A MIXED QUADRATURE FORMULA USING RULES OF LOWER ORDER

Transcription

1 Bulletin of the Marathwada Mathematical Society Vol.5, No., June 004, Pages 6-4 ABSTRACT A MIXED QUADRATURE FORMULA USING RULES OF LOWER ORDER Namita Das And Sudhir Kumar Pradhan P.G. Department of Mathematics Sambalpur University, Jyoti Vihar Burla, Sambalpur, Orissa, India. In this paper we have derived a mixed quadrature rule of precision seven using quadrature rules of lower precisions. Key words: symmetric, quadrature, Legendre polynomial 99 Mathematics subject classification: Primary, 6D6; secondary 6D99. INTRODUCTION The numerical methods for the approximate evalution of the real definite integrals of the form I (f) = b a f(x) dx () are broadly of two types, i.e. (i) Newton-Cotes-type and (ii) Gauss-type. In n-point Newt on-cotes-type methods, t he nodes are a (b a)k x k a, set of equidistant points x n k, k = 0,,..., n-, where,while in n-point Gauss-type methods the nodes are the zeros of the n th degree Legendre polynomial, which are irrational numbers in general. Among the above numerical integration methods, we find that computational efforts being equal, Gaussian-integration yields the most accurate results. However, Newton-Cotes-type methods are most suitable for hand calculation since the nodes and weights associated with the rules are simple rational numbers. Usually by increasing the value of n in both of the above

2 7 methods, we can optimise the accuracy of approximation of the definite integral in (). But in n-point Newton-Cotes-type methods, large value of n (n 8, n 9) is not recommended since as a consequence of the negativeness of some of the weights there may be a loss of significant digits in the result. In Gauss-type methods, the computational complexity for the evaluation of all the zeros of the n th degree Legendre polynomial increases for large n. Also in most of the cases either an appropriate derivative of the integrand does not exist or it is difficult to determine. Keeping these facts in view we desire to construct a symmetric quadrature rule of precision seven which is a linear combination of three other rules of equal precision three. Here we have considered the integrals of the type I(f) f(x) dx () instead of the integrals of the type in () in view of the fact that any two closed intervals in R are homoeomorphic to each other. The construction of the rule is outlined in the following section.. CONSTRUCTION OF THE RULE OF PRECISION SEVEN We choose Simpson s / rd rule Simpson s /8 rule f(x) dx R s (f) = [f (-) + 4 f (0) + f() ]. () f(x) dx R s (/8) (f) = [f (-) + f(- ) + f ( ) + f()], (4) 4 and the Gauss-Legendre two points rule f(x) dx R (f) f( ) f( ). Each of the rules (), (4) and (5), is of precision three. Let E s (f), E s(/8) (f) and E (f) denote the error in approximating the integral I (f) by the rules (), (4) and (5), respectively, Then I(f) = Rs(f) + Es(f) (6) (5)

3 8 and I(f) = Rs (/8) (f) + Es (/8) (f) (7) and I(f) = R (f) + E (f) (8) Now assuming f to be sufficiently differentiable in < x < we can express the errors associated with quadrature rules under reference as (8) (4) (6) 4 f (0) E (f) f (0) f (0).... s ! (6) (8) (4) 68 f (0) 608 f (0) E (f) f (0)... s(/8) ! 87 8! (6) (8) (4) 40 f (0) 6 f (0) E (f) f (0) ! 8 8! Now multiplying the equations (6), (7) and (8) by, -7 and 0 respectively and adding the results we obtain I(f) where R [R 5 S S 0R (f) E S 7R (f) ] 5 [E S E 7E R (f) R 0R 7R S 8 5 (9) S ( / ) S is the desired quadrature rule of precision seven for the approximate evaluation of (f), and the truncation error generated in this approximation is given by E (f) [E 0E 7E ] S 5 S (8) f ( 0)....( 0) The rule (9) may be called a mixed-type rule as it is constructed from different types of rules of the same precision. Remark. : The mixed quadrature rule R S (f) as described in (9) is a symmetric quadrature rule because R S 0R 7R 47 f (-) + 0 f = 8 - f f(0). ]

4 8 - f f f ().. ERROR ANALYSIS An asymptotic error estimate and an error bound of the rule (9) is given in theorem. and theorem., respectively. Theorem. : Let f(x) be a sufficiently differentiable function in the closed interval [-, ]. Then the error E S (f) associated with the rule R S (f) is given by (8) 64 f (0) E (f). S 85 8! () Proof : Follows from equation (0). Theorem. is given by where The bound for the truncation error E (f) = I(f) - R (f) S S E (f) 4M () S 5 M max x [,] f (5) (x). (4) Proof : E (f) f (η ), S 90 (4) E (f) f (η ), 5 (4) and E (f) f (η ), 405 (Refer to Conte and de Boor [].) η [,], η η [,], [,].

5 So E S (f) 0 [E (f) 0E 5 S (f) 7E 6 (4) (4) (4) f (η ) f (η ) f (η ) Let K = max f (4) (x) and k = min x x f (4) (x). As f 4 (x) is continuous and [-,] is compact, hence there exist points a and b in the interval [-,] such that K= f (4) (a) and k = f (4) (b). Thus E by S Hence 6 (4) (4) (4) (f) f (b) f (a) f (b) (4) (4) f (a) f (b) 5 (5) (a b)f ( ) for some ξ [,] 5 Mean value theorem, []. (f)] where E S (f) a b f ( 4M (5) ), 5 5 max M = f (5) (x). x [-,] 4. NUMERICAL VERIFICATION The approximate values of the integrals I (f) I (f) I (f) I (f) x e dx e x dx dx x sin x dx x

6 have been obtained by using the C-program given at the end of this section. In the program, R mixed (f) denotes the quadrature rule R The program evaluates simulataneously the values of the S (f). above integrals using the rules Rs(f), R, R s(/8) and R mixed (f). The approximate values of the integrals are shown in the Table and they are compared with the value V which is obtained by composite Simpson s / rd rule taking 500 sub-intervals. (For reference we have also given a C-program of composite Simpon s rule at the end of the section). In each case it is found that the approximation R mixed (f) is superior to the approximations R s (f), R (f), R s(/8) (f). Table- Quadratue Approximate values of rules I (f) I (f) I (f) I 4 (f) R s (f) R (f) R (f) R mixed (f) Value V

7 /* composite simpsons _rule */ # include < stdio.h > # include < math.h > # include < limits.h > # include < float.h > # include < conio.h > # define M 0 /* no. of sub intervals (even no.)*/ void main ( ) int a,b; double f; double simpson (int,int,int); clrscr ( ); print f ( input the limts of integration:a,b\n ); scan f ( % d % d,&a, &b); print f (The function in this programme is f(x) = exp(x)\n ); f = simpson (a, b, M); print f ( no of subintervals = % d and value of intetral = % 5.0f\ n,m,f); getch ( ); double simpson (int A, int B, int N) int I=; double funct ( double); double simp; double H, sum, x; float sum, sum; H = (B-A)/(double)N; sum = funct ((double) A)+funct ((double) B); sum = 0, 0; while (I < = N - ) x = A + I * H; sum + = funct (x); I + = ; I = ; sum = 0, 0;

8 while (I < = N - ) x = A + I * H; sum + = funct (x); + = ; simp = H * (sum * sum +.0 * sum )/.0; return(simp); double funct (double p) long double F; F = exp (p); return F; /* A mixed quadrature rule*/ # include < stdio.h > # include < math.h > # include < limits.h > # include < float.h > # include < conio.h > void main( ) double p, q, RS, R, RS_8, RMIXED; double funct (double); /* RS = value of the integral using Simpson s rule */ / *R = value of the integral using Gauss _ point rule*/ /* RS_8 = value of the integral using Simpson s _8th rule */ /* RMIXED = value of the integral using Mixed quadrature rule*/ clrscr( ); print f ( \the limits of integration is -to \n ); print f (\t (if necessary change the interval [a,b] to [-,].\n\n ); p = sqrt ((double)); q = (double); RS = (funct((double)-) + 4 * funct ((double)0) + funct((double))/.0; R = funct (-/p) + funct (/p); RS_8 = (funct((double)-) + * (funct(-/q) + funct(/q)) + funct ((double)))/.0; RMIXED = ( * RS + 0 * R - 7 * RS_8)/5.0;

9 4 print f ( \t \t \t f(x) = exp(x)\n ); print f ( \n\n ); print f ( \t Quadrature rules \t Approximate values of I(f) \n ); print f ( \n\ ); print f ( \t RS(f) \t \t \t % 6.f \n,rs); print f ( \t R \t \t \t % 6.f\n,R); print f ( \t RS_8 \t \t \t % 6.f\n,RS_8); print f ( \t RMIXED \t \t \t % 6.f\n,RMIXED); print f ( \t Exact value \t \t \n ); getch( ); double funct (double p) double F; F = exp(p); return F; REFERENCES Conte, S.D.and Boor, C.: Elementary Numerical Analysis, rd ed., McGraw Hill, Singapore, 98.. Atkinson, K.E.,: An Introduction to Numerical Analysis, nd ed., John Wiley and Sons, New York, 989.