AMTH247 Lecture 16. Numerical Integration I

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1 AMTH47 Lecture 16 Numericl Integrtion I 3 My 006 Reding: Heth , 8.3.1, 8.3., Contents 1 Numericl Integrtion 1.1 Monte-Crlo Integrtion Attinble Accurcy Interpoltory Methods Newton-Cotes Methods Trunction Error Choice of Nodes Gussin Methods All the Scilb functions used in this lecture re in the file l16.sci in the directory for the lecture. 1

2 1 Numericl Integrtion The problem we going to study is the pproximtion of definite integrls I = Almost ll pproximtions hve the form I f(x) dx. w i f(x i ). The points x i where the function is evluted re clled nodes nd the multipliers w i re clled weights. 1.1 Monte-Crlo Integrtion n nodes re chosen rndomly in the intervl [, b] with ll the weights equl to (b )/n, giving the pproximtion I b n f(x i ). To generte rndom numbers in n intervl [, b] we use the fct tht if x is uniformly distributed on [0, 1], then y = + (b )x is uniformly distributed on [, b]. function i = monte_int (f,, b, n) x = + (b-)*rnd(1,n) i = (b-)*sum(f(x))/n endfunction Exmple Let I = 1 0 e x dx. The exct vlue of the integrl cn be obtined from the error function, erf in Scilb, Erf(x) = x e t dt. π -->ii =erf(1)*sqrt(%pi)/ ii = Using Monte-Crlo to pproximte the integrl: -->function y = func (x) --> y = exp( -x.^) -->endfunction 0

3 -->monte_int(func, 0, 1, 100) ns = >monte_int(func, 0, 1, 10000) ns = Monte-Crlo methods re not prticulrly ccurte; our exmple gve n error of 0.3% with 10,000 nodes, but they re verstile. The mjor use of Monte- Crlo integrtion is for the pproximtion of integrls in severl dimensions. 1. Attinble Accurcy As with other numericl methods we need to understnd how rounding error ffects numericl integrtion. Consider n pproximtion I w i f(x i ). The rithmetic opertions performed t ech stge of forming the pproximting sum re, evlution of f(x i ), multipliction by w i nd ddition. Let the reltive error in these three opertions be ɛ i, so tht the contribution to the totl error is w i f(x i )ɛ i. The totl error is then E = w i f(x i ) ɛ i. Now, typiclly, we expect tht ɛ i is of the order of mgnitude ε mch, so tht the totl rounding error in the pproximte integrl is of the order of E w i f(x i ) ε mch I ε mch. Thus the reltive error, due to rounding error, is expected to be of the order of ε mch, which is s good s could be hoped for. 1.3 Interpoltory Methods The ide is very simple; for ny set of nodes x 1,..., x n, interpolte the dt (x i, f(x i )), i = 1,..., n, by function F (x) nd then use integrl of the interpolting function F (x) to pproximte the integrl. Consider the cse of polynomil interpoltion. The interpolting polynomil is, using the Lgrnge bsis polynomils, l i (x), p(x) = f(x i ) l i (x) 3

4 nd the pproximte integrl is p(x) dx = = f(x i ) w i f(x i ) l i (x) dx where the weights w i re the integrls of the Lgrnge bsis polynomils w i = l i (x) dx. Note tht the Lgrnge polynomils, nd hence the weights, depend on the choice of nodes. An lterntive methods for obtining the weights is the method of undetermined coefficients. By interpolting the dt t the n nodes, x 1,..., x n, by degree n 1 polynomil, we obtin n integrtion method which is exct for polynomils of degree (n 1) or less. In other words, the pproximtion f(x) dx w i f(x i ) is n equlity when f(x) is polynomil of degree (n 1) or less. This gives us system of liner equtions for the weights: w w w n 1 = w 1 x 1 + w x + + w n x n = w 1 x 1 + w x + + w n x n = w 1 x n w x n w n x n 1 n =. 1 dx = b x dx = (b )/ x dx = (b 3 3 )/3 x n 1 dx = (b n n )/n In mtrix form, these equtions re w 1 b x 1 x x 3... x n w (b )/ x 1 x x 3... x n w 3 = (b 3 3 )/ x1 n 1 x n 1 x3 n 1... xn n 1 w n (b n n )/n Note tht the mtrix in these equtions is the trnspose of the bsis mtrix for the monomil bsis used in interpoltion, see Lecture 11. Here is Scilb function to compute the weights for interpoltory integrtion by polynomil. nd b give the rnge of integrtion nd x is the vector of nodes: 4

5 function w = p_weights (, b, x) = monbsis(x) n = length(x) bb = zeros(n,1) for k = 1:n bb(k) = (b^k - ^k)/k end w = \bb endfunction We cn use this function to pproximte integrls by polynomil interpoltion: function i = poly_int (f,, b, x) w = p_weights (,b,x) i = sum(w.*f(x)) endfunction Exmple We will pproximte the integrl I = 1 0 e x dx using polynomil interpoltion with 6 eqully spced nodes on [0, 1]; -->x = (0:0.:1) x =! 0.!! 0.!! 0.4!! 0.6!! 0.8!! 1.! -->poly_int(func, 0, 1, x) ns = This hs n error of bout Newton-Cotes Methods Newton-Cotes methods use polynomil interpoltion with eqully spced nodes. These methods re clssified s closed or open depending on whether not the endpoints of the intervl re included in the nodes. The simplest nd most importnt exmples re: 5

6 ( + b)/ b 1. Midpoint Rule: Interpolting t the midpoint of the intervl by degree zero (constnt) polynomil gives the one point open Newton-Cotes rule ( ) + b M(f) = (b )f.. Trpezoidl Rule: Interpolting t the two end points of the intervl by degree one (stright line) polynomil b gives the two-point closed Newton-Cotes rule T (f) = (b ) (f() + f(b)). 3. Simpson s Rule: Interpolting t the two end points nd midpoint of the intervl by degree two (qudrtic) polynomil gives the three-point closed Newton-Cotes rule Exmple S(f) = (b ) 6 ( f() + 4f ( + b ) ) + f(b). We cn pply ech of three rules bove to our exmple, using poly_int nd estimte the error in ech cse: 6

7 -->i1 = poly_int(func, 0, 1, [0.5]) i1 = >e1 = i1 - ii e1 = >i = poly_int(func, 0, 1, [0 1] ) i = >e= i - ii e = >i3 = poly_int(func, 0, 1, [ ] ) i3 = >e3= i3 - ii e3 = Trunction Error The error in the midpoint rule cn be estimted s follows: expnd f(x) in Tylor series bout the midpoint m = ( + b)/ f(x) = f(m)+f (m)(x m)+ f (m) (x m) + f (3) (m) 6 (x m) 3 + f (4) (m) (x m) Integrting from to b term by term, the odd order terms drop out giving f(x) = f(m)(b ) + f (m) (b ) 3 + f (4) (m) (b ) = M(f) + E(f) + F (f) +... Here M(f) is the midpoint pproximtion, nd E(f) nd F (f) re the first two terms in its error expnsion. A similr clcultion for the trpezoidl method gives f(x) = T (f) E(f) 4F (f) +... where T (f) is the trpezoidl pproximtion, nd E(f) nd F (f) re the sme s those derived for the midpoint pproximtion. 7

8 There re number of interesting conclusions which cn be drwn from this nlysis: 1. The leding error term for the trpezoidl pproximtion is twice the size nd the opposite sign s tht for the midpoint pproximtion. Numericl evidence for this cn be seen in the previous exmple.. Ignoring the higher order terms, compring the midpoint nd trpezoidl errors gives M(f) T (f) E(f) 3 which cn be used to estimte the error in either pproximtion. Using our previous clcultions we cn compute this -->ee =(i1 - i)/3 ee = which gives n excellent estimte of the error. Note tht we hve obtined this error estimte from the pproximte integrls without knowing the exct vlue. 3. We cn rrnge for the terms involving E(f) in the midpoint nd trpezoidl pproximtion to cncel by dding twice the midpoint pproximtion to the trpezoidl pproximtion or 3 f(x) = M(f) + T (f) F (f) +... f(x) = 3 M(f) T (f) 3 F (f) Now 3 M(f) T (f) = ( ) + b (b )f + (b ) (f() + f(b)) 3 3 ( ( ) ) (b ) + b = f() + 4f + f(b) 6 which is just Simpson s rule. 4. From the derivtion of Simpson s rule bove, we hve the error estimte Exmple: f(x) = S(f) 3 F (f) We will pply Newton-Cotes methods to our exmple. We will use closed formuls with to 40 nodes nd compute the error in ech cse: 8

9 -->err = zeros(1,39); -->for k = :40 --> ik = poly_int(func, 0, 1, linspce(0,1,k) ); --> err(k-1) = ik-ii; -->end -->plotd(log10(bs(err))) We see tht decreses rpidly until we rech bout 17 nodes nd then increses slowly. This is cn hve two possible cuses (1) the monomil bsis is, s we hve seen, ill-conditioned nd this will cuse inccurcies in computing the weights, or () the interpolting polynomils of high degree will show lrge oscilltions. Rounding error in the pproximte integrtion cn be ignored becuse, s we sw erlier, this cn be expected to be of the order of ε mch, fct confirmed by the most ccurte pproximtions. Plotting the interpolting polynomil -->t =linspce(0,1,40) ; -->y = func(t); -->d = newt_interp(t, y); wrning mtrix is close to singulr or bdly scled. results my be inccurte. rcond = 3.18E-9 -->yy = newt_evl(t, d, tt); 9

10 -->plotd(tt, [yy func(tt)]) This shows tht oscilltions in the interpolting polynomil re not the problem in this cse, so the problem must be due to the ill-conditioning of the monomil bsis mtrix. 1.6 Choice of Nodes We sw erlier how, given ny set of nodes, x i, i = 1,..., n, to construct numericl integrtion method by polynomil interpoltion. The Newton-Cotes methods choose eqully spced nodes. The Clenshw-Curtis methods use the Chebyshev points s nodes. Both the Newton-Cotes nd Clenshw-Curtis methods re exct for polynomils of degree (n 1) or less. A numericl method is sid to be of degree d if it is exct for ll polynomils of degree d or less. Exmple We will pproximte I = 1 0 e x dx. using the 3 point Clenshw-Curtis method. The Chebyshev points on [ 1, 1] re given by ( ) (i 1)π x i = cos, i = 1,..., k. k for k = 3 we get the nodes -->i = (1:3) i =! 1.! 10

11 !.!! 3.! -->x = cos((*i-1)*%pi/6) x =! !! 6.13E-17!! ! The Chebyshev points lie in the intervl [ 1, 1]. To obtin the corresponding points in the intervl [, b] we need to perform the liner trnsformtion x = + b + b x The Chebyshev points for the intervl [0, 1] re -->x = (x+1)/ x =! !! 0.5!! ! The weights re: -->w = p_weights(0, 1, x) w =! 0.!! !! 0.! Now we cn use poly_int to pproximte the integrl -->function y = func(x) --> y = exp(- x.^) -->endfunction -->ii = sqrt(%pi)*erf(1)/ ii = >i= poly_int(func, 0, 1, x) i = >e = i - ii e = The error is bout 1/3 tht for Simpson s rule. 11

12 1.7 Gussin Methods Gussin methods choose the nodes so s to mximize the degree of the method. If we hve n nodes x i nd n corresponding weights w i to choose, then it seems plusible tht we cn choose these n prmeters to give method of degree (n 1). This wht Gussin methods do; the nodes nd weights re chosen to mximize the degree of the integrtion method. The nodes nd weights for Gussin methods re not esy to derive (see Heth, 8.3.3, for n indiction of how they cn be determined). Exmple For the two point Gussin method on [0, 1] the nodes re 3 ± 1 x = 3 with both weights 1/. For our exmple function we cn do the clcultion s follows: -->x1 x1 = = (sqrt(3)-1)/(*sqrt(3)) >x x = = (sqrt(3)+1)/(*sqrt(3)) >i = (func(x1)+func(x))/ i = >e = i - ii e = This is bout /3 of the error for Simpson s method, lthough we used only points rther thn 3. 1

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