4.4 Improper Integrals

Transcription

1 4.4 Improper Integrls 4 which contin no singulrities, nd where the endpoints re lso nonsingulr. qromb, in such circumstnces, tkes mny, mny fewer function evlutions thn either of the routines in 4.2. For exmple, the integrl 2 0 x 4 log(x + x 2 +)dx converges (with prmeters s shown bove) on the very first extrpoltion, fter just 5 clls to trpzd, while qsimp requires 8 clls (8 times s mny evlutions of the integrnd) nd qtrp requires 3 clls (mking 256 times s mny evlutions of the integrnd). CITED REFERENCES AND FURTHER READING: Stoer, J., nd Bulirsch, R. 980, Introduction to Numericl Anlysis (New York: Springer-Verlg), Dhlquist, G., nd Bjorck, A. 974, Numericl Methods (Englewood Cliffs, NJ: Prentice-Hll), Rlston, A., nd Rbinowitz, P. 978, A First Course in Numericl Anlysis, 2nd ed. (New York: McGrw-Hill), Improper Integrls For our present purposes, n integrl will be improper if it hs ny of the following problems: its integrnd goes to finite limiting vlue t finite upper nd lower limits, but cnnot be evluted right on one of those limits (e.g., sin x/x t x =0) its upper limit is, or its lower limit is it hs n integrble singulrity t either limit (e.g., x /2 t x =0) it hs n integrble singulrity t known plce between its upper nd lower limits it hs n integrble singulrity t n unknown plce between its upper nd lower limits If n integrl is infinite (e.g., x dx), or does not exist in limiting sense (e.g., cos xdx), we do not cll it improper; we cll it impossible. No mount of clever lgorithmics will return meningful nswer to n ill-posed problem. In this section we will generlize the techniques of the preceding two sections to cover the first four problems on the bove list. A more dvnced discussion of qudrture with integrble singulrities occurs in Chpter 8, notbly 8.3. The fifth problem, singulrity t unknown loction, cn relly only be hndled by the use of vrible stepsize differentil eqution integrtion routine, s will be given in Chpter 6. We need workhorse like the extended trpezoidl rule (eqution 4..), but one which is n open formul in the sense of 4., i.e., does not require the integrnd to be evluted t the endpoints. Eqution (4..9), the extended midpoint rule, is the best choice. The reson is tht (4..9) shres with (4..) the deep property

2 42 Chpter 4. Integrtion of Functions of hving n error series tht is entirely even in h. Indeed there is formul, not s well known s it ought to be, clled the Second Euler-Mclurin summtion formul, xn h[f 3/2 + f 5/2 + f 7/2 + +f N 3/2 +f N /2 ] x + B 2h 2 4 (f N f )+ + B 2kh 2k ( 2 2k+ )(f (2k ) N f (2k ) )+ (2k)! (4.4.) This eqution cn be derived by writing out (4.2.) with stepsize h, then writing it out gin with stepsize h/2, then subtrcting the first from twice the second. It is not possible to double the number of steps in the extended midpoint rule nd still hve the benefit of previous function evlutions (try it!). However, it is possible to triple the number of steps nd do so. Shll we do this, or double nd ccept the loss? On the verge, tripling does fctor 3 of unnecessry work, since the right number of steps for desired ccurcy criterion my in fct fll nywhere in the logrithmic intervl implied by tripling. For doubling, the fctor is only 2, but we lose n extr fctor of 2 in being unble to use ll the previous evlutions. Since.732 < 2.44, it is better to triple. Here is the resulting routine, which is directly comprble to trpzd. #define FUNC(x) ((*func)(x)) flot midpnt(flot (*func)(flot), flot, flot b, int n) This routine computes the nth stge of refinement of n extended midpoint rule. func is input s pointer to the function to be integrted between limits nd b, lso input. When clled with n=, the routine returns the crudest estimte of f(x)dx. Subsequent clls with n=2,3,... (in tht sequentil order) will improve the ccurcy of s by dding (2/3) 3 n- dditionl interior points. s should not be modified between sequentil clls. flot x,tnm,sum,del,ddel; sttic flot s; if (n == ) return (s=(b-)*func(0.5*(+b))); else for(it=,j=;j<n-;j++) it *= 3; tnm=it; del=(b-)/(3.0*tnm); ddel=del+del; x=+0.5*del; sum=0.0; for (j=;j<=it;j++) x += ddel; x += del; s=(s+(b-)*sum/tnm)/3.0; return s; The dded points lternte in spcing between del nd ddel. The new sum is combined with the old integrl to give refined integrl.

3 4.4 Improper Integrls 43 The routine midpnt cn exctly replce trpzd in driver routine like qtrp ( 4.2); one simply chnges trpzd(func,,b,j) to midpnt(func,,b, j), nd perhps lso decreses the prmeter JMAX since 3 JMAX (from step tripling) is much lrger number thn 2 JMAX (step doubling). The open formul implementtion nlogous to Simpson s rule (qsimp in 4.2) substitutes midpnt for trpzd nd decreses JMAX s bove, but now lso chnges the extrpoltion step to be s=(9.0*st-ost)/8.0; since, when the number of steps is tripled, the error decreses to /9th its size, not /4th s with step doubling. Either the modified qtrp or the modified qsimp will fix the first problem on the list t the beginning of this section. Yet more sophisticted is to generlize Romberg integrtion in like mnner: #include <mth.h> #define EPS.0e-6 #define JMAX 4 #define JMAXP (JMAX+) #define K 5 flot qromo(flot (*func)(flot), flot, flot b, flot (*choose)(flot(*)(flot), flot, flot, int)) Romberg integrtion on n open intervl. Returns the integrl of the function func from to b, using ny specified integrting function choose nd Romberg s method. Normlly choose will be n open formul, not evluting the function t the endpoints. It is ssumed tht choose triples the number of steps on ech cll, nd tht its error series contins only even powers of the number of steps. The routines midpnt, midinf, midsql, midsqu, midexp, re possible choices for choose. The prmeters hve the sme mening s in qromb. void polint(flot x[], flot y[], int n, flot x, flot *y, flot *dy); void nrerror(chr error_text[]); int j; flot ss,dss,h[jmaxp+],s[jmaxp]; h[]=.0; for (j=;j<=jmax;j++) s[j]=(*choose)(func,,b,j); if (j >= K) polint(&h[j-k],&s[j-k],k,0.0,&ss,&dss); if (fbs(dss) <= EPS*fbs(ss)) return ss; h[j+]=h[j]/9.0; This is where the ssumption of step tripling nd n even error series is used. nrerror("too mny steps in routing qromo"); return 0.0; Never get here. Don t be put off by qromo s complicted ANSI declrtion. A typicl invoction (integrting the Bessel function Y 0 (x) from 0 to 2) is simply #include "nr.h" flot nswer;... nswer=qromo(bessy0,0.0,2.0,midpnt);

4 44 Chpter 4. Integrtion of Functions The differences between qromo nd qromb ( 4.3) re so slighttht it is perhps grtuitous to list qromo in full. It, however, is n excellent driver routine for solving ll the other problems of improper integrls in our first list (except the intrctble fifth), s we shll now see. The bsic trick for improper integrls is to mke chnge of vribles to eliminte the singulrity, or to mp n infinite rnge of integrtion to finite one. For exmple, the identity / /b ( ) t 2 f dt b > 0 (4.4.2) t cn be used with either b nd positive, or with nd b negtive, nd works for ny function which decreses towrds infinity fster thn /x 2. You cn mke the chnge of vrible implied by (4.4.2) either nlyticlly nd then use (e.g.) qromo nd midpnt to do the numericl evlution, or you cn let the numericl lgorithm mke the chnge of vrible for you. We prefer the ltter method s being more trnsprent to the user. To implement eqution (4.4.2) we simply write modified version of midpnt, clled midinf, which llows b to be infinite (or, more precisely, very lrge number on your prticulr mchine, such s 0 30 ), or to be negtive nd infinite. #define FUNC(x) ((*funk)(.0/(x))/((x)*(x))) Effects the chnge of vrible. flot midinf(flot (*funk)(flot), flot, flot bb, int n) This routine is n exct replcement for midpnt, i.e., returns the nth stge of refinement of the integrl of funk from to bb, except tht the function is evluted t evenly spced points in /x rther thn in x. This llows the upper limit bb to be s lrge nd positive s the computer llows, or the lower limit to be s lrge nd negtive, but not both. nd bb must hve the sme sign. flot x,tnm,sum,del,ddel,b,; sttic flot s; b=.0/; These two sttements chnge the limits of integrtion. =.0/bb; if (n == ) From this point on, the routine is identicl to midpnt. return (s=(b-)*func(0.5*(+b))); else for(it=,j=;j<n-;j++) it *= 3; tnm=it; del=(b-)/(3.0*tnm); ddel=del+del; x=+0.5*del; sum=0.0; for (j=;j<=it;j++) x += ddel; x += del; return (s=(s+(b-)*sum/tnm)/3.0);

5 4.4 Improper Integrls 45 If you need to integrte from negtive lower limit to positive infinity, you do this by breking the integrl into two pieces t some positive vlue, for exmple, nswer=qromo(funk,-5.0,2.0,midpnt)+qromo(funk,2.0,.0e30,midinf); Where should you choose the brekpoint? At sufficiently lrge positive vlue so tht the function funk is t lest beginning to pproch its symptotic decrese to zero vlue t infinity. The polynomil extrpoltion implicit in the second cll to qromo dels with polynomil in /x, not in x. To del with n integrl tht hs n integrble power-lw singulrity t its lower limit, one lso mkes chnge of vrible. If the integrnd diverges s (x ) γ, 0 γ<, ner x =, use the identity (b ) γ t γ γ f(t γ + )dt (b >) (4.4.3) γ 0 If the singulrity is t the upper limit, use the identity (b ) γ t γ γ f(b t γ )dt (b >) (4.4.4) γ 0 If there is singulrity t both limits, divide the integrl t n interior brekpoint s in the exmple bove. Equtions (4.4.3) nd (4.4.4) re prticulrly simple in the cse of inverse squre-root singulrities, cse tht occurs frequently in prctice: for singulrity t, nd b 0 b 0 2tf( + t 2 )dt (b >) (4.4.5) 2tf(b t 2 )dt (b >) (4.4.6) for singulrity t b. Once gin, we cn implement these chnges of vrible trnsprently to the user by defining substitute routines for midpnt which mke the chnge of vrible utomticlly: #include <mth.h> #define FUNC(x) (2.0*(x)*(*funk)(+(x)*(x))) flot midsql(flot (*funk)(flot), flot, flot bb, int n) This routine is n exct replcement formidpnt, except tht it llows for n inverse squre-root singulrity in the integrnd t the lower limit. flot x,tnm,sum,del,ddel,,b; sttic flot s; b=sqrt(bb-); =0.0; if (n == ) The rest of the routine is exctly like midpnt nd is omitted.

6 46 Chpter 4. Integrtion of Functions Similrly, #include <mth.h> #define FUNC(x) (2.0*(x)*(*funk)(bb-(x)*(x))) flot midsqu(flot (*funk)(flot), flot, flot bb, int n) This routine is n exct replcement formidpnt, except tht it llows for n inverse squre-root singulrity in the integrnd t the upper limit bb. flot x,tnm,sum,del,ddel,,b; sttic flot s; b=sqrt(bb-); =0.0; if (n == ) The rest of the routine is exctly like midpnt nd is omitted. One lst exmple should suffice to show how these formuls re derived in generl. Suppose the upper limit of integrtion is infinite, nd the integrnd flls off exponentilly. Then we wnt chnge of vrible tht mps e x dx into (±)dt (with the sign chosen to keep the upper limit of the new vrible lrger thn the lower limit). Doing the integrtion gives by inspection so tht x= x= t = e x or x = log t (4.4.7) t=e t=0 The user-trnsprent implementtion would be #include <mth.h> #define FUNC(x) ((*funk)(-log(x))/(x)) f( log t) dt t (4.4.8) flot midexp(flot (*funk)(flot), flot, flot bb, int n) This routine is n exct replcement for midpnt, except tht bb is ssumed to be infinite (vlue pssed not ctully used). It is ssumed tht the function funk decreses exponentilly rpidly t infinity. flot x,tnm,sum,del,ddel,,b; sttic flot s; b=exp(-); =0.0; if (n == ) The rest of the routine is exctly like midpnt nd is omitted. CITED REFERENCES AND FURTHER READING: Acton, F.S. 970, Numericl Methods Tht Work; 990, corrected edition (Wshington: Mthemticl Assocition of Americ), Chpter 4.

7 4.5 Gussin Qudrtures nd Orthogonl Polynomils 47 Dhlquist, G., nd Bjorck, A. 974, Numericl Methods (Englewood Cliffs, NJ: Prentice-Hll), 7.4.3, p Stoer, J., nd Bulirsch, R. 980, Introduction to Numericl Anlysis (New York: Springer-Verlg), 3.7, p Gussin Qudrtures nd Orthogonl Polynomils In the formuls of 4., the integrl of function ws pproximted by the sum of its functionl vlues t set of eqully spced points, multiplied by certin ptly chosen weighting coefficients. We sw tht s we llowed ourselves more freedom in choosing the coefficients, we could chieve integrtion formuls of higher nd higher order. The ide of Gussin qudrtures is to give ourselves the freedom to choose not only the weighting coefficients, but lso the loction of the bscisss t which the function is to be evluted: They will no longer be eqully spced. Thus, we will hve twice the number of degrees of freedom t our disposl; it will turn out tht we cn chieve Gussin qudrture formuls whose order is, essentilly, twice tht of the Newton-Cotes formul with the sme number of function evlutions. Does this sound too good to be true? Well, in sense it is. The ctch is fmilir one, which cnnot be overemphsized: High order is not the sme s high ccurcy. High order trnsltes to high ccurcy only when the integrnd is very smooth, in the sense of being well-pproximted by polynomil. There is, however, one dditionl feture of Gussin qudrture formuls tht dds to their usefulness: We cn rrnge the choice of weights nd bscisss to mke the integrl exct for clss of integrnds polynomils times some known function W (x) rther thn for the usul clss of integrnds polynomils. The function W (x) cn then be chosen to remove integrble singulrities from the desired integrl. Given W (x), in other words, nd given n integer N, we cn find set of weights w j nd bscisss x j such tht the pproximtion W (x)f(x)dx N w j f(x j ) (4.5.) j= is exct if f(x) is polynomil. For exmple, to do the integrl exp( cos 2 x) dx (4.5.2) x 2 (not very nturl looking integrl, it must be dmitted), we might well be interested in Gussin qudrture formul bsed on the choice W (x) = x 2 (4.5.3) in the intervl (, ). (This prticulr choice is clled Guss-Chebyshev integrtion, for resons tht will become cler shortly.)