Numerical Integration

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Simpson’s 1/3 Rule • Simpson’s Rule is the second of the Newton- Cotes closed integration formulas (n=2) using Lagrange Interpolating Polynomials f x 2 2 ∫ f x dx ≅ ∫ I = ( ) f ( x ) dx 2 n x ( x) x x 0 0 n n x − = ∑ f ( x ) ∏ − i i= 0 j= 0 j≠i x x x i © 2003-2006 Roberto Muscedere 18 j j
Simpson’s 1/3 Rule ( x − x1 )( x − x2) ( x − x0)( x − x2) f 2 ( x ) = f ( x 0 ) + f ( x 1 ) + ( x − x )( x − x ) ( x − x )( x − x ) 0 1 0 ( x − ( x − 2 2 x x )( x )( x − x1 ) − x ) 0 f ( x2 0 2 1 • After integration ti and algebraic manipulation: I h 3 0 1 + x − 2 x h = 0 2 ≅ [ f ( x ) + 4 f ( x ) + f ( x ) ] © 2003-2006 Roberto Muscedere 19 1 0 ) 2 1 2
Page 1 and 2: Numerical Integration • Newton-Co
Page 3 and 4: Trapezoidal Rule • Trapezoidal Ru
Page 5 and 6: Trapezoidal Rule • Same result if
Page 7 and 8: Trapezoidal Rule: Example f (0) = 0
Page 9 and 10: Trapezoidal Rule • Add a series o
Page 11 and 12: • Grouping the terms: Trapezoidal
Page 13 and 14: Trapezoidal Rule: Example f (0) = 0
Page 15 and 16: Trapezoidal Code (p.2) double Integ
Page 17: Trapezoidal Code -Output n=1, I=0.1
Page 21 and 22: Simpson’s 1/3 Rule: Example • N
Page 23 and 24: Simpson’s 1/3 Rule • Add a seri
Page 25 and 26: Simpson’s 1/3 Rule • Grouping t
Page 27 and 28: I f Simpson’s 1/3 Rule: Example (
Page 29 and 30: Simpson’s 1/3 Code (p.2) double I
Page 31 and 32: Simpson’s 1/3 Code (p.4) double F

Numerical Integration

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