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UNIVERSITY OF ILLINOIS 
 GRADUATE COLLEGE 
 DIGITAL COMPUTER LABORATORY 
 
 INTERNAL REPORT NO. 59 
 
 A BOUND ON THE ERROR IN A GENERAL QUADRATURE 
 FORMULA WITH EQUIDISTANT ORDINATES 
 
 Robert T. Gregory 
 October 20, 195^ 
 
 This work has been supported in part by 
 
 the Office of Naval Research under 
 
 Contract NR 0^ 001 
 
A BOUND ON THE ERROR IN A GENERAL QUADRATURE 
 FORMULA WITH EQUIDISTANT ORDINATES 
 
 Consider the equation 
 
 f* n f(x)dx = g V f + 6, (1) 
 
 x i=l 
 
 where the summation on the right represents an approximation to the 
 
 integral and € is the error in this approximation. Here f . = f (x. ) and 
 
 x. n - x. = h for i = l,2,...,n-l. We will find a bound on € for an 
 l+l l 
 
 arbitrary quadrature formula, i.e., a bound in terms of the w. which 
 specify the particular formula under consideration. Equation (l) is a 
 generalization of the well known Euler-Maclaurin formula since that 
 formula is merely (l) with the w. chosen for the trapezoidal rule. 
 
 Let us introduce the step function 
 
 0(x) = X ± 
 
 
 X. < X < X. , 
 
 1 — 1+1 
 
 = x , 
 
 n-1 
 
 
 X = X 
 
 n 
 
 and then define 
 
 
 
 g(x) = x - <f{x 
 
 ) 
 
 X, < X < X 
 
 1 — — n 
 
 (i=l,...,n-l) 
 
 (2) 
 
 (?) 
 
 Thus we can write 
 n 
 
 .^ w.f. =Z ± w.f. + f n [g(x) + 0(x) - x] f(x) 
 
 dx 
 
 X l 
 
 n x 
 
 = - f n xf(x)dx + i Z 1 w.f. + f n g(x)f*(x)dx 
 X l X l 
 
 + / n 0(x)f(x)dx. 
 
 x l 
 
 Integrating the first integral by parts gives us 
 
 n x n x 
 
 Z w.f. = f n f(x)dx - xf(x) | X n +Z w.f. + f n g(x)f'(x)dx 
 . -, 1 i •* x, . . i i J 
 
 1=1 x 1 1=1 x 
 
 + Z x / i+1 f'(x)dx (k) 
 
 1=1 X 
 
 -1- 
 
Comparing (k) with (l) we see that 
 
 n x n-1 
 
 -6 - x,f, - x f + Z w.f . + C n g(x)f ' (x)dx + Z x. (f , . - f, ) 
 
 11 nn.,11 J d\/v/ . _ i x i+l \' 
 
 1=1 x. 1=1 
 
 x 
 
 n ,n 
 
 = w f + Z (w. - h) f. + J g(x) f'(x) dx (5) 
 
 n=2 x. 
 
 Since g(x) is a periodic function and satisfies the Dirichlet con- 
 ditions we can write the Fourier series expansion 
 
 CO 
 
 g(x) = h/2 - Z (h/itm) Cos itm Sin [ (2mn/h)(x - x - h/2 ) ] (6) 
 
 m=l 
 
 for the range x,< x < x . Hence, If x = X - x, - h/2, the last term in 
 1— — n 1 ' 
 
 (5) can be written 
 
 f n g(x)f'(x)dx = (h/2) f n f'(x)dx 
 X l X l 
 
 X _ 
 
 • n 
 
 J f'(x) [ Z (h/jtm) Cos itm Sin(2jtmx/h) ] dx 
 
 X l m=l 
 
 x 
 
 = (h/2)(f - f ) - Z (-l) m (h/jtm) J f'(x) Sin(2rtmx/h)dx 
 
 m=l 1 
 
 if we assume that f'(x) is of bounded variation. 
 
 In order to use (7) we need the relation, obtained by integrating by 
 
 parts t times , 
 
 x t 
 
 f n f'(x) Sin (2itrax/h) dx = Z (h^jtm) 1 A. [f 1 S.(x ) - fZ S.(xJ] 
 j 11 \ 1 x nin 111 
 
 x 1=1 
 
 - A t (h^m) 1 _f n f(x) s t (x) dx (8) 
 X l 
 
 where 
 
 A. = (_ 1 )( i, 2)(i+D/2 
 
 and 
 
 S.(x) = Sin (2itmx/h) if i is even 
 = Cos (2nmx/h) if i is odd. 
 
 - 2 - 
 
Substituting (8) into (7) gives us 
 
 / n g(x)f (x)dx = (h/2)(f - f ) - J, (-l)"(V«3a)I 2 A. (h/2,™) 1 
 
 •(fV(x ) - f*S.(x.)) - A + (h/2^m) t / n f t+1 S + (x)dxl 
 (_nin y 1 r 1 J t w x i V ' 
 
 In order to write the infinite series in (9) as the sum of two series 
 we must show that it is absolutely convergent. To do this we write the m 
 term in the form 
 
 nf 2 [(-l) m (h/«)f Z A.(h/2* )* m 1 " 1 (fVlxJ - fjs^x )) 
 *■ i=l 
 
 - A t (h/2*) 11 m 1_t f n f t+1 S t (x) dx ] (10) 
 
 1 
 
 If the first t+1 derivatives of f(x) are bounded then the term in brackets 
 
 00 -2 
 in (10) is bounded. Thus, since JS-. Mm converges (here M is a bound on 
 
 the term in brackets) the series in (9) is absolutely convergent by the 
 
 comparison test. Hence 
 
 / Xn g(x)f'(x)dx = (h/2)(f -f ) - Z (-l) m (hAm)[E A.Ch^nm) 1 
 1 m=l 1=1 
 
 •ffV(x ) - f*S<(x-)} ] 
 
 L n i N n 1 i v l'J 
 
 u x 
 
 + Z (-if 2A + (h/2*m) t+1 / " f t+ V(x) dx, (ll) 
 m=l t X l t 
 
 and substituting into (5 ) we get 
 
 n-1 
 
 -€ = (w -h/2) f . + (v - h/2) f + Z (w.-h) f. 
 1 ' 1 v n ' ' n . n x 1 ' 1 
 
 i=2 
 
 - Z (-l) m X 2A 1 (h/2«m) 1+1 /V S i( x n^ ' ^i^} 
 m=l J 
 
 + Z (-l) m 2A t (h/2nm) t+1 4 ^ f t+1 S (x) dx (12 ) 
 
 m=l 
 
 Examine the last term in (12). Since |S (x)| : 1 we know that if 
 
 |f I is bounded by K 
 
 I f\ f tfl s (x)dx | < f*n | f t+l| ^ <: K (n _ l} h (l5) 
 
 X l X " X l 
 
 - 3 - 
 
Thus 
 
 CO 
 
 ^ (-if 2A t (h/2^m) t+1 / n f t+1 S t (x)dx | < K(n-l) h t+2 / 2 V + ^ m - (t+l) 
 
 (l*0 
 
 'm _ 
 
 X l 
 
 and jf we assume t > 1 we have 
 
 X»" (1+t) 5* 2 /6- d5) 
 
 Hence with this restriction on t the term is hounded by 
 
 (1/3) (n-1) h t+ V-V (t+l) (16) 
 
 The other infinite series in (12 ) is absolutely convergent and so it 
 
 en as the sum of series . 
 , t , . r . 
 
 can be written as the sum of series . To do this we write 
 
 CD _, t 
 
 Z. (-l) m+1 .E, 2A.(h/27tm) 1+1 -(f 1 S.(x ) - fis.(xj) 
 i=l v ' 1=1 i v/ I n i v n 1 i v 1 j 
 
 = J^ (-l) m+1 [-(h 2 /2n 2 m 2 ) j r Cos(2n-3)rtm - f* Cos jtm} 
 
 + (h 5 /4it 5 m 5 )/f^ Sin(2n-3)jtm + f" Sin *ml 
 
 + (h /8it'm J-ff 5 Cos (2n-3)itm - f\ Cos jtm) + ... 
 
 ♦ 2A t . 1 (h/ 2 , m ) t {f*- 1 ! w (« n ) - f*" 1 ^l} 
 
 + 2A t (h/2ra ) t+ * |f *S t (x n ) - f *S t (x x ) ] ] . ( 17 ) 
 
 From the definitions of S.(x) it follows that one of the last two terms 
 
 i 
 
 will vanish since for either i = t or i = t-lj i is even and 
 
 S. (x ) = Sin (2n-3)«m = 
 
 r n x ' 
 
 S. (x, ) = Sin nm = 0. 
 i 1 
 
 We denote by T the odd integer (t or t-l) which gives 
 S T (x n ) = Cos(2n-3)«m = (-l) m ( 2n - 5) 
 
 (18) 
 
 S T (x 1 ) = Cos %m = (-l) m 
 
 (19) 
 
 k - 
 
Then the right side of (17) becomes 
 
 - (h 2 /* 2 ) [r £ m " 2 (-i) 2 "^ + ,. £ ^(-i) 2 - 1 ] 
 ♦ (hVa.S [f 5 £ -** (-D 2 "^- 1 ^ 1 - f? E. m - u (-D 2m+1 1 
 
 v ' n m=l 1 m=l 
 
 . (h 2 / 2 n 2 ) (r - f.) m | m " 2 - (hVa/) (f' - fj) J, m"* 
 
 + ... -2A T (h/2«) T+1 (f* - i\) X m " (T+l) • (20) 
 
 J, 00 
 
 Since Z. m~ 2p = (-l) P_1 B (2* ) p /2(2p)i for fixed integers p (here 
 
 ID— J- dp ao q p 
 
 B p are Bernoulli's numbers), we have IL El = it /6, 
 
 !_ ^ - «*/*>. and ^ m"( T+1 ) . (-l)^" 1 ^ b t+1 (2* ) T+1 /2 (Tfl ) I 
 
 Thus the right side of (20) becomes 
 
 - (h 2 /!2) (f« - f« ) - (hV720) (f 5 - fh + ... 
 
 + (-i)< T+1 ) / 2+T (h T+ \ +1 /(T + l)L ) (f T n - f\). (21) 
 
 Using (21), and (12) in (l) we get 
 
 /x n n-1 
 
 n f(x)dx - JL w.f. - [(w -h/2) f. i (w - h/2) f + .Z (w. - h) f. 
 
 i=l li 1 ' 1 n ' n i=2 i i 
 
 x l 
 
 (h 2 /l2)(f; - f£) - (hV720)(rJ - f^) + ... 
 
 (-l) (T " l)2/2+T (h T+1 B T|1 /(T + l)0(f^-f^ 
 
 2 m | x (-if A t (h/2«m) t+1 f* n f t+1 S t (x) dx ] (22) 
 
 ~1 
 
 2 
 which demonstrates the relation with the Euler-Maclaurin formula. 
 
 - 5 
 
If we use (2l) and (l6) with (12) we see that, under the assumptions 
 
 we have made, 
 
 n-1 
 
 |c| < |(w 1 - h/2) f x + (w n - h/2) f n +.^ (w. - h) f. 
 
 + (h 2 /l2) (r - fj!) - (h\/720) (f^ - f3) + ... 
 
 + ( _ l)( T + l) 2 /2 + T (h T + l BT+i/(T+l)i) (f T. f T } ! 
 
 + (K/3) (n-1) h^^^-Ct+l) (23) 
 
 Acknowledgement 
 
 The writer wishes to thank Dr. D. E. Muller and Mrs. Lily Seshu for 
 their helpful suggestions in the preparation of this report. 
 
 REFERENCES 
 
 1. E. Whittaker and G. Robinson, "The Calculus of Observations," 
 Blackie and Son Ltd., p. 135* 
 
 2. J. B. Scarborough, "Numerical Mathematical Analysis," Johns 
 Hopkins Press, Second Edition, p.. 157* 
 
 3. E. C. Titchmarsh, "The Theory of Functions," Oxford University 
 Press, Second Edition, p. ^21. 
 
 h. K. Khopp, "Theory and Application of Infinite Series," Blackie 
 and Son Ltd., p. 237- 
 
 RTG/hc 
 
 - 6 - 
 
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