1 Numerical integration

Transcription

1 1 Numericl integrtion 1.1 Introduction The term numericl integrtion reers to brod mily o lgorithms to compute numericl pproximtion to deinite (Riemnn) integrl. Generlly, the integrl is pproximted by weighted sum o unction vlues within the domin o integrtion, (x)dx w i (x i ). (1) Expression (1) is oten reerred to s qudrture (cubture or multidimensionl integrls) or rule. The bscisss x i (lso clled nodes) nd the weights w i o qudrture re usully optimized using one o lrge number o dierent strtegies to suit prticulr clss o integrtion problems. The best qudrture lgorithm or given problem depends on severl ctors, in prticulr on the integrnd. Dierent clsses o integrnds generlly require dierent qudrtures or the most eective clcultion. A populr numericl integrtion librry is QUADPACK [?]. It includes generl purpose routines like QAGS, bsed on n dptive GussKronrod qudrture with ccelertion s well s number o specilized routines. The GNU scientiic librry [?] (GSL) implements most o the QUADPACK routines nd in ddition includes modern generl-purpose dptive routine CQUAD bsed on Clenshw-Curtis qudrtures [?]. In the ollowing we shll consider some o the populr numericl integrtion lgorithms. 1. Rectngle nd trpezium rules In mthemtics, the Reimnn integrl is generlly deined in terms o Riemnn sums [?]. I the integrtion intervl [, b] is prtitioned into n subintervls, = t 0 < t 1 < t < < t n = b. () the Riemnn sum is deined s (x i ) x i, (3) 1

2 where x i [t i 1, t i ] nd x i = t i t i 1. Geometriclly Riemnn sum cn be interpreted s the re o collection o djucent rectngles with widths x i nd heights (x i ). The Riemn integrl is deined s the limit o Riemnn sum s the mesh the length o the lrgest subintervl o the prtition pproches zero. Speciiclly, the number denoted s (x)dx (4) is clled the Riemnn integrl, i or ny ɛ > 0 there exists δ > 0 such tht or ny prtition () with mx x i < δ we hve (x i ) x i (x)dx < ɛ. (5) A deinite integrl cn be interpreted s the net signed re bounded by the grph o the integrnd. Now, the n-point rectngle qudrture is simply the Riemnn sum (3), (x)dx (x i ) x i, (6) where the node x i is oten (but not lwys) tken in the middle o the corresponding subintervl, x i = t i x i, nd the subintervls re oten (but not lwys) chosen equl, x i = (b )/n. Geometriclly the n-point rectngle rule is n pproximtion to the integrl given by the re o collection o n djucent equl rectngles whose heights re determined by the vlues o the unction (t the middle o the rectngle). An n-point trpezium rule uses insted collection o trpezi itted under the grph, (t i 1 ) + (t i ) (x)dx x i. (7) Importntly, the trpezium rule is the verge o two Riemnn sums, (t i 1 ) + (t i ) x i = 1 (t i 1 ) x i + 1 (t i ) x i. (8) Rectngle nd trpezium qudrtures both hve the importnt eture o closely ollowing the very mthemticl deinition o the integrl s the limit

3 o the Riemnn sums. Thereore disregrding the round-o errors these two rules cnnot il i the integrl exists. For certin prtitions o the intervl the rectngle nd trpezium rules coincide. For exmple, or the nodes x i = + (b ) i 1 n, i = 1,..., n (9) both rules give the sme qudrture with equl weights, w i = (b )/n, (x)dx b n ( + (b ) i 1 ). (10) n Rectngle nd trpezium qudrtures re rrely used on their own becuse o the slow convergence but they oten serve s the bsis or more dvnced qudrtures, or exmple dptive qudrtures nd vrible trnsormtion qudrtures considered below. 1.3 Qudrtures with regulrly spced bscisss A qudrture (1) with n predeined nodes x i hs n ree prmeters: the weights w i. A set o n prmeters cn generlly be tuned to stisy n conditions. The rchetypl set o conditions in qudrtures is tht the qudrture integrtes exctly set o n unctions, This leds to set o n equtions, where the integrls {φ 1 (x),..., φ n (x)}. (11) k=1,...,n w i φ k (x i ) = I k, (1) I k. = φ k (x)dx (13) re ssumed to be known. Equtions (1) re liner in w i nd cn be esily solved. Since integrtion is liner opertion, the qudrture will then lso integrte exctly ny liner combintion o unctions (11). 3

4 A populr choice or predeined nodes is closed set tht is, including the end-points o the intervl o evenly spced bscisss, x i = + i 1 n 1 (b ),...,n. (14) However, in prctice it oten hppens tht the integrnd hs n integrble sinulrity t one or both ends o the intervl. In this cse one cn choose n open set o equidistnt nodes, x i = + i 1 n (b ),...,n. (15) The set o unctions to be integrted exctly is generlly chosen to suite the properties o the integrnds t hnd: the integrnds must be well represented by liner combintions o the chosen unctions Clssicl qudrtures Suppose the integrnd cn be well represented by the irst ew terms o its Tylor series, (k) () (x) = (x ) k, (16) k! k=0 where (k) is the k-th derivtive o the integrnd. This is oten the cse or nlytic tht is, ininitely dierentible unctions. For such integrnds one cn obviously choose polynomils {1, x, x,..., x n 1 } (17) s the set o unctions to be integrted exctly. This leds to the so clled clssicl qudrtures: qudrtures with regulrly spced bscisss nd polynomils s exctly integrble unctions. An n-point clssicl qudrture integrtes exctly the irst n terms o the unction s Tylor expnsion (16). The x n order term will not be integrted exctly nd will led to n error o the qudrture. Thus the error E n o the n-point clssicl qudrture is on the order o the integrl o the x n term in (16), E n (n) () (x ) n dx = (n) () n! (n + 1)! hn+1 h n+1, (18) 4

5 Tble 1: Mxim script to clculte nlyticlly the weights o n n-point clssicl qudrture with predeined bscisss in the intervl [0, 1]. n: 8; xs: mkelist((i-1)/(n-1),i,1,n); /* nodes: dpt to your needs */ ws: mkelist(conct(w,i),i,1,n); ps: mkelist(x^i,i,0,n-1); /* polynomils */ s: mkelist(buildq([i:i,ps:ps],lmbd([x],ps[i])),i,1,n); integ01: lmbd([],integrte((x),x,0,1)); Is: mplist(integ01,s); /* clculte the integrls */ eq: lmbd([],lreduce("+",mplist(,xs)*ws)); eqs: mplist(eq,s)-is; /* build equtions */ solve(eqs,ws); /* solve or the weights */ where h = b is the length o the integrtion intervl. A qudrture with the error o the order h n+1 is oten clled degree-n qudrture. I the integrnd is smooth enough nd the length h is smll enough clssicl qudrture with not so lrge n cn provide good pproximtion or the integrl. However, or lrge n the weights o clssicl qudrtures tend to hve lternting signs, which leds to lrge round-o errors, which in turn negtes the potentilly higher ccurcy o the qudrture. Agin, i the integrnd violtes the ssumption o Tylor expnsion or exmple by hving n integrble singulrity inside the integrtion intervl the higher order qudrtures my perorm poorly. Clssicl qudrtures re mostly o historicl interest nowdys. Alterntive methods such s qudrtures with optimized bscisss, dptive, nd vrible trnsormtion qudrtures re more stble nd ccurte nd re normlly preerred to clssicl qudrtures. Clssicl qudrtures with eqully spced bscisss both closed nd open sets re generlly reerred to s Newton-Cotes qudrtures. An interested reder cn generte Newton-Cotes qudrtures o ny degree n using the Mxim script in Tble (1). 1.4 Qudrtures with optimized bscisss In qudrtures with optimized bscisss not only the weights w i but lso the bscisss x i re chosen optimlly. The number o ree prmeters is thus n 5

6 nd one cn choose set o n unctions, {φ 1 (x),..., φ n (x)}, (19) to be integrted exctly. This gives system o n equtions, liner in w i nd non-liner in x i, k=1,...,n w i φ k (x i ) = I k, (0) where gin I k. = φ k (x)dx. (1) The weights nd bscisss o the qudrture cn be determined by solving this system o equtions 1. Although qudrtures with optimized bcisss re generlly o much higher order, n 1 compred to n 1 or non-optiml bscisss, the optiml points generlly cn not be reused t the next itertion in n dptive lgorithm Guss qudrtures Guss qudrtures del with slightly more generl orm o integrls, ω(x)(x)dx, (3) where ω(x) is positive weight unction. For ω(x) = 1 the problem is the sme s considered bove. Populr choices o the weight unction include ω(x) = (1 x ) ±1/, exp( x), exp( x ) nd others. The ide is to represent the integrnd s product ω(x)(x) such tht ll the diiculties go into the weight unction ω(x) while the remining ctor (x) is smooth nd well represented by polynomils. An N-point Guss qudrture is qudrture with optimized bcisss, ω(x)(x)dx N w i (x i ), (4) which integrtes exctly set o N polynomils o the orders 1,..., N 1 with the given weight ω(x). 1 Here is, or exmple, n n = qudrture with optimized bscisss, Z 1 q «q «1 1 (x)dx + +. ()

7 Fundmentl theorem There is theorem stting tht there exists set o polynomils p n (x), orthogonl on the intervl [, b] with the weight unction ω(x), ω(x)p n (x)p k (x) δ nk. (5) Now, one cn prove tht the optiml nodes or the N-point Guss qudrture re the roots o the polynomil p N (x), The ide behind the proo is to consider the integrl p N (x i ) = 0. (6) ω(x)q(x)p N (x)dx = 0, (7) where q(x) is n rbitrry polynomil o degree less thn N. The qudrture should represent this integrl exctly, N w i q(x i )p N (x i ) = 0. (8) Apprently this is only possible i x i re the roots o p N. Clcultion o nodes nd weights A net lgorithm usully reered to s Golub-Welsch [?] lgorithm or clcultion o the nodes nd weights o Guss qudrture is bsed on the symmetric orm o the three-term reccurence reltion or orthogonl polynomils, xp n 1 (x) = β n p n (x) + α n p n 1 (x) + β n 1 p n (x), (9) where p 1 (x). = 0, p 1 (x). = 1, nd n = 1,..., N. This reccurence reltion cn be written in the mtrix orm, xp(x) = Jp(x) + β N p N (x)e N, (30) where p(x) =. {p 0 (x),..., p N 1 (x)} T, e N = {0,..., 0, 1} T, nd the tridigonl mtrix J usully reered to s Jcobi mtrix or Jcobi opertor is given s α 1 β 1 β 1 α β J = β α 3 β 3. (31) β N 1 α N 7

8 Substituting the roots x i o p N tht is, the set {x i p N (x i ) = 0} into the mtrix eqution (30) leds to eigenvlue problem or the Jcobi mtrix, Jp(x i ) = x i p(x i ). (3) Thus, the nodes o n N-point Guss qudrture (the roots o the polynomil p N ) re the eigenvlues o the Jcobi mtrix J nd cn be clculted by stndrd digonliztion routine. The weights cn be obtined considering N integrls, ω(x)p n (x)dx = δ n0 ω(x)dx, n = 0,..., N 1. (33) Applying our qudrture gives the mtrix eqution, Pw = e 1 ω(x)dx, (34) where w =. {w 1,..., w N } T, e 1 = {1, 0,..., 0} T, nd p 0 (x 1 )... p 0 (x N ) P =. p 1 (x 1 )... p 1 (x N ) (35) p N 1 (x 1 )... p N 1 (x N ) Eqution (34) is liner in w i nd cn be solved directly. However, i digonliztion o the Jcobi mtrix provided the normlized eigenvectors, the weigths cn be redily obtined using the ollowing method. The mtrix P pprently consists o non-normlized column eigenvectors o the mtrix J. The eigenvectors re orthogonl nd thereore P T P is digonl mtrix with positive elements. Multiplying (34) by P T nd then by (P T P) 1 rom the let gives w = (P T P) 1 P T e 1 ω(x)dx. (36) From p 0 (x) = 1 it ollows tht P T e 1 = {1,..., 1} T nd thereore 1 w i = (P T P) ii ω(x)dx. (37) A symmetric tridigonl mtrix cn be digonlized very eectively using the QR/RL lgorithm. 8

9 Let the mtrix V be the set o the normlized column eigenvectors o the mtrix J. The mtrix V is then connected with the mtrix P through the normliztion eqution, V = (P T P) 1 P. (38) Thereore, gin tking into ccount tht p 0 (x) = 1, eqution (37) cn be written s w i = (V 1i ) ω(x)dx. (39) Exmple: Guss-Legendre qudrture Guss-Legendre qudrture dels with the weight ω(x) = 1 on the intervl [ 1, 1]. The ssocited polynomils re Legendre polynomils P n (x), hence the nme. Their reccurence reltion is usully given s (n 1)xP n 1 (x) = np n (x) + (n 1)P n (x). (40) Rescling the polynomils (preserving p 0 (x) = 1) s n + 1Pn (x) = p n (x) (41) reduces this reccurence reltion to the symmetric orm (9), xp n 1 (x) = 1 1 p n(x) (n) p n (x). (4) 1 ((n 1)) Correspondingly, the coeicients in the mtrix J re α n = 0, β n = 1 1. (43) 1 (n) The problem o inding the nodes nd the weights o the N-point Guss- Legendre qudrture is thus reduced to the eigenvlue problem or the Jcobi mtrix with coeicients (43). As n illustrtion o this lgorithm Tble () shows n Octve unction which clcultes the nodes nd the weights o the N-point Guss-Legendre qudrture nd then integrtes given unction. 9

10 Tble : An Octve unction which clcultes the nodes nd weights o the N-point Guss-Legendre qudrture nd then integrtes given unction. unction Q = g u s s l e g e n d r e (,, b,n) bet =. 5. / sqrt (1 ( (1:N 1 ) ). ˆ ( ) ) ; % r e c c u r e n c e r e l t i o n J = dig ( bet, 1 ) + dig ( bet, 1); % Jcobi mtrix [V,D] = eig ( J ) ; % d i g o n l i z t i o n o J x = dig (D) ; [ x, i ] = sort ( x ) ; % s o r t e d nodes w = V( 1, i ). ˆ ; % weights Q = w ( ( +b)/+(b )/ x ) ( b ) / ; % i n t e g r l endunction ; 1.4. Guss-Kronrod qudrtures Generlly, the error o numericl integrtion is estimted by compring the results rom two rules o dierent orders. However, or ordinry Guss qudrtures the nodes or two rules o dierent orders lmost never coinside. This mens tht one cn not reuse the points o the lower order rule when clculting the hihger order rule. Guss-Kronrod lgorithm [?] remedies this ineiciency. The points inherited rom the lower order rule re reused in the higher order rule s predeined nodes (with n weights s ree prmeters), nd then m more optiml points re dded (m bscisss nd m weights s ree prmeters). The order o the method is n + m 1. The lower order rule becomes embedded tht is, it uses subset o the nodes into the higher order rule. On the next itertion the procedure is repeted. Ptterson [?] hs tbulted nodes nd weigths or severl sequences o embedded Guss-Kronrod rules. 1.5 Adptive qudrtures Higher order qudrtures suer rom round-o errors s the weights w i generlly hve lternting signs. Agin, using high order polynomils is dngerous s they typiclly oscillte wildly nd my led to Runge s phenomenon. Thereore, i the error o the qudrture is yet too lrge or qudrture with suiciently lrge n, the best strtegy is to subdivide the intervl in two nd then use the qudrture on the hl-intervls. Indeed, i the error is o the order h k, the subdivision would led to reduced error, (h/) k < h k, i k > 1. An dptive qudrture is n lgorithm where the integrtion intervl is 10

11 subdivided into dptively reined subintervls until the given ccurcy gol is reched. Adptive lgorithms re usully built on pirs o qudrture rules higher order rule, Q = w i (x i ), (44) i where w i re the weights o the higher order rule nd Q is the higher order estimte o the integrl, nd lower order rule, q = i v i (x i ), (45) where v i re the weights o the lower order rule nd q is the the lower order estimte o the integrl. The dierence between the higher order rule nd the lower order rule gives n estimte o the error, δq = Q q. (46) The integrtion result is ccepted, i the error δq is smller thn tolernce, δq < δ + ɛ Q, (47) where δ is the bsolute ccurcy gol nd ɛ is the reltive ccurcy gol o the integrtion. I the error estimte is lrger thn tolernce, the intervl is subdivided into two hl-intervls nd the procedure pplies recursively to subintervls with the sme reltive ccurcy gol ɛ nd rescled bsolute ccurcy gol δ/. The points x i re usully chosen such tht the two qudrtures use the sme points, nd tht the points cn be reused in the subsequent recursive steps. The reuse o the unction evlutions mde t the previous step o dptive integrtion is very importnt or the eiciency o the lgorithm. The equllyspced bscisss nturlly provide or such reuse. As n exmple, Tble 3 shows n implementtion o the described lgorithm using { 1 x i = 6, 6, 4 6, 5 } (esily reusble points), (48) 6 { w i = 6, 1 6, 1 6, } (trpezium rule), (49) 6 { 1 v i = 4, 1 4, 1 4, 1 } (rectngle rule). (50) 4 11

12 During recursion the unction vlues t the points # nd #3 re inherited rom the previous step nd need not be reclculted. The points nd weights re cited or the rescled integrtion intervl [0, 1]. The trnsormtion o the points nd weights to the originl intervl [, b] is given s x i + (b )x i, w i (b )w i. (51) This implementtion clcultes directly the Riemnn sums nd cn thereore del with integrble singulrities, lthough rther ineiciently. More eicient dptive routines keep trck o the subdivisions o the intervl nd the locl errors [?]. This llows detection o singulrities nd switching in their vicinity to speciiclly tuned qudrtures. It lso llows better estimtes o locl nd globl errors. 1.6 Vrible trnsormtion qudrtures The ide behind vrible trnsormtion qudrtures is to pply the given qudrture either with optimimized or regulrly spced nodes not to the originl integrl, but to vrible trnsormed integrl [?], tb t ( g(t) ) g (t)dt N w i ( g(t i ) ) g (t i ), (5) where the trnsormtion x = g(t) is chosen such tht the trnsormed integrl better suits the given qudrture. Here g denotes the derivtive nd [t, t b ] is the corresponding intervl in the new vrible. For exmple, the Guss-Legendre qudrture ssumes the integrnd cn be well represented with polynomils nd perorms poorly on integrls with integrble singulrities like I = 0 1 dx. (53) x However, simple vrible trnsormtion x = t removes the singulrity, I = 0 dt, (54) 1

13 Tble 3: Recursive dptive integrtor in C #include<mth. h> #include< s s e r t. h> #include<s t d i o. h> double dpt4 ( double ( double ), double, double b, double cc, double eps, double, double 3, int nrec ) { s s e r t ( nrec < ); double 1= ( +(b ) / 6 ), 4= ( +5 (b ) / 6 ) ; double Q=( ) / 6 ( b ), q=( ) / 4 ( b ) ; double t o l e r n c e=cc+eps b s (Q), e r r o r= b s (Q q ) ; i ( e r r o r < t o l e r n c e ) return Q; else { double Q1=dpt4 (,, ( +b ) /, cc / s q r t (. ), eps, 1,, nrec +1); double Q=dpt4 (, ( +b ) /, b, cc / s q r t (. ), eps, 3, 4, nrec +1); return Q1+Q ; } } double dpt ( double ( double ), double, double b, double cc, double eps ) { double = ( + (b ) / 6 ), 3= ( +4 (b ) / 6 ) ; int nrec =0; return dpt4 (,, b, cc, eps,, 3, nrec ) ; } int min ( ) // uses gcc n e s t e d u n c t i o n s { int c l l s =0; double =0,b=1, cc =0.001, eps =0.001; double ( double x ){ c l l s ++; return 1/ s q r t ( x ) ; } ; // n e s t e d u n c t i o n double Q=dpt (,, b, cc, eps ) ; p r i n t ( Q=%g c l l s=%d\n,q, c l l s ) ; return 0 ; } nd the Guss-Legendre qudrture or the trnsormed integrl gives exct result. The price is tht the trnsormed qudrture perorms less eectively on smooth unctions. Some o the populr vrible trnsormtion qudrtures re Clenshw- Curtis [?], bsed on the trnsormtion 1 π 0 (cos θ) sin θdθ, (55) nd tnh-sinh qudrture [?], bsed on the trnsormtion 1 ( ( π )) π tnh sinh(t) cosh(t) cosh (. (56) π sinh(t))dt Generlly, the eqully spced trpezium rule is used ter the trnsormtion. 13

14 1.7 Ininite intervls One wy to clculte n integrl over ininite intervl is to trnsorm it by vrible sustitution into n integrl over inite intervl. The ltter cn then be evluted by ordinry integrtion methods. Tble 4 lists severl o such trnsormtion. Tble 4: Vrible trnsormtions reducing ininite intervl integrls into integrls over inite intervls ( ) t 1 + t 1 t (1 t dt, (57) ) ( ) ( 1 t + 1 t )) dt t t t, (58) ( ( + ( + 1 t ( t ) 1 dt, (59) 1 t (1 t) ) dt t t, (60) t ) 1 dt, (61) 1 + t (1 + t) ) dt t. (6) ( 1 t t 14