r N 1 (17) But now look t wht hppens when I pply the qudrture rule (15) to f(x): f(x)w (x)dx = N w n f(x n ) n=1 N [ ] w n Q N (x n ) p(x n ) + r(x n ) }{{} n=1 =0 The first term vnishes becuse the qudrture points re roots of Q N! This leves behind = N w n r(x n ) = n=1 r(x)w (x) dx (exctly) (18) 3 The opertion t work here is synthetic division do you remember this from high school?

11 18330 Lecture Notes 11 In other words, using our qudrture rule to integrte the function (17) is equivlent to integrting just the function r(x) But this function is exctly integrted by our qudrture rule becuse it hs degree N 1 nd our qudrture rule hndles ll such functions exctly Menwhile, we cn evlute the exct integrl f(x)w (x)dx nother wy, by expnding the function p(x) in (17) in the set of functions {Q n } [cf eqution (11b] Since p hs degree N 1, this expnsion includes only terms up to Q N 1 : nd hence (17) reds Integrting, we find f(x)w (x)dx = = p(x) = N 1 n=0 α n Q n (x) N 1 f(x) = Q N (x) α n Q n (x) + r(x) N 1 n=0 n=0 α n Q N (x)q n (x)w (x) dx + } {{ } =0 r(x)w (x) dx (19) r(x)w (x) dx (20) becuse Q N is orthogonl to Q n for ll n N 1 Compring (18) to (20) we see tht our qudrture rule is exct for ll functions which cn be decomposed in the form (17) tht is, for ll polynomils of degree 2N 1 Isn t this beutiful? I love this Guss vs Clenshw-Curtis For n interesting discussion of the reltive merits of Gussin vs Clenshw- Curtis qudrture, see the rticle Is Guss Qudrture Better thn Clenshw- Curtis?, by N Trefethen, SIAM Review 50 p 67, vilble online here: http: //epubssimorg/doi/pdf/101137/

12 18330 Lecture Notes 12 4 Integrl equtions nd Nyström s method Motivtion: The 1D Semiconductor In previous discussions, we considered the computtion of the electrosttic potentil in one-dimensionl crystlline ionic solid Let s now generlize this in two wys: we will tret the underlying chrge density s continuous rther thn discrete, nd we will consider the cse of semiconducting rther thn n ionic mteril In semiconductor, the locl chrge density ρ(x) depends strongly on the locl electrosttic potentil x A simple model of this dependence is furnished by ρ(x) = ρ 0 e φ(x)/v T (21) where the therml voltge V T is the temperture divided by the electron chrge, V T = kt e 0026 volts t room temperture Consider now 1D semiconductor of length L chrcterized by locl linechrge chrge density 4 λ(x) The electrosttic potentil φ(x) is determined by λ ccording to L/2 λ(x )dx φ(x) = x x L/2 However, from (21) we lso hve tht λ is determined by φ ccording to Combining, we obtin λ(x) = λe αφ(x) ( α = 1 V T ) φ(x) = ρ 0 L/2 L/2 e αφ(x ) dx x x (22) This is n integrl eqution for the electrosttic potentil Integrl equtions re much hrder thn differentil equtions, for the following reson: In the cse of differentil eqution, we cn lwys work loclly to figure out, for exmple, the next point on solution curve given just single point on tht curve Indeed, this is precisely the MO of the ODE solvers tht we discussed in the first unit of our course In doing this, we know nothing bout the globl behvior of the solution curve, know nothing bout wht the solution is doing fr wy from our given point, nd nonetheless cn infer incrementl knowledge from the locl informtion contined in the differentil eqution On the other hnd, in n eqution like (23) there is no notion of proceeding loclly: To do nything t ll with the RHS of the eqution requires globl knowledge of the function φ 4 The line-chrge density λ(x) is defined such tht the totl chrge in the intervl [x, x+dx] is λ(x)dx

13 18330 Lecture Notes 13 Nyström s Method Nystrom s method uses Gussin qudrture to convert n integrl eqution into liner system of equtions The most generl setting is to consider n integrl eqution of the form K(x, x )S(x )dx = F (x) (23) where K(x) is known kernel function, F (x) is n known forcing function, nd S(x) is n unknown source function for which we re trying to solve Nyström s method is to use n N-point qudrture rule for the intervl [, b]: K(x, x )S(x )dx N w n K(x, x n )S(x n ) We then require tht eqution (23) be stisfied t ech of the N qudrture points x n This gives us N equtions: w 1 K(x 1, x 1 )S(x 1 ) + w 2 K(x 1, x 2 )S(x 2 ) + + w N K(x 1, x N )S(x N ) = F (x 1 ) w 1 K(x 2, x 1 )S(x 1 ) + w 2 K(x 2, x 2 )S(x 2 ) + + w N K(x 2, x N )S(x N ) = F (x 2 ) nd so on down to w 1 K(x N, x 1 )S(x 1 ) + w 2 K(x N, x 2 )S(x 2 ) + + w N K(x N, x N )S(x N ) = F (x N ) n=1 This is n N N liner system of the form w 1 K(x 1, x 1 ) w 2 K(x 1, x 2 ) w N K(x 1, x N ) w 1 K(x 2, x 1 ) w 2 K(x 2, x 2 ) w N K(x 2, x N ) w 1 K(x N, x 1 ) w 2 K(x N, x 2 ) w N K(x N, x N ) S(x 1 ) S(x 2 ) S(x N ) = F (x 1 ) F (x 2 ) F (x N ) which we solve for the vlues of our unknown source distribution t the qudrture points