Transcription of 4.4 Composite Numerical Integration
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4.4 Composite Numerical Integration
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).
4.4 Composite Numerical Integration . Motivation: 1) on large interval, use low order -Cotes formulas Newton are not accurate. 2) on large interval, interpolation using high degree polynomial is unsuitable because of oscillatory nature of high degree polynomials.
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