6 Numerical Integration

Transcription

1 D. Levy 6 Numericl Integrtion 6.1 Bsic Concepts In this chpter we re going to explore vrious wys for pproximting the integrl of function over given domin. There re vrious resons s of why such pproximtions cn be useful. First, not every function cn be nlyticlly integrted. Second, even if closed integrtion formul exists, it might still not be the most efficient wy of clculting the integrl. In ddition, it cn hppen tht we need to integrte n unknown function, in which only some smples of the function re known. In order to gin some insight on numericl integrtion, it is nturl to review Riemnn integrtion, frmework tht cn be viewed s n pproch for pproximting integrls. We ssume tht f(x) is bounded function defined on [, b] nd tht {x 0,..., x n } is prtition (P ) of [, b]. For ech i we let nd M i (f) = m i (f) = sup f(x), x [x i 1,x i ] inf f(x), x [x i 1,x i ] Letting x i = x i x i 1, the upper (Drboux) sum of f(x) with respect to the prtition P is defined s U(f, P ) = M i x i, (6.1) while the lower (Drboux) sum of f(x) with respect to the prtition P is defined s L(f, P ) = m i x i. (6.) The upper integrl of f(x) on [, b] is defined s U(f) = inf(u(f, P )), nd the lower integrl of f(x) is defined s L(f) = sup(l(f, P )), where both the infimum nd the supremum re tken over ll possible prtitions, P, of the intervl [, b]. If the upper nd lower integrl of f(x) re equl to ech other, their common vlue is denoted by f(x)dx nd is referred to s the Riemnn integrl of f(x). 1

2 6.1 Bsic Concepts D. Levy For the purpose of the present discussion we cn think of the upper nd lower Drboux sums (6.1), (6.), s two pproximtions of the integrl (ssuming tht the function is indeed integrble). Of course, these sums re not defined in the most convenient wy for n pproximtion lgorithm. This is becuse we need to find the extrem of the function in every subintervl. Finding the extrem of the function, my be complicted tsk on its own, which we would like to void. A simpler pproch for pproximting the vlue of f(x)dx would be to compute the product of the vlue of the function t one of the end-points of the intervl by the length of the intervl. In cse we choose the end-point where the function is evluted to be x =, we obtin f(x)dx f()(b ). (6.3) This pproximtion (6.3) is clled the rectngulr method (see Figure 6.1). Numericl integrtion formuls re lso referred to s integrtion rules or qudrtures, nd hence we cn refer to (6.3) s the rectngulr rule or the rectngulr qudrture. The points x 0,... x n tht re used in the qudrture formul re clled qudrture points. f(b) f(x) f() x b Figure 6.1: A rectngulr qudrture A vrition on the rectngulr rule is the midpoint rule. Similrly to the rectngulr rule, we pproximte the vlue of the integrl f(x)dx by multiplying the length of the intervl by the vlue of the function t one point. Only this time, we replce the vlue of the function t n endpoint, by the vlue of the function t the center point

3 D. Levy 6.1 Bsic Concepts 1( + b), i.e., ( + b f(x)dx (b )f ). (6.4) (see lso Fig 6.). As we shll see below, the midpoint qudrture (6.4) is more ccurte qudrture thn the rectngulr rule (6.3). f(b) f((+b)/) f(x) f() (+b)/ b x Figure 6.: A midpoint qudrture In order to compute the qudrture error for the midpoint rule (6.4), we consider the primitive function F (x), F (x) = nd expnd +h x f(s)ds, f(s)ds = F ( + h) = F () + hf () + h F () + h3 6 F () + O(h 4 ) (6.5) = hf() + h f () + h3 6 f () + O(h 4 ) If we let b = + h, we hve (expnding f( + h/)) for the qudrture error, E, +h ( E = f(s)ds hf + h ) = hf() + h f () + h3 6 f () + O(h 4 ) h [f() + h ] f () + h 8 f () + O(h 3 ), 3

4 6.1 Bsic Concepts D. Levy which mens tht the error term is of order O(h 3 ). Hving n error of order h 3 does not men tht this is third-order method. In our cse, the prmeter h equls to b. It is not prmeter tht should be vied s smll vlue tht goes to zero. It is fixed. The error of the midpoint method is of the order of O((b ) 3 ). Unfortuntely, these clcultions cnnot directly provide us with n ccurte estimte of the error. This is the cse since when truncting two Tylor pproximtions, we re left with n error terms tht re evluted t two (generlly different) intermedite points. Hence we cnnot directly combine the error term 1 6 h3 f (ξ 1 ) with 1 8 h3 f (ξ ). This cn still be done, but we hve to use better pproch. The min difficulty in evluting the difference between the exct vlue, f(x)dx, nd its midpoint rule pproximtion, (b )f ( ) +b, is due to hving n integrl in one term nd no integrl in the second term. The pproch will be to replce the midpoint pproximtion with n integrl expression. Indeed, if we denote the midpoint by c, i.e., c = + b, then the tngent line to f(x) t x = c is given by Clerly, P 1 (x) = f(c) + f (c)(x c). P 1 (x)dx = (b )f(c), nd hence ( ) + b f(x)dx (b )f = (f(x) P 1 (x))dx. To estimte the difference between f(x) nd P 1 (x) we cn expnd f(x) round x = c. Assuming tht x [, b], we hve f(x) = f(c + (x c)) = f(c) + f (c)(x c) + 1 f (ξ)(x c), ξ (, b). Hence (f(x) P 1 (x))dx = 1 f (ξ x )(x c) dx. In view of the midvlue theorem for integrls, the lst integrl cn be replces by 1 f (ξ) (x c) dx = 1 4 (b )3 f (ξ), < ξ < b. (6.6) Remrk. Throughout this section we ssumed tht ll functions we re interested in integrting re ctully integrble in the domin of interest. We lso ssumed tht they re bounded nd tht they re defined t every point, so tht whenever we need to evlute function t point, we cn do it. We will go on nd use these ssumptions throughout the chpter. 4

5 D. Levy 6. Integrtion vi Interpoltion 6. Integrtion vi Interpoltion One direct wy of obtining qudrtures from given smples of function is by integrting n interpolnt. As lwys, our gol is to evlute I = f(x)dx. We ssume tht the vlues of the function f(x) re given t n + 1 points: x 0,..., x n [, b]. Note tht we do not require the first point x 0 to be equl to, nd the sme holds for the right side of the intervl. Given the vlues f(x 0 ),... f(x n ), we cn write the interpolting polynomil of degree n, which in the Lrgenge form is with P n (x) = l i (x) = f(x i )l i (x), n j=0 j i x x j x i x j, 0 i n. The integrl of f(x) cn then be pproximted by the integrl of P n (x), i.e., f(x)dx P n (x)dx = f(x i ) The qudrture coefficients A i in (6.7) re given by l i (x)dx = A i f(x i ). (6.7) A i = l i (x)dx. (6.8) Note tht if we wnt to integrte severl different functions, nd use their vlues t the sme points (x 0,..., x n ), the qudrture coefficients (6.8) should be computed only once, since they do not depend on the function tht is being integrted. If we chnge the interpoltion/integrtion points, then we must recompute the qudrture coefficients. For eqully spced points, x 0,..., x n, numericl integrtion formul of the form f(x)dx A i f(x i ), (6.9) is clled Newton-Cotes formul. Exmple 6.1 We let n = 1 nd consider two interpoltion points which we set s x 0 =, x 1 = b. In this cse l 0 (x) = b x b, l 1(x) = x b. 5

6 6. Integrtion vi Interpoltion D. Levy Hence A 0 = Similrly, A 1 = l 0 (x) = l 1 (x) = b x b dx = b. x b dx = b = A 0. The resulting qudrture is the so-clled trpezoidl rule, (see Figure 6.3). dx b [f() + f(b)], (6.10) f(b) f(x) f() x b Figure 6.3: A trpezoidl qudrture We cn now use the interpoltion error to compute the error in the qudrture (6.10). The interpoltion error is f(x) P 1 (x) = 1 f (ξ x )(x )(x b), ξ x (, b). We recll tht ccording to the midvlue theorem for integrls, if u(x) nd v(x) re continuous on [, b] nd if v 0, then there exists ξ (, b) such tht u(x)v(x)dx = u(ξ) v(x)dx. 6

7 D. Levy 6. Integrtion vi Interpoltion Hence, the interpoltion error is given by E = with ξ (, b). 1 f (ξ x )(x )(x b) = f (ξ) (x )(x b)dx = f (ξ) 1 (b )3, (6.11) Remrks. 1. We note tht the qudrtures (6.7),(6.8), re exct for polynomils of degree n. For if p(x) is polynomil of degree n, it cn be written s p(x) = p(x i )l i (x). (Two polynomils of degree n tht gree with ech other t n + 1 points must be identicl). Hence p(x)dx = p(x i ) l i (x)dx = A i p(x i ).. As of the opposite direction. Assume tht the qudrture f(x)dx A i f(x i ), is exct for ll polynomils of degree n. We know tht nd hence deg(l j (x)) = n, l j (x)dx = A i l j (x i ) = A i δ ij = A j. This mens tht the qudrture coefficients must be given by A j = l j (x)dx. 7

8 6.3 Composite Integrtion Rules D. Levy 6.3 Composite Integrtion Rules In composite qudrture, we divide the intervl into subintervls nd pply n integrtion rule to ech subintervl. We demonstrte this ide with couple of exmples. Exmple 6. Consider the points = x 0 < x 1 < < x n = b. The composite trpezoidl rule is obtined by pplying the trpezoidl rule in ech subintervl [x i 1, x i ], i = 1,..., n, i.e., f(x)dx = (see Figure 6.4). xi x i 1 f(x)dx 1 (x i x i 1 )[f(x i 1 ) + f(x i )], (6.1) f(x) x 0 x 1 x x n 1 x n x Figure 6.4: A composite trpezoidl rule A prticulr cse is when these points re uniformly spced, i.e., when ll intervls hve n equl length. For exmple, if where x i = + ih, h = b n, 8

9 D. Levy 6.3 Composite Integrtion Rules then [ ] f(x)dx h n 1 f() + f( + ih) + f(b) = h f( + ih). (6.13) The nottion of sum with two primes,, mens tht we sum over ll the terms with the exception of the first nd lst terms tht re being divided by. We cn lso compute the error term s function of the distnce between neighboring points, h. We know from (6.11) tht in every subintervl the qudrture error is h3 1 f (ξ x ). Hence, the overll error is obtined by summing over n such terms: [ ] h3 1 f (ξ i ) = h3 n 1 f (ξ i ). 1 n Here, we use the nottion ξ i to denote n intermedite point tht belongs to the i th intervl. Let Clerly M = 1 n f (ξ i ). min f (x) M mx f (x) x [,b] x [,b] If we ssume tht f (x) is continuous in [, b] (which we nyhow do in order for the interpoltion error formul to be vlid) then there exists point ξ [, b] such tht f (ξ) = M. Hence (reclling tht (b )/n = h, we hve E = (b )h f (ξ), ξ [, b]. (6.14) 1 This mens tht the composite trpezoidl rule is second-order ccurte. Exmple 6.3 In the intervl [, b] we ssume n subintervls nd let h = b n. 9

10 6.4 Additionl Integrtion Techniques D. Levy The qudrture points re ( x j = + j 1 ) h, j = 1,,..., n. The composite midpoint rule is given by pplying the midpoint rule (6.4) in every subintervl, i.e., f(x)dx h f(x j ). (6.15) j=1 Eqution (6.15) is known s the composite midpoint rule. In order to obtin the qudrture error in the pproximtion (6.15) we recll tht in ech subintervl the error is given ccording to (6.6), i.e., Hence E j = h3 4 f (ξ j ), ξ j E = j=1 E j = h3 4 j=1 ( x j h, x j + h [ f (ξ j ) = h3 4 n 1 n ). ] f (ξ j ) j=1 = h (b ) f (ξ), (6.16) 4 where ξ (, b). This mens tht the composite midpoint rule is lso second-order ccurte (just like the composite trpezoidl rule). 6.4 Additionl Integrtion Techniques The method of undetermined coefficients The methods of undetermined coefficients for deriving qudrtures is the following: 1. Select the qudrture points.. Write qudrture s liner combintion of the vlues of the function t the chosen qudrture points. 3. Determine the coefficients of the liner combintion by requiring tht the qudrture is exct for s mny polynomils s possible from the the ordered set {1, x, x,...}. We demonstrte this technique with the following exmple. Exmple 6.4 Problem: Find qudrture of the form ( ) 1 f(x)dx A 0 f(0) + A 1 f + A f(1), 0 tht is exct for ll polynomils of degree. 10

11 D. Levy 6.4 Additionl Integrtion Techniques Solution: Since the qudrture hs to be exct for ll polynomils of degree, it hs to be exct for the polynomils 1, x, nd x. Hence we obtin the system of liner equtions 1 = 1 = 1 3 = dx = A 0 + A 1 + A, xdx = 1 A 1 + A, x dx = 1 4 A 1 + A. Therefore, A 0 = A = 1 nd A 6 1 =, nd the desired qudrture is 3 0 f(x)dx f(0) + 4f ( 1 ) + f(1). (6.17) 6 Since the resulting formul (6.17) is liner, its being exct for 1, x, nd x, implies tht it is exct for ny polynomil of degree. In fct, we will show in Section tht this pproximtion is ctully exct for polynomils of degree Chnge of n intervl Suppose tht we hve qudrture formul on the intervl [c, d] of the form d f(t)dt A i f(t i ). (6.18) c We would like to to use (6.18) to find qudrture on the intervl [, b], tht pproximtes for f(x)dx. The mpping between the intervls [c, d] [, b] cn be written s liner trnsformtion of the form λ(t) = b d bc t + d c d c. Hence This mens tht f(x)dx = b d c f(x)dx b d c d c f(λ(t))dt b d c A i f ( b d c t i + A i f(λ(t i )). ) d bc. (6.19) d c We note tht if the qudrture (6.18) ws exct for polynomils of degree m, so is (6.19). 11

12 6.5 Simpson s Integrtion D. Levy Exmple 6.5 We wnt to write the result of the previous exmple 0 f(x)dx f(0) + 4f ( 1 ) + f(1), 6 s qudrture on the intervl [, b]. According to (6.19) f(x)dx b 6 [ f() + 4f ( ) + b ] + f(b). (6.0) The pproximtion (6.0) is known s the Simpson qudrture. 6.5 Simpson s Integrtion In the lst exmple we obtined Simpson s qudrture (6.0). An lterntive derivtion is the following: strt with polynomil Q (x) tht interpoltes f(x) t the points, ( + b)/, nd b. Then pproximte [ ] (x c)(x b) (x )(x b) (x )(x c) f(x)dx f() + f(c) + ( c)( b) (c )(c b) (b )(b c) f(b) dx =... = b [ ( ) ] + b f() + 4f + f(b), 6 which is Simpson s rule (6.0). Figure 6.5 demonstrtes this process of deriving Simpson s qudrture for the specific choice of pproximting 3 sin xdx The qudrture error It turns out tht Simpson s qudrture is exct for polynomils of degree 3 nd not only for polynomils of degree, s expected by the wy it ws constructed. We will obtin this result by studying the error term. In order to derive the error term for Simpson s method, we discuss n error nlysis technique tht is vlid for qudrtures tht re obtined through integrtion. In ll such cses, the qudrture error is the difference between the integrl of the function nd the integrl of its interpolnt, i.e., E = (f(t) p n (t))dt = f (n+1) (ξ x ) ω(t)dt, (6.1) (n + 1)! where ω(t) = n (t t i )dt. 1

13 D. Levy 6.5 Simpson s Integrtion P (x) sinx x Figure 6.5: An exmple of Simpson s qudrture. The pproximtion of 3 sin xdx is 1 obtined by integrting the qudrtic interpolnt Q (x) over [1, 3] It ω(t) is lwys non-negtive or non-positive between nd b, then ccording to the midvlue theorem for integrls, the error in (6.1) becomes E = f (n+1) (ξ) (n + 1)! ω(t)dt, ξ (, b). Exmples for such cses re the trpezoidl rule for which E = f (ξ) 1 (b )3, nd the rectngle rule for which E = f (ξ) 1 (t )dt = f (ξ) (b ). Another cse which is rther esy to nlyze is the cse in which ω(t)dt = 0. Exmples for the cse in (6.) include the midpoint rule for which ( ω(t)dt = t + b ) dt = 0, (6.) 13

14 6.5 Simpson s Integrtion D. Levy nd Simpson s rule for which ( ω(t)dt = (t ) t + b ) (t b)dt = 0. In this cse, we cn dd nother interpoltion point without chnging the integrl of the interpolnt. This is the cse since we replce f(x) by p n (x) nd integrte f(t)dt p n (x)dx. Adding n rbitrry interpoltion point, x n+1, to p n (x) turns it into n interpolting polynomil of higher order, p n+1 (x), tht is given by p n+1 (x) = p n (x) + f[x 0,..., x n+1 ]ω(x). (6.3) Since ω(x)dx = 0, when integrting (6.3) in order to obtin qudrture, we observe tht f(t)dt p n+1 (x)dx = p n (x)dx, so the originl qudrture does not chnge by dding n rbitrry interpoltion point. We now hve ll the required tools in order to derive qudrture for Simpson s method. Since in this cse +b ω(t)dt = 0, we dd to,, b n rbitrry interpoltion point which we choose s +b gin. The function ω(t) becomes ( ω(t) = (t ) t + b ) (t b). Hence, for t [, b], our new ω(t) stisfies ω(t) 0. By the midvlue theorem for integrls the error in Simpson s method cn be written s E = f (4) (ξ) b ( (t ) t + b ) (t b)dt = 1 ( ) 5 b f (4) (ξ), (6.4) 4 90 for ξ (, b). Since the fourth derivtive of ny polynomil of degree 3 is identiclly zero, the qudrture error formul (6.4) implies tht Simpson s qudrture is exct for polynomils of degree Composite Simpson rule To derive composite version of Simpson s qudrture, we divide the intervl [, b] into n even number of subintervls, n, nd let x i = + ih, 0 i n, 14

15 D. Levy 6.6 Weighted Qudrtures where h = b n. Hence, if we replce the integrl in every subintervls by Simpson s rule (6.0), we obtin f(x)dx = x x 0 f(x)dx xn n/ f(x)dx = x n h n/ [f(x i ) + 4f(x i 1 ) + f(x i )]. 3 xi x i f(x)dx The composite Simpson qudrture is thus given by f(x)dx h n/ n/ f(x 0 ) + f(x i ) + 4 f(x i 1 ) + f(x n ). (6.5) 3 Summing the error terms (tht re given by (6.4)) over ll sub-intervls, the qudrture error tkes the form Since E = h5 90 n/ min x [,b] f (4) (x) n we cn conclude tht f (4) (ξ i ) = h5 90 n n n/ n/ f (4) (ξ i ). f (4) (ξ i ) mx f (4) (x), x [,b] E = h5 n 90 f (4) (ξ) = h4 180 f (4) (ξ), ξ [, b], (6.6) i.e., the composite Simpson qudrture is fourth-order ccurte. 6.6 Weighted Qudrtures We recll tht weight function is continuous, non-negtive function with positive mss. We ssume tht such weight function w(x) is given nd would like to write qudrture of the form f(x)w(x)dx A i f(x i ). (6.7) 15

16 6.7 Gussin Qudrture D. Levy Such qudrtures re clled generl (weighted) qudrtures. Previously, for the cse w(x) 1, we wrote qudrture of the form where A i = f(x)dx A i f(x i ), l i (x)dx. Repeting the derivtion we crried out in Section 6., we construct n interpolnt Q n (x) of degree n tht psses through the points x 0,..., x n. Its Lgrnge form is Q n (x) = f(x i )l i (x), with the usul n l i (x) = j=0 j i x x j x i x j, 0 i n. Hence f(x)w(x)dx Q n (x)w(x)dx = where the coefficients A i re given by f(x i ) l i (x)w(x)dx = A i f(x i ), A i = l i (x)w(x)dx. (6.8) To summrize, the generl qudrture is f(x)w(x)dx A i f(x i ), (6.9) with qudrture coefficients, A i, tht re given by (6.8). 6.7 Gussin Qudrture Mximizing the qudrture s ccurcy So fr, ll the qudrtures we encountered were of the form f(x)dx A i f(x i ). (6.30) 16

17 D. Levy 6.7 Gussin Qudrture An pproximtion of the form (6.30) ws shown to be exct for polynomils of degree n for n pproprite choice of the qudrture coefficients A i. In ll cses, the qudrture points x 0,..., x n were given up front. In other words, given set of nodes x 0,..., x n, the coefficients {A i } n were determined such tht the pproximtion ws exct in Π n. We re now interested in investigting the possibility of writing more ccurte qudrtures without incresing the totl number of qudrture points. This will be possible if we llow for the freedom of choosing the qudrture points. The qudrture problem becomes now problem of choosing the qudrture points in ddition to determining the corresponding coefficients in wy tht the qudrture is exct for polynomils of mximl degree. Qudrtures tht re obtined tht wy re clled Gussin qudrtures. Exmple 6.6 The qudrture formul 1 f(x)dx f ( 1 3 ) + f ( 1 3 ), is exct for polynomils of degree 3(!) We will revisit this problem nd prove this result in Exmple 6.9 below. An equivlent problem cn be stted for the more generl weighted qudrture cse. Here, f(x)w(x)dx A i f(x i ), (6.31) where w(x) 0 is weight function. Eqution (6.31) is exct for f Π n if nd only if A i = w(x) j=0 j i x x j x i x j dx. (6.3) In both cses (6.30) nd (6.31), the number of qudrture nodes, x 0,..., x n, is n+1, nd so is the number of qudrture coefficients, A i. Hence, if we hve the flexibility of determining the loction of the points in ddition to determining the coefficients, we hve ltogether n + degrees of freedom, nd hence we cn expect to be ble to derive qudrtures tht re exct for polynomils in Π n+1. This is indeed the cse s we shll see below. We will show tht the generl solution of this integrtion problem is connected with the roots of orthogonl polynomils. We strt with the following theorem. Theorem 6.7 Let q(x) be nonzero polynomil of degree n + 1 tht is w-orthogonl to Π n, i.e., p(x) Π n, p(x)q(x)w(x)dx = 0. 17

18 6.7 Gussin Qudrture D. Levy If x 0,..., x n re the zeros of q(x) then (6.31), with A i given by (6.3), is exct f Π n+1. Proof. For f(x) Π n+1, write f(x) = q(x)p(x) + r(x). We note tht p(x), r(x) Π n. Since x 0,..., x n re the zeros of q(x) then Hence, f(x i ) = r(x i ). f(x)w(x)dx = = A i r(x i ) = [q(x)p(x) + r(x)]w(x)dx = A i f(x i ). r(x)w(x)dx (6.33) The second equlity in (6.33) holds since q(x) is w-orthogonl to Π n. The third equlity (6.33) holds since (6.31), with A i given by (6.3), is exct for polynomils in Π n. According to Theorem 6.7 we lredy know tht the qudrture points tht will provide the most ccurte qudrture rule re the n+1 roots of n orthogonl polynomil of degree n + 1 (where the orthogonlity is with respect to the weight function w(x)). We recll tht the roots of q(x) re rel, simple nd lie in (, b), something we know from our previous discussion on orthogonl polynomils (see Theorem??). In other words, we need n + 1 qudrture points in the intervl, nd n orthogonl polynomil of degree n + 1 does hve n + 1 distinct roots in the intervl. We now restte the result regrding the roots of orthogonl functions with n lterntive proof. Theorem 6.8 Let w(x) be weight function. Assume tht f(x) is continuous in [, b] tht is not the zero function, nd tht f(x) is w-orthogonl to Π n. Then f(x) chnges sign t lest n + 1 times on (, b). Proof. Since 1 Π n, f(x)w(x)dx = 0. Hence, f(x) chnges sign t lest once. Now suppose tht f(x) chnges size only r times, where r n. Choose {t i } i 0 such tht = t 0 < t 1 < < t r = b, nd f(x) is of one sign on (t 0, t 1 ), (t 1, t ),..., (t r 1, t r ). The polynomil r 1 p(x) = (x t i ), 18

19 D. Levy 6.7 Gussin Qudrture hs the sme sign property. Hence f(x)p(x)w(x)dx 0, which leds to contrdiction since p(x) Π n. Exmple 6.9 We re looking for qudrture of the form 1 f(x)dx A 0 f(x 0 ) + A 1 f(x 1 ). A strightforwrd computtion will mount to mking this qudrture exct for the polynomils of degree 3. The linerity of the qudrture mens tht it is sufficient to mke the qudrture exct for 1, x, x, nd x 3. Hence we write the system of equtions 1 f(x)dx = 1 From this we cn write A 0 + A 1 =, A 0 x 0 + A 1 x 1 = 0, A 0 x 0 + A 1 x 1 =, 3 A 0 x A 1 x 3 1 = 0. Solving for A 1, A, x 0, nd x 1 we get A 1 = A = 1, x 0 = x 1 = 1 3, x i dx = A 0 x i 0 + A 1 x i 1, i = 0, 1,, 3. so tht the desired qudrture is 1 f(x)dx f ( 1 ) ( ) 1 + f 3. (6.34) 3 Exmple 6.10 We repet the previous problem using orthogonl polynomils. Since n = 1, we expect to find qudrture tht is exct for polynomils of degree n + 1 = 3. The polynomil of degree n+1 = which is orthogonl to Π n = Π 1 with weight w(x) 1 is the Legendre polynomil of degree, i.e., P (x) = 1 (3x 1). 19

20 6.7 Gussin Qudrture D. Levy The integrtion points will then be the zeros of P (x), i.e., x 0 = 1 3, x 1 = 1 3. All tht remins is to determine the coefficients A 1, A. This is done in the usul wy, ssuming tht the qudrture 1 f(x)dx A 0 f(x 0 ) + A 1 f(x 1 ), is exct for polynomils of degree 1. The simplest will be to use 1 nd x, i.e., nd = 0 = 1 1 1dx = A 0 + A 1, xdx = A A Hence A 0 = A 1 = 1, nd the qudrture is the sme s (6.34) (s should be) Convergence nd error nlysis Lemm 6.11 In Gussin qudrture formul, the coefficients re positive nd their sum is w(x)dx. Proof. Fix n. Let q(x) Π n+1 be w-orthogonl to Π n. Also ssume tht q(x i ) = 0 for i = 0,..., n, nd tke {x i } n to be the qudrture points, i.e., f(x)w(x)dx A i f(x i ). (6.35) Fix 0 j n. Let p(x) Π n be defined s p(x) = q(x) x x j. Since x j is root of q(x), p(x) is indeed polynomil of degree n. The degree of p (x) n which mens tht the Gussin qudrture (6.35) is exct for it. Hence 0 < p (x)w(x)dx = A i p (x i ) = A i q (x i ) (x i x j ) = A jp (x j ), 0

21 D. Levy 6.7 Gussin Qudrture which mens tht j, A j > 0. In ddition, since the Gussin qudrture is exct for f(x) 1, we hve w(x)dx = A i. In order to estimte the error in the Gussin qudrture we would first like to present n lterntive wy of deriving the Gussin qudrture. Our strting point is the Lgrnge form of the Hermite polynomil tht interpoltes f(x) nd f (x) t x 0,..., x n. It is given by (??), i.e., with p(x) = f(x i ) i (x) + f (x i )b i (x), i (x) = (l i (x)) [1 + l i(x i )(x i x)], b i (x) = (x x i )l i (x), 0 i n, nd l i (x) = n j=0 j i x x j x i x j. We now ssume tht w(x) is weight function in [, b] nd pproximte where nd A i = B i = w(x)f(x)dx w(x)p n+1 (x)dx = A i f(x i ) + B i f (x i ), (6.36) w(x) i (x)dx, (6.37) w(x)b i (x)dx. (6.38) In some sense, it seems to be rther strnge to del with the Hermite interpolnt when we do not explicitly know the vlues of f (x) t the interpoltion points. However, we cn eliminte the derivtives from the qudrture (6.36) by setting B i = 0 in (6.38). Indeed (ssuming n 0): B i = w(x)(x x i )l i (x)dx = n (x i x j ) j=0 j i w(x) n (x x j )l i (x)dx. j=0 1

22 6.7 Gussin Qudrture D. Levy Hence, B i = 0, if the product n j=0 (x x j) is orthogonl to l i (x). Since l i (x) is polynomil in Π n, ll tht we need is to set the points x 0,..., x n s the roots of polynomil of degree n+1 tht is w-orthogonl to Π n. This is precisely wht we defined s Gussin qudrture. We re now redy to formlly estblish the fct tht the Gussin qudrture is exct for polynomils of degree n + 1. Theorem 6.1 Let f C n+ [, b] nd let w(x) be weight function. Consider the Gussin qudrture f(x)w(x)dx A i f(x i ). Then there exists ζ (, b) such tht f(x)w(x)dx A i f(x i ) = f (n+) (ζ) (n + )! n (x x j ) w(x)dx. Proof. We use the chrcteriztion of the Gussin qudrture s the integrl of Hermite interpolnt. We recll tht the error formul for the Hermite interpoltion is given by (??), f(x) p n+1 (x) = f (n+) (ξ) (n + )! j=0 n (x x j ), ξ (, b). j=0 Hence ccording to (6.36) we hve f(x)w(x)dx A i f(x i ) = = f(x)w(x)dx w(x) f (n+) (ξ) (n + )! p n+1 w(x)dx n (x x j ) dx. The integrl men vlue theorem then implies tht there exists ζ (, b) such tht f(x)w(x)dx A i f(x i ) = f (n+) (ζ) (n + )! j=0 n (x x j ) (x)w(x)dx. We conclude this section with convergence theorem tht sttes tht for continuous functions, the Gussin qudrture converges to the exct vlue of the integrl s the number of qudrture points tends to infinity. This theorem is not of gret prcticl vlue becuse it does not provide n estimte on the rte of convergence. A proof of the theorem tht is bsed on the Weierstrss pproximtion theorem cn be found in, e.g., in [?]. j=0

23 D. Levy 6.8 Romberg Integrtion Theorem 6.13 We let w(x) be weight function nd ssuming tht f(x) is continuous function on [, b]. For ech n N we let {x ni } n be the n + 1 roots of the polynomil of degree n + 1 tht is w-orthogonl to Π n, nd consider the corresponding Gussin qudrture: f(x)w(x)dx A ni f(x ni ). (6.39) Then the right-hnd-side of (6.39) converges to the left-hnd-side s n. 6.8 Romberg Integrtion We hve introduced Richrdson s extrpoltion in Section?? in the context of numericl differentition. We cn use similr principle with numericl integrtion. We will demonstrte this principle with prticulr exmple. Let I denote the exct integrl tht we would like to pproximte, i.e., I = f(x)dx. Let s ssume tht this integrl is pproximted with composite trpezoidl rule on uniform grid with mesh spcing h (6.13), T (h) = h f( + ih). We know tht the composite trpezoidl rule is second-order ccurte (see (6.14)). A more detiled study of the qudrture error revels tht the difference between I nd T (h) cn be written s I = T (h) + c 1 h + c h c k h k + O(h k+ ). The exct vlues of the coefficients, c k, re of no interest to us s long s they do not depend on h (which is indeed the cse). We cn now write similr qudrture tht is bsed on hlf the number of points, i.e., T (h). Hence I = T (h) + c 1 (h) + c (h) This enbles us to eliminte the h error term: I = 4T (h) T (h) 3 + ĉ h

24 6.8 Romberg Integrtion D. Levy Therefore 4T (h) T (h) 3 = 1 [ ( 1 4h 3 f 0 + f f n ) f n ( 1 h f 0 + f f n + 1 )] f n = h 3 (f 0 + 4f 1 + f f n + 4f n 1 + f n ) = S(n). Here, S(n) denotes the composite Simpson s rule with n subintervls. The procedure of incresing the ccurcy of the qudrture by eliminting the leding error term is known s Romberg integrtion. In some plces, Romberg integrtion is used to describe the specific cse of turning the composite trpezoidl rule into Simpson s rule (nd so on). The qudrture tht is obtined from Simpson s rule by eliminting the leding error term is known s the super Simpson rule. 4