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Romberg_EI

Calculates Complete Elliptic Integral of the Second Kind Using Romberg's Method

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Dr. Ronald W. Satz - Show more from this author Transpower Corporation http://www.transpowercorp.com

  Function definition: (Copy & paste into FileMaker's Edit Custom Function window)

/* This function calculates the Complete Elliptical Integral of the Second Kind using Romberg's Method */ Let ( [ he = (theta2 - theta1)/32 ; y01 = EI ( ecc; theta1 ) ; y02 = EI ( ecc; theta1 + he ) ; y03 = EI ( ecc; theta1 + 2 * he ) ; y04 = EI ( ecc; theta1 + 3 * he ) ; y05 = EI ( ecc; theta1 + 4 * he ) ; y06 = EI ( ecc; theta1 + 5 * he ) ; y07 = EI ( ecc; theta1 + 6 * he ) ; y08 = EI ( ecc; theta1 + 7 * he ) ; y09 = EI ( ecc; theta1 + 8 * he ) ; y10 = EI ( ecc; theta1 + 9 * he ) ; y11 = EI ( ecc; theta1 + 10 * he ) ; y12 = EI ( ecc; theta1 + 11 * he ) ; y13 = EI ( ecc; theta1 + 12 * he ) ; y14 = EI ( ecc; theta1 + 13 * he ) ; y15 = EI ( ecc; theta1 + 14 * he ) ; y16 = EI ( ecc; theta1 + 15 * he ) ; y17 = EI ( ecc; theta1 + 16 * he ) ; y18 = EI ( ecc; theta1 + 17 * he ) ; y19 = EI ( ecc; theta1 + 18 * he ) ; y20 = EI ( ecc; theta1 + 19 * he ) ; y21 = EI ( ecc; theta1 + 20 * he ) ; y22 = EI ( ecc; theta1 + 21 * he ) ; y23 = EI ( ecc; theta1 + 22 * he ) ; y24 = EI ( ecc; theta1 + 23 * he ) ; y25 = EI ( ecc; theta1 + 24 * he ) ; y26 = EI ( ecc; theta1 + 25 * he ) ; y27 = EI ( ecc; theta1 + 26 * he ) ; y28 = EI ( ecc; theta1 + 27 * he ) ; y29 = EI ( ecc; theta1 + 28 * he ) ; y30 = EI ( ecc; theta1 + 29 * he ) ; y31 = EI ( ecc; theta1 + 30 * he ) ; y32 = EI ( ecc; theta1 + 31 * he ) ; y33 = EI ( ecc; theta1 + 32 * he ) ; ha = (theta2 - theta1) / 2 ; aa = .5 * ha * ( y01 + 2 * y17 +y33 ) ; hb = ( theta2 - theta1 ) / 4 ; ab = .5 * hb * ( y01 + 2 * y09 + 2 * y17 + 2 * y25 + y33 ) ; hc = ( theta2 - theta1 ) / 8 ; ac = .5 * hc * ( y01 + 2 * y05 + 2 * y09 + 2 * y13 + 2 * y17 + 2 * y21 + 2 * y25 + 2 * y29 + y33) ; hd = ( theta2 - theta1 ) / 16 ; ad = .5 * hd * (y01 + 2 * y03 + 2 * y05 + 2 * y07 + 2 * y09 + 2 * y11 + 2 * y13 + 2 * y15 + 2 * y17 + 2 * y19 + 2 * y21 + 2 * y23 + 2 * y25 + 2 * y27 + 2 * y29 + 2 * y31 + y33 ) ; ae = .5 * he * (y01 + 2 * y02 + 2 * y03 + 2 * y04 + 2 * y05 + 2 * y06 + 2 * y07 + 2 * y08 + 2 * y09 + 2 * y10 + 2 * y11 + 2 * y12 + 2 * y13 + 2 * y14 + 2 * y15 + 2 * y16 + 2 * y17 + 2 * y18 + 2 * y19 + 2 * y20 + 2 * y21 + 2 * y22 + 2 * y23 + 2 * y24 + 2 * y25 + 2 * y26 + 2 * y27 + 2 * y28 + 2 * y29 + 2 * y30 + 2 * y31 + 2 * y32 + y33 ) ; ba = ab + (1/3) * (ab - aa) ; bb = ac + (1/3) * (ac - ab) ; bc = ad + (1/3) * (ad - ac) ; bd = ae + (1/3) * (ae - ad) ; ca = bb + (1/15) * (bb - ba) ; cb = bc + (1/15) * (bc - bb) ; cc = bd + (1/15) * (bd - bc) ; da = cb + (1/255) * (cb - ca) ; db = cc + (1/255) * (cc - cb) ]; db + (1/65535) * (db - da) ) Click to copy

This function calls EI (ecc ; theta ) (which is equal to Sqrt ( 1- (ecc ^ 2 ) * ( ( Sin ( theta ) ) ^ 2 ) ) ) (where ecc = ellipse eccentricity and theta = angle in radians) to compute the integrand of the Complete Elliptic Integral of the Second kind at 33 points. It then takes these values to compute the Romberg value for the Complete Elliptic Integral of the Second kind--for instance, the perimeter of an elliptic orbit (which would be 4 x a x the value computed, where a = semi-major axis of the ellipse).

 

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