Transcription of 4.4 Composite Numerical Integration
1 Composite Numerical Integration Motivation: 1) on large interval, use low order Newton-Cotes formulas are not accurate. 2) on large interval, interpolation using high degree polynomial is unsuitable because of oscillatory nature of high degree polynomials. Main idea: divide Integration interval [ , ] into subintervals and use simple Integration rule for each subinterval. Example 1. a) Use Simpson s rule to approximate 40. The exact value is b) Divide [0,4 ] into [0,1]+[1,2]+[2,3]+[3,4 ]. Use Simpson s rule to approximate 10, 21, 32 and 43. Then approximate 40 by adding approximations for 10, 21, 32 and 43. Compare with accurate value. Solution: a) =4 02. 40 23( 0+4 2+ 4)= Error= | | = b) 40= 10+ 21+ 32+ 43 ( 0+4 + 1)+ ( 1+4 + 2)+ ( 2+4 + 3)+ ( 3+4 + 4)= Error=| | = b) is much more accurate than a).
2 Composite Trapezoidal rule Let 2[ , ], = , and = + for =0, , . On each subinterval 1, , for for =1, , , apply Trapezoidal rule: Figure 1 Composite Trapezoidal Rule ( ) = 2 ( 0)+ ( 1) 312 ( 1) + 2 ( 1)+ ( 2) 312 ( 2) + + 2 ( 1)+ ( ) 312 ( ) = 2 ( )+2 1 =1+ ( ) 312 ( ) =1= 2 ( )+2 1 =1+ ( ) 12 2 ( ) Error, which can be simplified Theorem Let 2[ , ], = , and = + for each =0, , . There exists a ( , ) for which Composite Trapezoidal rule with its error term is ( ) = 2 ( )+2 1 =1+ ( ) 12 2 ( ) Error term Composite Simpson s rule Let 2[ , ], , = , and = + for =0.
3 On each consecutive pair of subintervals (for example [ 0, 2], [ 2, 4], and 2 2, 2 ) for each =1, , /2, apply a Simpson s rule: Figure 2 Composite Simpson's rule ( ) = ( ) 2 2 2 /2 =1= 3 2 2 +4 2 1 + 2 590 (4) /2 =1= 3 ( 0)+2 2 2 1 =1+4 2 1 2 =1+ ( ) 590 (4) 2 =1 Error, which can be simplified Theorem Let 4[ , ], , = , and = + for each =0, , . There exists a ( , ) for which Composite Simpson s rule with its error term is ( ) = 3 ( )+2 2 ( 2) 1 =1+4 2 1 ( 2) =1+ ( ) 180 4 (4)( ) Error Term Composite Midpoint rule Theorem Let 2[ , ], , = +2, and = +( +1) for each = 1, 0, , , +1.
4 There exists a ( , ) for which Composite Midpoint rule with its error term is ( ) =2 2 ( 2) =0 + 6 2 ( ) Figure 3 Composite Midpoint rule Exercise 13. Determine the values of and required to approximate 1 +4 20 to within 10 5 and compute the approximation. Use a. Composite Trapezoidal rule. b. Composite Simpson s rule. c. Composite Midpoint rule.
