Moment-free numerical integration of highly oscillatory functions

Transcription

1 Moment-free numericl integrtion of highly oscilltory functions Sheehn Olver September 4, 005 Abstrct The im of this pper is to derive new methods for numericlly pproximting the integrl of highly oscilltory function. We begin with review of the symptotic nd Filon-type methods developed by Iserles nd Nørsett. Using method developed by Levin s point of deprture, we construct new method tht utilizes the sme informtion s the Filon-type method, nd obtins the sme symptotic order, while not requiring the computtion of moments. We lso show tht specil cse of this method hs the property tht the symptotic order increses with the ddition of smple points within the intervl of integrtion, unlike ll the preceding methods whose orders depend only on the endpoints. Introduction A highly oscilltory integrl is defined s I[f] = b f(x) e iωg(x) dx for f, g C nd frequency ω. In this pper we only consider the cse of g (x) 0 for x b, in other words, when g hs no sttionry points. The most immedite cndidte for numericlly pproximting this integrl might be Gussin qudrture. Unfortuntely, if we subdivide [, b] into pnels of length h > 0 we cn choose ω lrge enough such tht the pproximtion is completely meningless, s the node points re essentilly rndom smples in the rnge of oscilltion. The error of this pproximtion is O() s ω, which compres to n error of O ( ω ) when pproximting I[f] by zero (Iserles nd Nørsett, 004). It is sfe to sy tht ny pproximtion tht is less ccurte thn equting the integrl to zero is firly useless. Letting h depend on ω, on the other hnd, results in n enormous mount of computtion for lrge ω. Fortuntely, there is nother wy. We begin with review of two methods described in (Iserles nd Nørsett, 004), the symptotic method nd the Filon-type method, which both hve n error of order O ( ω s ) for ny fixed positive integer s nd incresing frequency ω. The impliction is tht, in strk contrst to Gussin qudrture, the lrger the frequency the more ccurte the pproximtion. Using these two methods s n inspirtion nd extending the work of Dvid Levin, we derive nother method tht lso hs n error of order O ( ω s ). Like the Filon-type method, this new method uses interior points s well s the endpoints for deriving the pproximtion, while unlike the Filon-type method it does not require moments. We lso show tht specil cse of this method hs the property tht using interior points, in ddition to endpoints, further increses the order of error. Deprtment of Applied Mthemtics nd Theoreticl Physics, Centre for Mthemticl Sciences, Wilberforce Rd, Cmbridge CB3 0WA, UK, emil: S.Olver@dmtp.cm.c.uk.

2 The symptotic method The foundtion of the proofs in this pper lies in the observtion, s described in (Iserles nd Nørsett, 004), tht I[f] = b = iω f(x) e iωg(x) dx = iω [ ] f(x) b g (x) eiωg(x) iω b b f(x) d g (x) dx eiωg(x) dx [ d f(x) dx g (x) ] e iωg(x) dx = Q A [f] iω I [ d f dx g [ ] b where Q A [f] is defined s f(x) iω g (x) eiωg(x). Since we ssume tht g (x) 0 for ll x [, b] there is no issue with dividing by g. This eqution sttes tht if we pproximte I[f] by Q A [f] we hve n order of error of O ( ω ), gin using the fct tht I[f] = O ( ω ) for bounded f (Iserles nd Nørsett, 004). But the error term of this pproximtion cn likewise be pproximted, now by [ d iω QA dx ] f g ], [ d iω QA f dx. Hence we hve derived new pproximtion of I[f], nmely Q A [f] g ], which hs n order of error of O ( ω 3). Clerly, if we continue this process of pproximting the error terms using Q A, fter s steps we obtin error O ( ω s ). Thus we hve derived n symptotic expnsion: Theorem Let f C nd g (x) 0 for x b. Define σ k s Then, for ω, σ [f](x) = f(x) g (x), I[f] k= σ k+[f](x) = σ k[f] (x) g, k. (x) { ( iω) k σ k [f](b) e iωg(b) σ k [f]() e iωg()}. A forml proof of this theorem cn be found in (Iserles nd Nørsett, 004). We define Q A s [f] = s k= { ( iω) k σ k [f](b) e iωg(b) σ k [f]() e iωg()}, i.e. the s-step prtil sum of the symptotic expnsion. From the omitted proof of the theorem we know tht the error I[f] Q A s [f] is equl to ( iω) s b g (x) σ s+ [f](x) e iωg(x) dx nd by the definition of symptotic expnsions we know tht I[f] Q A s [f] O ( ω s ). Hence the error of the pproximtion tends to zero s ω s. In other words, the more oscilltory the integrnd, the more ccurtely we cn pproximte the integrl! This flies in the fce of the common intuition, bsed on the problems ssocited with Gussin qudrture, tht oscilltions mke numericl integrtion difficult. From this theorem we now derive corollry tht will be used to find the order of error for the Filon-type nd Levin-type methods. Corollry Suppose 0 = f() = f(b) = f () = f (b) = = f (s ) () = f (s ) (b) for some positive integer s. Furthermore, llow f to depend on ω, nd suppose tht every function in the set { f,, f (s+) } is of symptotic order O(ω n ), ω, for some fixed n. Then, s ω, I[f] O ( ω n s ).

3 Proof Fix s. By simple inductive resoning, we see tht ech σ k [f] is liner combintion of terms independent of ω multiplied by functions in the set { f,, f (k )}. As result 0 = σ k [f]() = σ k [f](b) for ll k s, nd it follows tht Q A s [f] is 0. Hence { I[f] = ( iω) s+ σ s+ [f](b) e iωg(b) σ s+ [f]() e iωg()} + ( iω) s+ b g (x) σ s+ [f](x) e iωg(x) dx. The first two terms re O ( ω n s ). We know tht σ s+ [f] K ω for some constnt K, since n σ s+ [f] is combintion of f nd its first s + derivtives. Thus the integrl term is lso O ( ω n s ), since ( iω) s+ b g (x) σ s+ [f](x) e iωg(x) dx K ω n+s+ (b ) = O( ω n s ). Note tht this corollry pplies eqully well when f nd its derivtives re independent of ω, in which cse we tke n = 0. Unless otherwise stted, we ssume n = 0 when this corollry is used. Remrk In Corollry it is necessry to impose the order requirement on both f nd its first s derivtives: f = O(ω n ) with no restriction on its derivtives is not sufficient. For exmple consider f(x) = ω n e iωnx. Then f (k) (x) = i k ω (k )n e iωnx = O ( ω (k )n) nd the corollry does not hold. 3 The Filon-type method The shortcoming with using n symptotic expnsion s n pproximtion is tht in generl Q A s [f] diverges for fixed ω s s. In other words, for fixed ω the ccurcy of pproximting n integrl by the prtil sums Q A s is limited. To work round this wekness we derive Filon-type method, which extends the work of Filon s described in (Iserles nd Nørsett, 004). Theorem Let s be some positive integer, let {x k } ν 0 be set of node points such tht = x 0 < x < < x ν = b, nd let {m k } ν 0 be set of multiplicities ssocited with those node points such tht m 0, m ν s. Suppose tht v(x) = n k=0 c kx k, where n = ν k=0 m k, is the solution to the system of equtions for every integer 0 k ν. Then v(x k ) = f(x k ) v (x k ) = f (x k ). v (m k ) (x k ) = f (m k ) (x k ) I[f] Q F [f] O ( ω s ), where Q F [f] I[v] = n k=0 [ c k I x k]. 3

4 Figure : The error scled by ω 3 of Q A [f] (left figure, top), QF [f] with only endpoints nd multiplicities ll (left figure, bottom)/(right figure, top), nd Q F [f] with nodes { 0,, } nd multiplicities {,, } (right figure, bottom) for I[f] = 0 cos(x) eiωx dx. Proof Note tht I[f] Q F [f] = I[f] I[v] = I[f v]. By the definition of v, the hypotheses of Corollry hold for the function f v, hence I[f v] O ( ω s ). In other words, we interpolte f by polynomil v using Hermite interpoltion. Since we re ssuming tht moments re vilble nd Q F [f] is liner combintion of moments, we know tht Q F [f] cn be computed. The obvious question then is if it hs the sme order of error s the symptotic method, s well s requiring the sme number of derivtives, why bother? The nswer is tht in mny situtions the ccurcy of the Filon-type method is significntly higher thn tht of the symptotic method, even though it is of the sme order. We lso hve the bility to dd interior node points to further increse the ccurcy, nd it is cler tht Q F [f] converges to I[f] whenever the interpolting polynomil v converges uniformly to f. We now compre symptotic nd Filon-type methods numericlly. For exmple, consider the cse of the Fourier oscilltor g(x) = x, nd let f(x) = cos x over the intervl [0, ]. In Figure we compre severl methods of order 3: Q A [f], QF [f] with nodes {0, } nd multiplicities {, }, nd Q F [f] with nodes { 0,, } nd multiplicities {,, }. Even when smpling f only t the endpoints of the intervl, the Filon-type method represents significnt improvement over the symptotic method, hving pproximtely one twelfth the error, while using exctly the sme informtion. Adding single interpoltion point resulted in n error lmost indistinguishble from zero when compred to the symptotic method. Adding dditionl node points continues to hve similr effect. Unfortuntely, it is not lwys true tht the Filon-type method is more ccurte thn the symptotic method. Tke the cse of the Fourier oscilltor nd f(x) =, now over the +0x intervl [, ]. This suffers from Runge s phenomenon, s described in (Powell, 98), where certin non-oscilltory functions hve oscillting interpoltion polynomils. Since the Filon-type method is bsed on interpoltion, it is logicl tht the ccurcy of Q F [f] is directly relted to how ccurte the interpoltion is. In Figure we see tht dding dditionl nodes ctully reduces the ccurcy of Q F [f]. It should be noted tht in this exmple Q F [f] with only endpoints nd Q A [f] re equivlent, which cn be trivilly proved by finding the explicit formul for Q F [f]. Thus Q A [f] is the best method of the three tried. We know tht using Chebyshev interpoltion points, lso described in (Powell, 98), helps reduce the mgnitude of Runge s phenomenon. Using this choice for nodes, long with the required 4

5 Figure : The error scled by ω of Q F [f] with only endpoints (right figure), endpoints nd two dditionl evenly spced points (left figure, bottom), nd endpoints nd four dditionl evenly spced points (left figure, top), where ll multiplicities re for I[f] = 0 e iωx dx. +0x Figure 3: The error scled by ω of Q F [f] with only endpoints (right figure), endpoints nd dditionl Chebyshev interpoltion points (left figure, top), nd endpoints nd 4 dditionl Chebyshev interpoltion points (left figure, bottom), where ll multiplicities re for I[f] = 0 e iωx dx. +0x endpoint nodes, results in the errors seen in Figure 3. Now dding dditionl node points results in more ccurte pproximtion. This certinly is huge improvement over Figure, but the Filon-type methods definitely do not hve the sme mgnitude of improvement over the symptotic method tht they did in Figure. Another option, with regrds to Runge s phenomenon, is to use cubic splines in plce of interpoltion. Unfortuntely this suffers from the fct tht cubic spline cn only mtch up to the first derivtive t the endpoints, hence the order is t most O ( ω 3) in the present frmework. Since we re only considering methods with rbitrrily high order of convergence for incresing ω, we will not explore the use of cubic or higher-degree splines. 4 The Levin-type method The Filon-type method requires tht moments re esily computble, which is not necessrily the cse. Fortuntely, we cn work round this problem by expnding on the method developed by 5

6 Dvid Levin in (Levin, 997). Wht follows is brief, nd simplified, synopsis of the method described in tht pper. Suppose we hve function F (x) such tht d [ F (x) e iωg(x)] = f(x) e iωg(x). (4.) dx It follows immeditely tht I[f] = [ F (x) e iωg(x)] b. If we pproximte F by some function v, then we cn pproximte the integrl by Q L [f] = [ v(x) e iωg(x)] b. By expnding out the derivtive on the left hnd side of (4.) nd cnceling the e iωg(x) terms we obtin the eqution L[F ](x) = f(x), where L is the opertor defined by L[F ] = F + iωg F. Now let v(x) = n k=0 c kx k be the colloction polynomil which is the solution to the system of equtions L[v](x k ) = f(x k ) t points = x 0 < x < < x ν = b. Then Q L [f] pproximtes I[f] with error O ( ω ). The nturl extension to the Levin method is to emulte the Filon-type method of the preceding section nd mtch not only the vlue of f nd L[v] t the node points, but lso the vlues of the derivtives of f nd L[v], up to given multiplicity. We prove in this section tht, if we gin mtch the function vlues f nd L[v] nd the first s derivtives t the endpoints, then we obtin n order of error of O ( ω s ). Since the proof of the following theorem does not rely on v being polynomil, we llow v to be liner combintion of set of bsis functions mtching certin criteri generliztion tht will be exploited in Section 5. Theorem 3 Suppose tht g (x) 0 for x [, b]. Let {ψ k } n 0 be bsis of functions independent of ω nd let s be some positive integer. Furthermore let {x k } ν 0 be set of node points such tht = x 0 < x < < x ν = b nd {m k } ν 0 set of multiplicities ssocited with those node points such tht m 0, m ν s. Suppose tht v = n k=0 c kψ k, where n = ν k=0 m k, is the solution to the system of colloction equtions L[v](x k ) = f(x k ) dl[v] dx (x k) = f (x k ). d mk L[v] dx m k (x k) = f (mk ) (x k ) for every integer 0 k n nd L[v] = v + iωg v. Define [ (g g k = ) ψ k (x0 ),, ( g ) (m0 ) ψ k (x0 ),, ( g ) ψ k (xν ),, ( g ) ] (mν ) ψ k (xν ). If the vectors {g 0,, g n } re linerly independent, then for sufficiently lrge ω the system hs unique solution nd I[f] Q L [f] O ( ω s ), where Q L [f] [v(x) e iωg(x)] b = v(b) eiωg(b) v() e iωg(). (4.) Proof We know tht I[f] Q L [f] = I[f] I[L[v]] = I[f L[v]]. Hence we use Corollry, in mnner similr to the proof of { Theorem. Unfortuntely, L[v] depends on ω so we need to show tht ll the functions in the set f L[v],, f (s+) L[v] (s+)} re bounded for incresing ω. Since { f,, f (s+) } re by definition independent of ω, we need only show tht {L[v],, L[v] (s+)} re O(). 6

7 The vector of coefficients c = [c 0,, c n ] solves the system of eqution Ac = f, where L[ψ 0 ](x 0 ) L[ψ n ](x 0 )..... L[ψ 0 ] (m0 ) (x 0 ) L[ψ n ] (m0 ) (x 0 ) A =....., f = L[ψ 0 ](x ν ) L[ψ n ](x ν )..... L[ψ 0 ] (mν ) (x ν ) L[ψ n ] (mν ) (x ν ) f(x 0 ). f (m 0 ) (x 0 ) f(x ν ). f (mν ) (x ν ) For nottionl brevity we regrd mtrices s row vectors whose entries re column vectors. If we let [ p k = ψ k (x 0),, ψ (m 0) k (x 0 ),, ψ k (x ν),, ψ (mν) k (x ν )], then A = [p 0 + iωg 0,, p n + iωg n ]. In lter proofs we will lso use k = p k + iωg k to denote the (k + )th column of A. Using Crmer s rule we find tht c k = det D k det A for 0 k n, where D k is the mtrix A with the (k + )th column replced by f. Note tht ll the entries of the mtrix D k, except for single column, re of order O(ω). Hence it is obvious from the definition of the determinnt tht det D k = O(ω n ). We now show tht det A = O( ω n ). We know tht det A = det [iωg 0,, iωg n ] + O(ω n ) = (iω) n+ det [g 0,, g n ] + O(ω n ) But, by the hypothesis, the columns of this determinnt re linerly independent, hence this determinnt is not zero nd det A is n (n + )th degree polynomil in ω. If ω is sufficiently lrge, then the ω n+ term overwhelms the O(ω n ) term nd det A 0, which proves tht the system hs unique solution. Furthermore det A = O( ω n ), nd we hve shown tht c k = O ( ω ). Since ech ψ k is independent of ω, it follows tht v nd its derivtives re lso O ( ω ). Thus, L[v] (j) = v (j+) + iω j g (k+) v (j k) = O ( ω ) + O() = O() k=0 for ll 0 j s +. We hve stisfied the hypotheses of Corollry nd the theorem follows. Theorem 4 provides somewht simpler conditions on the bsis {ψ k } n 0 in the preceding theorem. It is especilly helpful s it ensures tht the stndrd polynomil bsis cn be used with the Levintype method nd ny choice of nodes nd multiplicities. Recll from (Powell, 98) tht stting tht bsis is Chebyshev set is equivlent to stting tht it spns set M tht stisfies the Hr condition, or in other words tht every function u M hs less thn n + roots to the equtions u(x) = 0 in the intervl [, b].. Theorem 4 Suppose tht the bsis {ψ k } n 0 is Chebyshev set. Then the conditions on {g k} n 0 preceding theorem re stisfied for ll choices of {x k } ν 0 nd {m k} ν 0. of the Proof Let M be equl to the spn of {ψ k } n 0. We begin by showing tht {g ψ k } n 0 is Chebyshev set. Note tht {g ψ k } n 0 is fmily of linerly independent functions, since c k g ψ k = g c k ψ k 7

8 Figure 4: The error scled by ω 3 of Q A [f] (left figure, top), QL [f] (left figure, bottom)/(right figure, top) nd Q F [f] (right figure, bottom) both with only endpoints nd multiplicities two for I[f] = 0 cos(x) eiω(x +x) dx. nd g 0. Let M = spn {g ψ k } n 0 nd ũ M, where ũ is not identiclly zero. We know tht ũ = g u for some u M, nd u is equl to zero less thn n+ times. But if u(x) 0 then ũ(x) 0. Thus M stisfies the Hr condition. It follows tht the vectors [g (y 0 ) ψ k (y 0 ),, g (y n ) ψ k (y n )] for 0 k n re linerly independent for ny choice of nodes {y k } n 0 (Powell, 98). Thus, by trivil limiting rgument, we know tht {g 0,, g n } re linerly independent. The simplest nd most obvious choice for {ψ k } is the stndrd bsis of polynomils, which we know is Chebyshev set. In fct this choice is equivlent to the Filon-type method for the Fourier oscilltor cse. This ws proved in (Xing, 005) for the originl Levin method (i.e. multiplicities ll one), nd the proof is trivil to generlize for the preceding Levin-type method. For the reminder of this section we ssume tht {ψ k } is the stndrd bsis of polynomils. How does the Levin-type method compre numericlly to the symptotic nd Filon-type methods? Consider the cse with g(x) = x + x nd f(x) = cos x. We fix s equl to two, hence the endpoints for the Filon-type nd Levin-type methods must hve multiplicity t lest two, nd we obtin Figure 4. This figure suggests tht, in resonble situtions, the Levin-type method is cler improvement over the symptotic method, though not quite s ccurte s the Filon-type method. Figure 5 compres the Levin-type method nd the Filon-type method with the ddition of two smple points. This grph helps emphsize the effectiveness of dding node points within the intervl of integrtion. With just two node points, only one of which hs multiplicity greter thn one, the error of Q L [f] is less thn sixth of wht it ws. In fct it is firly close to the former Q F [f] while still not requiring the knowledge of moments. On the other hnd, dding the sme node points nd multiplicities to Q F [f] results in n error indistinguishble from zero in comprison to the originl Q L [f]. It should be emphsized tht even Q L [f] with only endpoints is still very effective method, s ll the vlues in this grph re divided by ω = It comes s no surprise tht the hierrchy of ccurcy between the symptotic, Filon-type, nd Levin-type methods depends on the choice of f nd g. After ll, we hve lredy seen tht the Filon-type method cn be less ccurte thn the symptotic method when f exhibits Runge s phenomenon. Further in this pper, we will see n exmple where the Levin-type method with polynomil bsis is significntly more ccurte thn the Filon-type method, nd oscilltes between more ccurte nd less ccurte thn the symptotic method, for incresing ω. 8

9 Figure 5: The error scled by ω 3 of Q L [f] (left figure, top) nd Q F [f] both with only endpoints nd multiplicities two (left figure, bottom) compred to Q L [f] (left figure, middle) nd Q F [f] (right figure) both with nodes { 0, 4, 3, } nd multiplicities {,,, } for I[f] = 0 cos(x) eiω(x +x) dx. 5 Choosing bsis It is importnt to note tht, for the Levin-type method, there is no prticulr reson to use polynomils for {ψ k }. Not only cn we gretly improve the ccurcy of the pproximtion by choosing the bsis wisely, but surprisingly we cn even obtin higher symptotic order. The ide is to choose {ψ k } so tht L[v] is qulittively similr in shpe to f within the intervl of integrtion. Pretend for moment tht Q A s [f] is equl to I[f] = [ F (x) e iωg(x)] b. Then F (x) = s k= (iω) k σ k [f](x). This suggests tht resonble choice for {ψ k } is to define ψ 0 = nd ψ k = σ k [f] for k. Provided tht this choice for {ψ k } stisfies the hypotheses of the Levin-type method, it turns out tht we obtin n error of order O ( ω n s ) for n = ν k=0 m k, i.e. where n + is the number of equtions in the Levin-type method system. This is very significnt improvement since, unlike in the cse of the Filon-type method s well s the Levin-type method with polynomil bsis, node points within the intervl increse the order nd dding n dditionl multiplicity to ech endpoint increses the order by three. To prove the order of the error we rely hevily on Crmer s rule. As result we need to do severl determinnt mnipultions. Hence we derive the following lemm, which we use repetedly in the proof of Theorem 6. Lemm 5 Let ψ 0 =,ψ = f g, nd ψ k+ = ψ k g for integer k. If k then det [g k, k,, k+j, B] = det [g k, g k+,, g k+j+, B] (5.3) where g k nd k were defined in Theorem 3 nd B represents ny dditionl columns tht render the mtrix squre. Proof Note tht ψ k = g ψ k+, for k. Hence we cn rewrite p k s [ ] p k = ψ k (x 0),, ψ (m 0) k (x 0 ),, ψ k (x ν),, ψ (mν) k (x ν ) [ (g = ) ψ k+ (x0 ),, ( g ) (m0 ) ψ k+ (x0 ),, ( g ) ψ k+ (xν ),, ( g ) (mν ) ψ k+ (xν )] = g k+ 9

10 Recll tht k = p k + iωg k. Thus multiplying the first column of the determinnt on the left hnd side of (5.3) by iω nd subtrcting it from the second results in second column equl to k iωg k = p k = g k+. Clerly we cn repet this process on the remining columns, hence the lemm follows by n inductive rgument. Note tht Lemm 5 holds for ny column interchnge tht occurs on both sides of the equlity. Using this lemm we cn prove the following theorem: Theorem 6 Let {ψ k } be defined s in the preceding lemm, nd suppose tht {x k } ν 0, {m k} ν 0, nd {ψ k } n 0 stisfy the conditions for Theorem 3. Then, if m 0, m ν s, where s before n = ν k=0 m k. I[f] Q L [f] O ( ω n s ), Proof } Provided tht we cn show tht the functions in the set {L[v] f,, L[v] (s+) f (s+) re of order O(ω n ), the theorem will follow from Corollry. If we fix 0 j s +, then L[v] (j) f (j) = n c k L[ψ k ] (j) f (j) = k=0 = iωc 0 g (j+) + n k= n k=0 c k ( ψ (j+) k + iω ( g ψ k ) (j) ) f (j) c k [ (g ψ k+ ) (j) + iω ( g ψ k ) (j) ] ( g ψ ) (j) = iωc 0 g (j+) + (iωc ) ( g ) (j) n ψ + (c k + iωc k ) ( g ) (j) ( ψ k + cn g ) (j) ψ n+ = [ iω det D 0 g (j+) + (iω det D det A) ( g ) (j) ψ det A n + (det D k + iω det D k ) ( ] g ) (j) ( ψ k + det Dn g ) (j) ψ n+. k= Recll tht we showed in the proof of Theorem 3 tht det A = O( ω n ). Hence it is sufficient to show tht the numertor of the preceding frction is O(ω). There re four types of terms we need to hndle: iω det D 0, iω det D det A, det D k + iω det D k for integer k n, nd det D n. The first of these cses follows immeditely from Lemm 5. For the second cse note tht k= det A iω det D = det [ 0, p + iωg,,, n ] iω det D = det [ 0, p,,, n ] + iω det [ 0, g,,, n ] iω det D = iω det [g 0, g,,, n ], where we use the fct tht 0 = iωg 0 nd g = f, since g (x k ) ψ (x k ) = f(x k ). After pplying Lemm 5 to this determinnt, it is cler tht this cse is lso O(ω). The third cse is hndled in very similr mnner. Like before we begin by rewriting determinnts: det D k + iω det D k = det [ 0,, k, g, p k + iωg k, k+,, n ] +iω det [ 0,, k, p k + iωg k, g, k+,, n ] = det [ 0,, k, g, g k+, k+,, n ] 0

11 Figure 6: The error scled by ω 4 of Q A 3 [f] (left figure, top), QF [f] with endpoints for nodes nd multiplicities (left figure, bottom), nd Q B [f] in with nodes { 0,, } nd multiplicities one (right figure) for I[f] = 0 log(x + ) eiωx dx. +iω det [ 0,, k, g, g k, k+,, n ] +iω det [ 0,, k, g k, g, k+,, n ] + (iω) det [ 0,, k, g k, g, k+,, n ] = det [ 0,, k, g, g k+, k+,, n ] + (iω) det [ 0,, k, g k, g, k+,, n ]. After using Lemm 5 twice the first of these determinnts is clerly O(ω). But using the lemm on the second determinnt results in two columns being equl to g k, hence the determinnt is zero. The fourth nd finl cse, much like the first cse, is O(ω) due to Lemm 5. Hence we hve shown tht L[v] (j) f (j) is of order O(ω n ) for ll 0 j s +, nd the proof is complete. To emphsize the distinction, we denote the Levin-type method with the stndrd polynomil bsis s Q L [f] nd the Levin-type method with the bsis of the preceding theorem, which we cll the symptotic bsis, s Q B [f]. Clerly, when the sme node points nd multiplicities re used nd ω is sufficiently lrge, Q B [f] is substntive improvement over Q L [f] nd Q F [f]. Of course, it lso requires f (k) for k up to n + s, where the Filon-type method only requires f (k) for k up to s. Thus in some sense it is more pproprite to compre Q B [f] with other methods of the sme order. Consider the Fourier oscilltor nd let f(x) = log(x + ). We compre methods of order O ( ω 4), hence fix s = 3. This includes Q A 3 [f], QF [f] (which is equivlent to Q L [f]) with nodes {0, } nd multiplicities {, }, nd Q B [f] using nodes { 0,, } nd multiplicities ll one. With this set up we obtin Figure 6. The results re decent, with Q B [f] being slightly more ccurte thn Q F [f] on verge. The problem with Q A s [f] nd Q F [f] is tht in generl s s these methods diverge. Hence nother worthwhile comprison is to see how Q B [f] compres to these two methods for fixed ω nd incresing s. Thus fix ω = 50, chosen purposely reltively smll since the lrger ω, the longer it tkes for incresing s to cuse the pproximtion to diverge. This choice results in Figure 7, where we tke the bse-0 logrithms of the error. This figure clerly shows the benefit of using Q B [f] for this prticulr cse. Though t lower orders the error of Q F [f] nd Q B [f] re very similr, t higher orders they differ by orders of mgnitude. For O ( ω 9), the error of Q B [f] is slightly better thn 0 6 while the error of Q F [f] is slightly better thn 0, nd the error of Q A [f] is not

12 s 4 6 Figure 7: The bse-0 logrithm of the error of Q A s [f] (top), Q F [f] with endpoints for nodes nd multiplicities s (middle), nd Q B [f] with nodes {k/ (s )} s k=0 nd multiplicities ll one (bottom) for I[f] = 0 log(x + ) eiωx dx ω Figure 8: The error scled by ω 6 of Q B [f] with nodes { 0, 4,, 3 4, } nd multiplicities ll one (bottom), nodes { 0,, } nd multiplicities {, 3, } (middle), nd nodes {0, } nd multiplicities both equl to two (top) for I[f] = 0 log(x + ) eiωx dx. even 0. At round 0 6 we rech IEEE mchine precision, hence it would be meningless to extend this grph to higher orders. We cn lso compre Q B [f] with itself under different choices of node points. Though we retin the sme f nd g, we compre different methods of order O ( ω 6) to increse the number of possible node choices. We consider three choices of nodes nd multiplicities: nodes { 0, 4,, 3 4, } nd multiplicities ll one, nodes { 0,, } nd multiplicities {, 3, }, nd nodes {0, } nd multiplicities both equl to two. This results in Figure 8. We tke reltively mild vlues for ω s ny vlue significntly lrger nd the ccurcy reches IEEE mchine precision. It is not entirely suprising tht the more concentrted the smpling the less ccurte the pproximtion. Though they re not displyed in the preceding figure, for comprison Q A 5 [f] performed horribly, oscillting between 3 nd 5, while Q F [f] with nodes {0, } nd multiplicities five performed roughly in the middle of the pck, oscillting between 0. nd 0.8.

13 ω Figure 9: The error scled by ω 6 of Q A 5 [f] (top) nd QB [f] with nodes {, 5, 0, 0, } nd multiplicities ll one (bottom) for I[f] = 0 x eiωx dx compred to E (iω) ω Figure 0: The error scled by ω 6 of Re Q A 5 [f] (top) nd Re QB [f] with nodes {, 5, 0, 0, } nd multiplicities ll one (bottom) for I[f] = 0 x eiωx dx compred to Ci(ω). Now consider the cse of E ( iω) = e iωx x dx where E is the exponentil integrl s defined in (Abrmowitz nd Stegun, 964). This function is importnt since we cn derive the sine nd cosine integrls from its rel nd imginry prts. Note tht though we never explicitly hndled the cse of b =, ll the proofs up to this point re vlid for this sitution s long s we cn integrte by prts. Thus we cn use the symptotic method with σ k [f](x) = f (k ) (x) = ( ) k (k )!x k, mening tht σ k [f]( ) = 0 nd σ k [f]() = ( ) k (k )! for ll k. Since σ k [f]( ) is lwys zero, we hve the dded benefit tht it is only necessry to evlute f nd its derivtives t one of the endpoints to obtin the desired order. Thus we derive the following symptotic expnsion: E ( iω) e iω k= ( ) k (k )! ( iω) k. It should come s no surprise tht this is equivlent to the expnsion in (Abrmowitz nd Stegun, 964). 3

14 Figure : The error scled by ω 4 of Q A 3 [f] (left figure, bottom), QF [f] with only endpoints nd multiplicities two (left figure, top), nd Q B [f] with end points nd two Chebyshev nodes, ll with multiplicity one (right figure) for I[f] = 0 e iωx dx. +0x Clerly neither Filon-type method nor Levin-type method with polynomil bsis cn hndle this sitution since polynomils diverge t. On the other hnd, we cn use the symptotic bsis with the Levin-type method to derive n pproximtion. Consider the cse of rbitrrily chosen nodes {, 5, 0, 0, } with multiplicities ll one. This hs order of error O ( ω 6), thus we compre it to the symptotic method with s = 5 in Figure 9. Even with rbitrrily chosen nodes, Q B [f] is substntilly more ccurte thn simply using the symptotic expnsion; in this cse it hs less thn tenth of the error on verge. We cn lso compre the rel prts of ech pproximtion to Ci(ω), where Ci is the cosine integrl s defined in (Abrmowitz nd Stegun, 964). This results in Figure 0. We now consider gin the function tht suffered from Runge s phenomenon. Since Q B [f] is not polynomil interpoltion, there is good chnce tht Runge s phenomenon will not ffect us in the sme wy. In fct, numericl tests show tht Q B [f] hs significntly less error thn its polynomil counterprts. Direct computtion shows tht det A is polynomil in ω of degree n, not of degree n +. Fortuntely, the proof of Theorem 6 holds s is, except tht Q B [f] now hs error of order O(ω n s ). Agin we compre methods of similr order in Figure, which shows tht Q B [f] is the best of the three methods tried. Another sitution somewht similr to Runge s phenomenon is when f increses much too fst to be ccurtely pproximted by polynomils. Let f(x) = e 0x nd g(x) = x + x. Note tht this ppers to be ludicrously difficult exmple not only do we hve high oscilltions but f exceeds, 000 in the intervl of integrtion! Amzingly, we will see tht the methods described within this pper re still very ccurte, especilly the Levin-type method with symptotic bsis. We compre Q B [f] which hs only endpoints for nodes nd multiplicities ll one to Q A [f] nd QF [f] with only endpoints for nodes nd multiplicities both two in Figure. We omit the proof tht the vectors {g 0,, g n } ssocited with Q B [f] re linerly independent, s it is simple exercise in liner lgebr. In this exmple Q F [f] produces tremendously bd pproximtion, due to the difficulty in interpolting n exponentil by polynomil. As seen in Tble, the ctul error for ω = 00 is bout On the other hnd, Q A [f] performed significntly better thn the Filon-type method, though still not spectculrly, with n error of pproximtely for ω = 00. The str of this show is clerly Q B [f], where the ctul error for ω = 00 is bout ; less thn tenth of the error of Q A [f]. 4

15 Figure : The error scled by ω 3 of Q F [f] with endpoints nd multiplicities both two (left figure, top), Q L [f] with endpoints nd multiplicities both two (left figure, bottom), Q A [f] (right figure, top), nd Q B [f] with endpoints nd multiplicities ll zero (right figure, bottom) for I[f] = 0 e0x e iω(x +x) dx. s Q A s [f] Q F [f] Q L [f] Q B [f] Tble : The bsolute vlue of the errors for ω = 00 of the following methods of order O ( ω s ) : Q A s [f], Q F [f] nd Q L [f] with endpoints nd multiplicities both s, nd Q B [f] with nodes {k/ (s )} s k=0 nd multiplicities ll one for I[f] = 0 e0x e 00i(x +x) dx. Adding dditionl nodes to Q B [f] increses the ccurcy further. For exmple, gin with ω = 00, dding single node t the midpoint decreses the error to while dding nodes t 4, the midpoint, nd 3 4 further decreses the error to the stoundingly smll This exmple demonstrtes just how powerful these qudrture techniques re compred to Gussin qudrture: even with 00, 000 pnels Gussin qudrture hd n error of 0., not even close to the ccurcy of the Filon-type method, to sy nothing of Q B [f]. Acknowledgements The uthor wishes to thnk Arieh Iserles, Dvid Levin, Alexei Shdrin, Gtes Cmbridge Trust, University of Cmbridge, nd University of Tel Aviv. References Abrmowitz, M. nd Stegun, I. A., editors (964). Hndbook of Mthemticl Functions with Formuls, Grphs, nd Mthemticl Tbles. U. S. Government Printing Office, Wshington, D. C. Iserles, A. nd Nørsett, S. P. (004). Efficient qudrture of highly oscilltory integrls using derivtives. Technicl Report NA004/03, DAMTP, University of Cmbridge, To pper in Proceedings Royl Soc. A. 5

16 Levin, D. (997). Anlysis of colloction method for integrting rpidly oscilltory functions. J. Comput. Appl. Mths, 78:3 38. Powell, M. J. D. (98). Approximtion Theory nd Methods. Cmbridge University Press, Cmbridge. Xing, S.-H. (005). On the numericl qudrture of highly oscilltory functions. Technicl report, Centrl South University, Hunn, Chin. 6