7.3 Recursive Rules and Romberg Integration

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1 SEC. 7. RECURSIVE RULES AND ROMBERG INTEGRATION Recursive Rules nd Romberg Integrtion In this section we show how to compute Simpson pproximtions with specil liner combintion of trpezoidl s. The pproximtion will hve greter ccurcy if one uses lrger number of subintervls. How mny should we choose? The sequentil process helps nswer this question by trying two subintervls, four subintervls, nd so on, until the desired ccurcy is obtined. First, sequence {T (J)} of trpezoidl pproximtions must be generted. As the number of subintervls is doubled, the number of function vlues is roughly doubled, becuse the function must be evluted t ll the previous points nd t the midpoints of the previous subintervls (see Figure 7.8). Theorem 7.4 explins how to eliminte redundnt function evlutions nd dditions. Theorem 7.4 (Successive Trpezoidl Rules). Suppose tht J 1 nd the points {x k = + kh} subdivide [, b] into J = M subintervls of equl width h = (b )/ J. The trpezoidl s T ( f, h) nd T ( f, h) obey the reltionship (1) T ( f, h) = T ( f, h) + h M f (x k 1 ). k=1

2 78 CHAP. 7 NUMERICAL INTEGRATION y = f(x) y = f(x) b b () (b) y = f(x) y = f(x) b b (c) (d) Figure 7.8 () T (0) is the re under 0 = 1 trpezoid. (b) T (1) is the re under 1 = trpezoids. (c) T () is the re under = 4 trpezoids. (d) T () is the re under = 8 trpezoids. Definition 7. (Sequence of Trpezoidl Rules). Define T (0) = (h/)( f () + f (b)), which is the trpezoidl with step size h = b. Then for ech J 1 define T (J) = T ( f, h), where T ( f, h) is the trpezoidl with step size h = (b )/ J. Corollry 7.4 (Recursive Trpezoidl Rule). Strt with T (0) = (h/)( f () + f (b)). Then sequence of trpezoidl s {T (J)} is generted by the recursive formul () T (J) = T (J 1) + h M k=1 f (x k 1 ) for J = 1,,..., where h = (b )/ J nd {x k = + kh}. Proof. For the even nodes x 0 < x < < x M < x M, we use the trpezoidl with step size h: () T (J 1) = h ( f 0 + f + f f M 4 + f M + f M ). For ll of the nodes x 0 < x 1 < x < < x M 1 < x M, we use the trpezoidl

3 SEC. 7. RECURSIVE RULES AND ROMBERG INTEGRATION 79 with step size h: (4) T (J) = h ( f 0 + f 1 + f + + f M + f M 1 + f M ). Collecting the even nd odd subscripts in (4) yields (5) T (J) = h ( f 0 + f + + f M + f M ) + h M k=1 f k 1. Substituting () into (5) results in T (J) = T (J 1)/ + h M k=1 f k 1, nd the proof of the theorem is complete. Exmple Use the sequentil trpezoidl to compute the pproximtions T (0), T (1), T (), nd T () for the integrl 5 1 dx/x = ln(5) ln(1) = Tble 7.4 shows the nine vlues required to compute T () nd the midpoints required to compute T (1), T (), nd T (). Detils for obtining the results re s follows: When h = 4: T (0) = 4 ( ) = When h = : T (1) = T (0) + (0.) = = When h = 1: T () = T (1) + 1( ) = = When h = 1 T () : T () = + 1 ( ) = = Our next result shows n importnt reltionship between the trpezoidl nd Simpson s. When the trpezoidl is computed using step sizes h nd h, the result is T ( f, h) nd T ( f, h), respectively. These vlues re combined to obtin Simpson s : 4T ( f, h) T ( f, h) (6) S( f, h) =. Theorem 7.5 (Recursive Simpson Rules). Suppose tht {T ( J)} is the sequence of trpezoidl s generted by Corollry 7.4. If J 1 nd S(J) is Simpson s for J subintervls of [, b], then S(J) nd the trpezoidl s T (J 1) nd T (J) obey the reltionship (7) S(J) = 4T (J) T (J 1) for J = 1,,...

4 80 CHAP. 7 NUMERICAL INTEGRATION Tble 7.4 The Nine Points Used to Compute T () nd the Midpoints Required to Compute T (1), T (), ndt () x f (x) = 1 x Endpoints for computing T (0) Midpoints for computing T (1) Midpoints for computing T () Midpoints for computing T () Proof. (8) The trpezoidl T (J) with step size h yields the pproximtion f (x) dx h ( f 0 + f 1 + f + + f M + f M 1 + f M ) = T (J). The trpezoidl T (J 1) with step size h produces (9) f (x) dx h( f 0 + f + + f M + f M ) = T (J 1). Multiplying reltion (8) by 4 yields (10) 4 f (x) dx h( f f f + +4 f M + 4 f M 1 + f M ) = 4T (J). Now subtrct (9) from (10) nd the result is (11) f (x) dx h( f f 1 + f + + f M + 4 f M 1 + f M ) = 4T (J) T (J 1). This cn be rerrnged to obtin f (x) dx h (1) ( f f 1 + f + + f M + 4 f M 1 + f M ) 4T (J) T (J 1) =. The middle term in (1) is Simpson s S(J) = S( f, h) nd hence the theorem is proved.

5 SEC. 7. RECURSIVE RULES AND ROMBERG INTEGRATION 81 Exmple 7.1. Use the sequentil Simpson to compute the pproximtions S(1), S(), nd S() for the integrl of Exmple Using the results of Exmple 7.11 nd formul (7) with J = 1,, nd, we compute 4T (1) T (0) 4( ) S(1) = = = , 4T () T (1) 4(1.68) S() = = = 1.6, 4T () T () 4( ) 1.68 S() = = = In Section 7.1 the formul for Boole s ws given in Theorem 7.1. It ws obtined by integrting the Lgrnge polynomil of degree 4 bsed on the nodes x 0, x 1, x, x, nd x 4. An lterntive method for estblishing Boole s is mentioned in the exercises. When it is pplied M times over 4M eqully spced subintervls of [, b] of step size h = (b )/(4M), we cll it the composite Boole : (1) B( f, h) = h 45 M (7 f 4k 4 + f 4k + 1 f 4k + f 4k f 4k ). k=1 The next result gives the reltionship between the sequentil Boole nd Simpson s. Theorem 7.6 (Recursive Boole Rules). Suppose tht {S( J)} is the sequence of Simpson s s generted by Theorem 7.5. If J nd B(J) is Boole s for J subintervls of [, b], then B(J) nd Simpson s s S(J 1) nd S(J) obey the reltionship 16S(J) S(J 1) (14) B(J) = for J =,,... Proof. The proof is left s n exercise for the reder. Exmple 7.1. Use the sequentil Boole to compute the pproximtions B() nd B() for the integrl of Exmple Using the results of Exmple 7.1 nd formul (14) with J = nd, we compute 16S() S(1) 16(1.6) B() = = = , 16S() S() 16( ) 1.6 B() = = = The reder my wonder wht we re leding up to. We will now show tht formuls (7) nd (14) re specil cses of the process of Romberg integrtion. Let us nnounce tht the next level of pproximtion for the integrl of Exmple 7.11 is 64B() B() 6 = 64( ) nd this nswer gives n ccurcy of five deciml plces. = ,

6 8 CHAP. 7 NUMERICAL INTEGRATION Romberg Integrtion In Section 7. we sw tht the error terms E T ( f, h) nd E S ( f, h) for the composite trpezoidl nd composite Simpson re of order O(h ) nd O(h 4 ), respectively. It is not difficult to show tht the error term E B ( f, h) for the composite Boole is of the order O(h 6 ). Thus we hve the pttern () (16) (17) f (x) dx = T ( f, h) + O(h ), f (x) dx = S( f, h) + O(h 4 ), f (x) dx = B( f, h) + O(h 6 ). The pttern for the reminders in () through (17) is extended in the following sense. Suppose tht n pproximtion is used with step sizes h nd h; then n lgebric mnipultion of the two nswers is used to produce n improved nswer. Ech successive level of improvement increses the order of the error term from O(h N ) to O(h N+ ). This process, clled Romberg integrtion, hs its strengths nd weknesses. The Newton-Cotes s re seldom used pst Boole s. This is becuse the nine-point Newton-Cotes qudrture involves negtive weights, nd ll the s pst the 10-point involve negtive weights. This could introduce loss of significnce error due to round off. The Romberg method hs the dvntges tht ll the weights re positive nd the eqully spced bscisss re esy to compute. A computtionl wekness of Romberg integrtion is tht twice s mny function evlutions re needed to decrese the error from O(h N ) to O(h N+ ). The use of the sequentil s will help keep the number of computtions down. The development of Romberg integrtion relies on the theoreticl ssumption tht, if f C N [, b] for ll N, then the error term for the trpezoidl cn be represented in series involving only even powers of h; tht is, (18) f (x) dx = T ( f, h) + E T ( f, h), where (19) E T ( f, h) = 1 h + h 4 + h 6 +. Since only even powers of h cn occur in (19), the Richrdson improvement process is used successively first to eliminte 1, next to eliminte, then to eliminte, nd so on. This process genertes qudrture formuls whose error terms hve even orders O(h 4 ), O(h 6 ), O(h 8 ), nd so on. We shll show tht the first improvement is Simpson s for M intervls. Strt with T ( f, h) nd T ( f, h) nd the equtions (0) f (x) dx = T ( f, h) + 1 4h + 16h h 6 +

7 SEC. 7. RECURSIVE RULES AND ROMBERG INTEGRATION 8 nd (1) f (x) dx = T ( f, h) + 1 h + h 4 + h 6 +. Multiply eqution (1) by 4 nd obtin () 4 f (x) dx = 4T ( f, h) + 1 4h + 4h 4 + 4h 6 +. Eliminte 1 by subtrcting (0) from (). The result is () f (x) dx = 4T ( f, h) T ( f, h) 1h 4 60h 6. Now divide eqution () by nd renme the coefficients in the series: (4) f (x) dx = 4T ( f, h) T ( f, h) + b 1 h 4 + b h 6 +. As noted in (6), the first quntity on the right side of (4) is Simpson s S( f, h). This shows tht E S ( f, h) involves only even powers of h: (5) f (x) dx = S( f, h) + b 1 h 4 + b h 6 + b h 8 +. To show tht the second improvement is Boole s, strt with (5) nd write down the formul involving S( f, h): (6) f (x) dx = S( f, h) + b 1 16h 4 + b 64h 6 + b 56h 8 +. When b 1 is eliminted from (5) nd (6), the result involves Boole s : (7) f (x) dx = 16S( f, h) S( f, h) = B( f, h) b 48h 6 b 48h 6 b 40h 8 b 40h 8.

8 84 CHAP. 7 NUMERICAL INTEGRATION The generl pttern for Romberg integrtion relies on Lemm 7.1. Lemm 7.1 (Richrdson s Improvement for Romberg Integrtion). Given two pproximtions R(h, K 1) nd R(h, K 1) for the quntity Q tht stisfy (8) Q = R(h, K 1) + c 1 h K + c h K + + nd (9) Q = R(h, K 1) + c 1 4 K h K + c 4 K +1 h K + +, n improved pproximtion hs the form (0) Q = 4K R(h, K 1) R(h, K 1) 4 K 1 + O(h K + ). Proof. The proof is strightforwrd nd is left for the reder. Definition 7.4. Define the sequence {R(J, K ) : J K } J=0 of qudrture formuls for f (x) over [, b] s follows (1) R(J, 0) = T (J) R(J, 1) = S(J) R(J, ) = B(J) for J 0, is the sequentil trpezoidl. for J 1, is the sequentil Simpson. for J, is the sequentil Boole s. The strting s, {R(J, 0)}, re used to generte the first improvement, {R(J, 1)}, which in turn is used to generte the second improvement, {R(J, )}. We hve lredy seen the ptterns () R(J, 1) = 41 R(J, 0) R(J 1, 0) R(J, ) = 4 R(J, 1) R(J 1, 1) 4 1 for J 1 for J, which re the s in (4) nd (7) stted using the nottion in (1). The generl for constructing improvements is () R(J, K ) = 4K R(J, K 1) R(J 1, K 1) 4 K 1 for J K.

9 SEC. 7. RECURSIVE RULES AND ROMBERG INTEGRATION 85 Tble 7.5 Romberg Integrtion Tbleu J R(J, 0) Trpezoidl R(J, 1) Simpson s R(J, ) Boole s R(J, ) Third improvement R(J, 4) Fourth improvement 0 R(0, 0) 1 R(1, 0) R(1, 1) R(, 0) R(, 1) R(, ) R(, 0) R(, 1) R(, ) R(, ) 4 R(4, 0) R(4, 1) R(4, ) R(4, ) R(4, 4) Tble 7.6 Romberg Integrtion Tbleu for Exmple 7.14 J R(J, 0) Trpezoidl R(J, 1) Simpson s R(J, ) Boole s R(J, ) Third improvement For computtionl purposes, the vlues R(J, K ) re rrnged in the Romberg integrtion tbleu given in Tble 7.5. Exmple Use Romberg integrtion to find pproximtions for the definite integrl π/ 0 (x + x + 1) cos(x) dx = + π + π = The computtions re given in Tble 7.6. In ech column the numbers re converging to the vlue The vlues in the Simpson s column converge fster thn the vlues in the trpezoidl column. For this exmple, convergence in columns to the right is fster thn the djcent column to the left. Convergence of the Romberg vlues in Tble 7.6 is esier to see if we look t the error terms E(J, K ) = +π/+π /4 R(J, K ). Suppose tht the intervl width is h = b nd tht the higher derivtives of f (x) re of the sme mgnitude. The error in column K of the Romberg tble diminishes by bout fctor of 1/ K + = 1/4 K +1 s one progresses down its rows. The errors E(J, 0) diminish by fctor of 1/4, the errors E(J, 1) diminish by fctor of 1/16, nd so on. This cn be observed by inspecting the entries {E(J, K )} in Tble 7.7.

10 86 CHAP. 7 NUMERICAL INTEGRATION Tble 7.7 Romberg Error Tbleu for Exmple 7.14 J h E(J, 0) = O(h ) E(J, 1) = O(h 4 ) E(J, ) = O(h 6 ) E(J, ) = O(h 8 ) 0 b b b 4 b 8 b 16 b Theorem 7.7 (Precision of Romberg Integrtion). Assume tht f C K + [, b]. Then the trunction error term for the Romberg pproximtion is given in the formul (4) f (x) dx = R(J, K ) + b K h K + f (K +) (c J,K ) = R(J, K ) + O(h K + ), where h = (b )/ J, b K is constnt tht depends on K, nd c J,K [, b]. Exmple 7.. Apply Theorem 7.7 nd show tht 0 10x 9 dx = 104 R(4, 4). The integrnd is f (x) = 10x 9, nd f (10) (x) 0. Thus the vlue K = 4 will mke the error term identiclly zero. A numericl computtion will produce R(4, 4) = 104.

11 Numericl Methods Using Mtlb, 4 th Edition, 004 John H. Mthews nd Kurtis K. Fink ISBN: Prentice-Hll Inc. Upper Sddle River, New Jersey, USA