integrate

integrate - Numerical Integration estimate A2 = h (f0 + f1...

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Numerical Integration estimate polynomial he l name A 2 = h 2 ( f 0 + f 1 )l i n e a r b - a - 1 12 f ± h 3 trapezoidal rule A 3 = h 3 ( f 0 +4 f 1 + f 2 )q u a d r a t i c ( b - a ) / 2 - 1 90 f (4) h 5 Simpson rule A 4 = 3 h 8 ( f 0 +3 f 1 +3 f 2 + f 3 )c u b i c ( b - a ) / 3 - 3 80 f (4) h 5 Table 1: Newton-Cotes quadrature rules. To verify these formulas, make a Taylor series expansion of f ( x )about x 0 : f ( x )= f 0 +( x - x 0 ) f ± 0 +( x - x 0 ) 2 f ± 0 / 2+( x - x 0 ) 3 f ± 0 / 3!+( x - x 0 ) 4 f (4) 0 / 4!+ ··· Then the exact integral I is I = ± f ( x ) dx = ² ( x - x 0 ) f 0 +( x - x 0 ) 2 f ± 0 / 2+( x - x 0 ) 3 f ± 0 / 3! + ( x - x 0 ) 4 f ± 0 / 4!+ ( x - x 0 ) 5 f (4) 0 / 5! + ··· ³ In Simpson’s rule, A 3 = h 3 ( f ( x 0 )+4 f ( x 0 + h )+ f ( x 0 +2 h )) 2 hf 0 +2 h 2 f ± 0 + 4 h 3 3 f ± 0 + 2 h 4 3 f ± 0 + 5 h 5 18 f (4) 0 From the formula for the exact integral (setting x = x 0 +2 h ), I 2 hf 0 +2 h 2 f ± 0 + 4 h 3 3 f ± 0 + 2 h 4 3 f ± 0 + 4 h 5 15 f (4) 0 and the local error is e l = I - A 3 ≈- h 5 90 f (4) 0
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Romberg Integration
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integrate - Numerical Integration estimate A2 = h (f0 + f1...

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