Undergrdute Reserch A Trigonometric Simpson s Rule By Ctherine Cusimno Kirby nd Sony Stnley Biogrphicl Sketch Ctherine Cusimno Kirby is the dughter of Donn nd Sm Cusimno. Originlly from Vestvi Hills, Albm, she is 1999 grdute of John Crroll Ctholic High School. She grduted with honors from Smford University in 003 with mjors in mthemtics nd Spnish. She credits Dr. Sony Stnley, who directed her reserch for this pper, with hving considerble influence on her mthemticl development. Ctherine now resides in Chttnoog, Tennessee. [15]
16 Albm Journl of Mthemtics Introduction There re some integrble functions f for which the definite integrl R b f (x) dx cnnot be clculted exctly. Becuse of this, numerous methods of pproximting definite integrls exist, including Simpson s rule nd the trpezoidl rule. These well-known numericl integrtion methods re bsed on polynomil interpoltion nd re well-suited for computer implementtion. In this pper, we develop similr methods using trigonometric polynomils nd trigonometric splines nd present comprisons with existing numericl integrtion methods. Simpson s Rule The scheme for pproximting n integrl using Simpson s rule isbsedonthefctthtifp (x) is ny polynomil of degree three or less, then R b b P (x) dx = 6 P ()+4P +b + P (b). Since there exists unique qudrtic polynomil Ax + Bx + C tht interpoltes the function f t the points, (, f()), +b,f +b, nd (b, f(b)), the integrl of this qudrtic polynomil serves s n pproximtion to R b f (x) dx. Inotherwords, Z b Z b f (x) dx Ax + Bx + C dx. Hence, Z b (Eq. 1) f (x) dx b + b f ()+4f + f (b). 6 The error term ssocited with Simpson s rule is 1 5 b f (4) (ξ), ξ (, b). 90 As motivting exmple, we will derive Simpson s rule using the method of undetermined coefficients. This method, illustrted here with =0ndb =1, yields the formul Z 1 1 f (x) dx A 0 f (0) + A 1 f + A f (1). 0 which is exct if f (x) is polynomil of degree less thn or equl to 3. To obtin formul tht clcultes integrls of qudrtic polynomils exctly, we simply force the integrls of the bsis functions,
Spring 004 17 f 1 (x) =1,f (x) =x, nd f 3 (x) =x to be represented exctly. Doing this we obtin the following: 1= R 1 0 dx = A 0 + A 1 + A 1 = R 1 0 xdx = 1 A 1 + A 1 3 = R 1 0 x dx = 1 4 A 1 + A. The solution of the system is A 0 = 1 6,A 1 = 3,A = 1 6, thereby yielding Z 1 0 f (x) dx 1 6 f (0) + 3 f 1 + 1 6 f (1). A more generl ppliction of the foregoing procedure on the intervl [, b] cn be used to obtin the form of Simpson s rule given in Eq. 1. Integrtion Using Trigonometric Polynomils. It is nturl to sk whether numericl integrtion formul bsed on trigonometric polynomils would be better suited to pproximte the integrl of trigonometric function thn n pproch bsed on qudrtic polynomils. In the cse of trigonometric polynomils, we hve developed n pproximtion for the integrl R b f (x) dx by imitting the method of undetermined coefficients described bove in the derivtion of Simpson s rule. In our cse we use the following bsis functions, first introduced in [1] f 1 (x) = sin(t t) sin(t t 1), f (x) = sin(t t)sin(t t1) (sin(t t 1 )), f 3 (x) = sin(t t1 ) sin(t t 1). If we let h = b, the formul tht results is: R b f (x) dx csc (h) 1 + h cot (h)+ 1 csc (h) h 1 4 sin (4h) f () + csc (h) 1 +cot(h)+csc(h) f +b + csc(h) 1 + h cot (h)+csc(h) h 1 4 sin (4h) f (b).
18 Albm Journl of Mthemtics This formul is exct for functions of the form f 1 (x)+bf (x)+ cf 3 (x). We demonstrte this in exmples in Tble 1. Also, even though this formul ws derived on the intervl [ h, h], it is vlid on ny intervl of length h. For exmple, if h =1ndb =, the result is R b f (x) dx 0.385095f ()+1.981f +b +0.385095f (b). Trigonometric-Spline-Bsed Integrtion Approximtion Method Now we construct n pproximtion method bsed on trigonometric splines. The first step in this process is to choose n pproprite representtion for the spline used in the construction. We define second-degree trigonometric spline in the following wy: Definition 1. Let t [t 1,t ], 0 <t t 1 < π nd let A 1, B 1,C 1,A,B, nd C be rel numbers. Then, the function p (t) given by p (t) = A 1 nd p (t) = A sin( t t) sin(t t) sin(t t) + B 1 + C 1 + B + C sin( t t) sin(t t 1 ) sin(t t) sin(t t) sin(t t) sin(t t 1 ), for t 1 t< t sin(t t) sin(t t) sin(t t), for t t t is second-degree trigonometric spline. It is our intention to use this trigonometric spline to develop numericl integrtion formul similr to Simpson s rule nd to compre the performnce of this new formul in pproximting integrls of severl fmilies of functions with the performnce of Simpson s rule. In prticulr, we will investigte whether our trigonometric version of Simpson s rule pproximtes trigonometric integrls better thn the polynomil-bsed Simpson s rule. Using the definition of spline in definition 1, we will develop trigonometric-spline-bsed integrtion method to pproximte R b f (x) dx. Thevlueofthesplinewillbeclcultedtt = h, t =0, nd t = h nd set equl to y 0 = f ( h),y 1 = f (0), nd y = f (h) respectively. We set the vlue of the left-hnd side of the spline equl to the vlue of the right-hnd side t t =0, nd we set
Spring 004 19 the slopevlue tt = 0 equl from both the left nd the right. Also, we let = t 1 = h nd b = t = h. Using this informtion, the following system of equtions is set up nd solved for A 1,B 1, C 1,A,B, nd C : p (t 1 ) = A sin( t1 ) sin( t1 ) sin(0) 1 sin( t +B1 1) sin( t 1) sin( t 1) + C 1 sin(0) sin( t 1) = A1 = y 0 p 0 (0) = csc(h) B 1 +cot(h) C 1 = s p (0) = A sin(0) sin(0) sin( t1) 1 sin( t 1 ) +B1 sin( t 1 ) sin( t 1 ) + C 1 sin( t1) sin( t 1 ) = C1 = y 1 p (0) = A sin(t) sin(t) sin(0) sin(t ) +B sin(t ) sin(t ) + C sin(0) sin(t ) = A = y 1 p 0 (0) = cot(h) A +csc(h) B = s p (t ) = A sin(0) sin(0) sin(t ) sin(t ) +B sin(t ) sin(t ) + C sin(t ) sin(t ) = C = y where y 0 = f ( h),y 1 = f (0),y = f (h), nd s = δ1δ δ 1 +δ with δ 1 = y 1 y 0 h nd δ = y y 1 h.this system cn be represented in mtrix form s follows: 1 0 0 0 0 0 0 csc(h) cot(h) 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 cot(h) csc(h) 0 0 0 0 0 0 1 A 1 B 1 C 1 A B C = y 0 s y 1 y 1 s y
0 Albm Journl of Mthemtics From this it follows tht R h h f (x) dx = R 0 h "A 1 sin( t t) +B 1 sin( t t) sin(t t 1) + C 1 sin(t t 1 ) # dt + R h 0 "A sin(t t) sin(t t) +B sin(t t) sin(t t) sin(t t) sin(t t) + C sin(t t) sin(t t) # dt which results in Z b f (x) dx 1 4 csc (h)[ h (y 0 + y )+4hy 1 cos(h)+(y 0 y 1 + y )sin(h)]. Comprison of Simpson s Rule nd Trigonometric Integrtion Methods We now present comprison of exct integrl vlues with pproximtions clculted through the use of Simpson s Rule, the trigonometric polynomil-bsed method, nd the trigonometric spline-bsed method. These comprisons re shown for vrious functions on the sme intervl nd for severl functions on different intervls.
Spring 004 1 Function Exct Simpson s Trigonometric Trigonometf (x) = Integrl Rule Polynomil- ric Spline- Vlue Approx- bsed Ap- bsed Apimtion proximtion proximtion 5 10 10 10 10 5x +7 14 14 14 14 x 0. 6 0. 6 0.77019 0.77019 3x + 6 6 6.31057 6.31057 x 3 0 0 0 0 3x 3 + 4 4 4 4 x 4 0.4 0. 6 0.77019 0.77019 x 4 +5 10.4 10. 6 10.770 10.770 x 5 0 0 0 0 (x +1) 5 10. 6 1 13.559 13.559 sin (x) 0 0 0 0 cos (x) 1.6894 1.69353 1.64595 1.64595 sin (x) 0.54535 0.47049 0.54535 0.54535 cos (x) 1.45465 1.5795 1.45465 1.45465 sin 3 (x) 0 0 0 0 cos 3 (x) 1.8573 1.43849 1.3519 1.3519 cos 3 (x) 1.83108 1.91053 1.89664 1.89664 +sin (x) Tble 1. Comprisons of exct nd pproximte integrl vlues for R 1 f (x) dx. 1 Integrl Exct Simpson s Trigonometric Integrl Rule Polynomil- Vlue Approx- bsed Apimtion proximtion R 1 1 x dx 0.66667 0.66667 0.77019 R 3 1 x dx 18.6667 18.6667 18.770 R 4 x dx 18.6667 18.6667 18.770 R 1 1 cos (x)+sin 3 (x) dx 1.60497 1.61679 1.566 R 3 R1 6 cos (x)+sin 3 (x) dx 1.85711 1.85689 1.85796 cos (x)+sin 3 (x) dx 0.99783 1.0019 0.69508 Tble. Comprisons of exct nd pproximte integrl vlues over vrious intervls. From these exmples, it cn be seen tht there ws no observble difference in the trigonometric polynomil-bsed pproximtion method nd the trigonometric-spline pproximtion method. The reson for this hs not been investigted t this time. Also,
Albm Journl of Mthemtics it cn be seen in these exmples tht in both the liner nd constnt functions, the trigonometric pproximtions gree with the exct integrl vlues. Wheres the trigonometric bsed methods give exct integrl vlues for odd polynomil functions, they give only close pproximtions for the even polynomil functions. Also, in the cses of the trigonometric functions, the trigonometric-bsed methods often give closer integrl pproximtions thn Simpson s rule. References [1] Alfeld, P., Nemtu, M., nd Schumker, L., Circulr Bernstein-Bezier Polynomils, Mthemticl Methods in CAGD, M.Dehlen,T.Lyche,ndL.L.Schumker (eds), Vnderbilt University Press, (1995), 1-10. [] Cusimno, C. nd Stnley, S., Circulr Berstein-Bezier Polynomil Splines with Knot Removl, Journl of Computtionl nd Applied Mthemtics, 155 / 1, (003), 177-185. [3] Kincid, D. nd Cheney, W., Numericl Anlysis: Mthemtics of Scientific Computing.nd. ed. Pcific Grove, CA: Brooks/Cole, 1996. [4] Schumker, L. nd Stnley, S., Shpe-Preserving Knot Removl. Computer Aided Geometric Design. 13 (1996). 851-87. [5] Schumker, L., On Shpe-Preserving Qudrtic Spline Interpoltion, SIAM Journl of Numericl Anlysis. 0 (1983), 854-864. [6] Stewrt, J., Clculus: Concepts nd Contexts, Brooks/Cole, 001. Deprtment of Mthemtics & Computer Science Smford University 800 Lkeshore Drive Birminghm, AL 359 ssstnle@smford.edu (05)76-009 fx: (05)76-471