3.4 Numerical integration


 Lee Hunter
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1 3.4. Numericl integrtion Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [, b], i.e. I( f )= w(x) f (x) dx. The weight function my be the identity function w(x) =1, so tht the integrl represents the re under the function f in the intervl. In other pplictions the weight function could lso be the probbility density function of continuous rndom vrible x with support [, b], sothti( f ), represents the expected vlue of f ( x). In the following we discuss soclled numericl qudrture methods. In order to pproximte the bove integrl, qudrture methods choose discrete set of qudrture nodes x i nd pproprite weights w i in wy tht n I( f ) w i f (x i ). The integrl is therefore pproximted by the sum of function vlues t the nodes x 0 < < x n b nd the respective qudrture weights w i. Of course, the pproximtion improves with rising number n. Different qudrture methods only differ with respect to the chosen nodes x i nd weights w i. In the following we will concentrte on NewtonCotes method nd Gussin qudrture method. The former pproximtes the integrnd f between nodes using loworder polynomils nd sum the integrls of the polynomils to estimte the integrl of f. The ltter methods choose the nodes nd weights in order to mtch specific moments(such s expected vlue, vrince etc.) of the pproximted function Summed NewtonCotes methods In this section we set w(x) =1. A weighted integrl cn then simply be computed by setting g(x) =w(x) f (x) nd pproximting g(x) dx. Summed NewtonCotes formuls prtition the intervl [, b] into n subintervls of equl length h by computing the qudrture nodes x i := + ih, i = 0, 1,..., n, mit h := b n. A summed NewtonCotes formul of degree k then interpoltes the function f on every subintervl [x i, x i+1 ] with polynomil of degree k. Wefinlly integrte the interpolting polynomil on every subintervl nd sum up ll the resulting subres. Figure 3.10 shows this pproch for k = 0ndk = 1.
2 64 Chpter 3 Numericl Solution Methods f(x) f(x) f(x) f(x) = x 1 x x 3 x 4 x 5 b = x 6 x = x 1 x x 3 x 4 x 5 b = x 6 x Figure 3.10: Summed NewtonCotes formuls for k = 0 nd k = 1 Summed rectngle rule If k = 0, the interpolting functions re polynomils of degree 0, i.e. constnt functions. We cn write ny re below this constnt functions on the intervl [x i, x i+1 ] s I (0) [x i,x i+1 ] ( f )=(x i+1 x i ) f (x i )=hf(x i ), which is the surfce of rectngle. The degree 0 formul is therefore clled the summed rectngle rule, the explicit form of which is given by I (0) n 1 ( f )= hf(x i ). The weights of the summed rectngle rule consequently re w i = h for i = 0,...,n 1 nd w n = 0. Summed trpezoid rule If k = 1, the function f in the ith subintervl [x i, x i+1 ] is pproximted by the line segment pssing through the two points (x i, f (x i )) nd (x i+1, f (x i+1 )). The re under this line segment is the surfce of trpezoid, i.e. I (1) [x i,x i+1 ] ( f )=hf(x i)+ 1 h[ f (x i+1) f (x i )] = h [ f (x i)+ f (x i+1 )]. It is immeditely cler tht the summed trpezoid rule improves the pproximtion of I( f ) compred to the summed rectngle rule discussed bove. Summing up the res of the trpezoids cross subintervls yields the pproximtion { } I (1) = h n 1 f ()+ f (x i )+ f (b) i=1 of the whole integrl I( f ). The weights of the rule therefore re w 0 = w n = h nd w i = h for i = 1,...,n 1. (3.5)
3 3.4. Numericl integrtion 65 Summed rectngle nd trpezoid rule re simple nd robust. Hence, computtion does not need much effort. In ddition, the ccurcy of the pproximtion of I( f ) increses with rising n. It is lso cler tht the trpezoid rule will exctly compute the integrl of ny firstorder polynomil, i.e. line. Progrm 3.11 shows how to pply the summed trpezoid rule to the function cos(x). We Progrm 3.11: Summed trpezoid rule with cos(x) progrm NewtonCotes! vrible declrtion implicit none integer, prmeter :: n = 10 rel*8, prmeter :: = 0d0, b = d0 rel*8 :: h, x(0:n), w(0:n), f(0:n) integer :: i! clculte qudrture nodes h = (b)/dble(n) x = (/( + dble(i)*h,,n)/)! get weights w(0) = h/d0 w(n) = h/d0 w(1:n1) = h! clculte function vlues t nodes f = cos(x)! Output numericl nd nlyticl solution write(*, (,f10.6) ) Numericl:,sum(w*f, 1) write(*, (,f10.6) ) Anlyticl:,sin(d0)sin(0d0) end progrm thereby first declre ll the vribles needed. Specil ttention should be devoted to the declrtion of x, w nd f. These rrys will store the qudrture nodes x i, the weights w i nd the function vlues t the nodes f (x i ), respectively. We then first compute h nd the nodes x i = + ih nd the weights w i s in (3.5). With the respective function vlues, the pproximtion to the integrl 0 cos(x) dx = sin() sin(0) is the given by sum(w*f, 1). Note tht the pproximtion of the integrl is quite bd. If we increse the number of qudrture nodes n increses the ccurcy, however, we need bout 600 nodes in order to perfectly mtch numericl nd nlyticl result on 6 digits.
4 66 Chpter 3 Numericl Solution Methods Summed Simpson rule Finlly if k =, the integrnd f in the ith subintervl [x i, x i+1 ] is pproximted by secondorder polynomil function c 0 + c 1 x + c x.nowthreegrph points re required to specify the polynomil prmeters c i in the subintervls. Given these prmeters, one cn compute the integrl of the qudrtic function in the subintervl [x i, x i+1 ] s [ ( ) ] I () [x i,x i+1 ] ( f )=h xi + x f (x i )+4f i+1 + f (x i+1 ). 3 Summing up the different res on the subintervls yields { I () ( f )= h n 1 n 1 ( ) } xi + x f ()+ 6 f (x i )+4 f i+1 + f (b). i=1 The summed Simpson rule is lmost s simple to implement s the trpezoid rule. If the integrnd is smooth, Simpson s rule yields pproximtion error tht with rising n flls twice s fst s tht of the trpezoid rule. For this reson Simpson s rule is usully preferred to the trpezoid nd rectngle rule. Note, however, tht the trpezoid rule will often be more ccurte thn Simpson s rule if the integrnd exhibits discontinuities in its first derivtive, which cn occur in economic pplictions exhibiting corner solutions. Of course, summed NewtonCotes rules lso exist for higher order piecewise polynomil pproximtions, but they re more difficult to work with nd thus rrely used Gussin Qudrture In opposite to the NewtonCotes methods, Gussin qudrture tkes explicitly into ccount weight functions w(x). Here obviously, qudrture nodes x i nd weights w i re computed differently. GussLegendre qudrture Suppose gin, for the moment, tht w(x) =1. The GussLegendre qudrture nodes x i [, b] nd weights w i re computed in wy tht they stisfy the n + momentmtching conditions x k w(x) dx = x k n dx = w i xi k for k = 0, 1,..., n 1. (3.6) With the bove conditions holding, we cn write every integrl over polynomil with degree m n 1s p(x) dx = c 0 p(x) =c 0 + c 1 x c m x m 1 dx +c 1 xdx } {{} = n w ix i }{{} = n w i c m x m dx } {{} = n w ixi m
5 3.4. Numericl integrtion 67 n = w i [c 0 + c 1 x i c m x m n i ] = w i p(x i ). Consequently, qudrture nodes nd weights tht stisfy the momentmtching conditions in (3.6) re ble to integrte ny polynomil p of degree m n 1exctly. Clculting the nodes nd weights of the GussLegendre qudrture formul is not so esy. One method is to set up the nonliner eqution system defined in (3.6). For n = 3 e.g. we hve w 0 x 0 + w 1 x 1 + w x = 1 ) (b = 1 dx, w 0 x 0 + w 1 x 1 + w x = 1 ( b ) = xdx,. w 0 x0 5 + w 1x1 5 + w x 5 = 1 ( b 6 6) = x 5 dx. 6 This nonliner eqution system cn be solved by rootfinding lgorithm like fzero. However, there is more efficient, but lso less intuitive wy to clculte qudrture nodes nd weights of the GussLegendre qudrture implemented in the subroutine legendre in the module gussin_int. Progrm3.1 demonstrtes how to use it. The Progrm 3.1: GussLegendre qudrture with cos(x) progrm GussLegendre! module use use gussin_int! vrible declrtion implicit none integer, prmeter :: n = 10 rel*8, prmeter :: = 0d0, b = d0 rel*8 :: x(0:n), w(0:n), f(0:n)! clculte nodes nd weights cll legendre(0d0, d0, x, w)! clculte function vlues t nodes f = cos(x)! Output numericl nd nlyticl solution write(*, (,f10.6) ) Numericl:,sum(w*f, 1) write(*, (,f10.6) ) Anlyticl:,sin(d0)sin(0d0) end progrm subroutine tkes two rel*8 rguments defining the left nd right intervl borders nd
6 68 Chpter 3 Numericl Solution Methods b. In ddition, we hve to pss two rrys of equl length to the routine. The first of these will be filled with the nodes x i, wheres the second will be given the weights w i. After hving clculted the function vlues f (x i ), we cn gin clculte the numericl pproximtion of the integrl like in Progrm Here, with 10 qudrture nodes, we lredy mtch the nlyticl integrl vlue by 6 digits. Note tht we needed bout 600 nodes with the trpezoid rule. GussHermite qudrture If we set the weight function to w(x) = 1 π exp( x ),weresultinthegusshermite qudrture method. Note tht the weight function now is equl to the density function of the stndrd norml distribution. Now, with m n + 1 nd normlly distributed rndom vrible x, wehve E( x) = 1 exp( x )x m n dx = π w i xi m = I GH (x m ). Consequently, with GussHermite qudrture, we re ble to perfectly mtch the first n 1 moments of normlly distributed rndom vrible x. An pproximtion procedure for normlly distributed rndom vribles is included in the module normlprob. The procedure norml_discrete(x, w, mu, sig) is exctly bsed on the GussHermite qudrture method. The subroutine receives four input rguments, the lst of which re expected vlue nd vrince of the norml distribution tht should be pproximted. The routine then stores pproprite nodes x i nd weights w i in the rrys x nd w (tht should be of sme length) which cn be used to compute moments of normlly distributed rndom vrible x with expecttion mu nd vrince sig. For pplying this subroutine we consider the following exmple: Exmple Consider n griculturl commodity mrket, where plnting decisions re bsed on the price expected t hrvest A = E(p), (3.7) with A denoting crege supply nd E(p) defining expected price. After the crege is plnted, normlly distributed rndom yield y N(1, 0.1) is relized, giving rise to the quntity q s = Ay which is sold t the mrket clering price p = 3 q s. In order to solve this system we substitute nd therefore q s = [ E(p)] y p = 3 [ E(p)] y.
7 3.4. Numericl integrtion 69 Tking expecttions on both sides leds to E(p) =3 [ E(p)] E(y) nd therefore E(p) = 1. Consequently, equilibrium crege is A = 1. Finlly, the equilibrium price distribution hs vrince of Vr(p) =4 [ E(p)] Vr(y) =4Vr(y) =0.4. Suppose now tht the government introduces price support progrm which gurntees ech producer minimum price of 1. If the mrket price flls below this level, the government pys the producer the difference per unit produced. Consequently, the producer now receives n effective price of mx(p,1) nd the expected price in (3.7) thenis clculted vi E(p) =E [mx (3 Ay,1)]. (3.8) The equilibrium crege supply finlly is the supply A tht fulfills (3.7) with the bove price expecttion. Agin, this problem cnnot be solved nlyticlly. In order to compute the solution of the bove crege problem, one hs to use two modules. normlprob on the one hnd provides the method to discretize the normlly distributed rndom vrible y. The solution to (3.7) cn on the other hnd be clculted by using the method fzero from the rootfinding module. Progrm 3.13 shows how to do this. Due to spce restrictions, we do not show the whole progrm. For running the progrm, we lso need module globls in which we store the expecttion mu nd vrince sig of the normlly distributed rndom vrible y s well s the minimum price minp gurnteed by the government. In ddition, the module stores the qudrture nodes y nd weights w obtined by discretizing the norml distribution for y. In the progrm, we first discretize the distribution for y by mens of the subroutine norml_discrete. This routine receives the expected vlue nd vrince of the norml distribution nd stores the respective nodes y nd weights w in two rrys of sme size. Next we set strting guess for crege supply. We then let fzero find the root of the function mrket tht clcultes the mrket equilibrium condition in (3.7). This function only gets A s n input. From A we cn clculte the expected price E(p) by mens of (3.8) nd finlly the mrket clering condition. Hving found the root of mrket, i.e. the equilibrium crege supply, we cn clculte the expected vlue nd vrince of the price p. Inorder to showthe effects of minimum price, wefirst set the minimum price gurntee minp to lrge negtive vlue, sy 100. This lrge negtive minimum price will never be binding, hence, the outcome is exctly the sme s in the model without price gurntee, see the bove exmple description. If we now set the minimum price t 1, equilibrium crege supply increses by bout 9.7percent. This is becuse the expected effective price
8 70 Chpter 3 Numericl Solution Methods Progrm 3.13: Agriculturl problem progrm griculture...! discretize y cll norml_discrete(y, w, mu, sig)! initilize vribles A = 1d0! get optimum cll fzero(a, mrket, check)! get expecttion nd vrince of price Ep = sum(w*mx(3d0d0*a*y, minp)) Vrp = sum(w*(mx(3d0d0*a*y, minp)  Ep)**)... end progrm function mrket(a) use globls rel*8, intent(in) :: A rel*8 :: mrket rel*8 :: Ep! clculte expected price Ep = sum(w*mx(3d0d0*a*y, minp))! get equilibrium eqution mrket = A  (0.5d0+0.5d0*Ep) end function for the producer rises from the minimum price gurntee. In ddition, the uncertinty for the producers is reduced, since the vrince of the price distribution decreses from 0.4 to bout 0.115, which is gin due to the fct tht the possible price reliztion now hve lower limit of 1.
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