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Trisection Algorithms Presented by Namir Shammas 1

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To Joe Horn, the “pinball wizard” or number rounding! Dedication 2

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“The much better is also the enemy of the good” -- Namir My better Creed 3

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In your dreams!! 4

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Trisection Method Each iteration divide the root-bracketing interval [A, B] into three parts, instead of two as does the Bisection. Chooses the first point X 1 within the interval [A, B] closest to the end point A, or B, that has the smallest absolute function value (call this point Z). If f(X 1 ) and f(Z) have opposite signs, the iteration has finished its task. If f(X 1 ) and f(Z) have the same signs, we calculate X 2 which closer to the other interval end point (call it Y). 5

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Trisection Method (cont.) Determines whether the interval [X 1, X 2 ] or [X 2, Y] is the new root-bracketing interval. 6

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Trisection Plus Method Adds inverse linear interpolation to the Trisection method. Set Fa=f(A) and Fb=f(B) and start iterations as shown next until [A, B]<tolerance. If |Fa| < |Fb| calculate X 1 = A + (B - A) / 3 and evaluate Fx1=f(X 1 ). This is case 1 where [A, X 1 ] has the root. 7

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Trisection Plus Method If Fx1 * Fa < 0 Calculate X3 as the inverse linear interpolation using [A, Fa] and [X 1, Fx1]. Also calculate Fx3=f(X 3 ). If Fa * Fx3 < 0 then set point B = point X 3, else set point A = point X 3 and point B = point X 1. 8

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Trisection Plus Method (cont.) 9

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Trisection Plus Method (cont.) If Fx1 * Fa > 0 Calculate X 2 = A + 2 * (B - A) / 3. Also calculate Fx2=f(X 2 ). This is case 2 where [X 1, X 2 ] has the root. If Fx1 * Fx2 < 0 Calculate X 3 as the inverse linear interpolation using [X 1, Fx1] and [X 2, Fx2]. Also calculate Fx3=f(X 3 ). If Fx1 * Fx3 < 0 then set point A=point X 1 and point B=point X 3, else set A=point X 3 and point B=point X 2. 11

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Trisection Plus Method (cont.) 12

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Trisection Plus Method (cont.) If Fx1 * Fx2 > 0 Calculate X 3 as the inverse linear interpolation using [X 2, Fx2] and [B, Fb]. Also calculate Fx3=f(X 3 ). If Fx2 * Fx3 < 0 then set point A = point X 2 and point B = point X 3, else set point A = point 3. If |Fa| > |Fb| we deal with symmetrical 3 cases as above. 13

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Trisection Plus Method (cont.) 14

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Trisection Plus Method (cont.) 15

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Trisection Plus Method (cont.) Included in HHC material are HP Prime listings for the Bisection Plus, Trisection, and Trisection Plus algorithms. Also included are version 2 of the above listings. Version 2 of the listings store, in matrix M0, the values for [A, B] along with X 1, X 2, and X 3 (in case of the Trisection Plus) for each iterations. Viewing matrix M0 allows you to inspect the programs of each iteration. 16

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Trisection Plus Method (cont.) 17

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Trisection Plus Method (cont.) Function[A, B] Toler / FxToler RootIterationsNum Fx Calls Exp(X) – X^3[1, 2] 1E–10 1E–7 1.857183 Bisec= 24 Trisec= 16 Trisec+ = 4 Newton= 7 Bisec= 26 Trisec = 24 Trisec+= 10 Newton= 14 Exp(X) –3*X^2[3, 4] 1E–10 1E–7 3.73307 Bisec= 26 Trisec= 17 Trisec+ = 5 Newton= 7 Bisec= 28 Trisec= 24 Trisec+ = 12 Newton= 14 Cos(X) – X[0, 1] 1E–10 1E–7 0.73908 Bisec= 23 Trisec= 14 Trisec+ = 4 Newton= 5 Bisec= 25 Trisec= 21 Trisec+ = 10 Newton= 10 (X–1.234) * (X–5.678) * (X+12.345)[5, 6] 1E–10 1E–7 5.678 Bisec= 28 Trisec= 18 Trisec+ = 4 Newton=6 Bisec= 30 Trisec= 23 Trisec+ = 10 Newton=12 (X–1.234) * (X–5.678) * (X+12.345)[1, 2] 1E–10 1E–7 1.234 Bisec= 28 Trisec= 17 Trisec+ = 4 Newton= 5 Bisec= 30 Trisec= 23 Trisec+ = 10 Newton= 10 (X–1.234) * (X–5.678) * (X+12.345)[5,11] 1E–10 1E–7 5.678 Bisec= 29 Trisec= 19 Trisec+ = 6 Newton= 7 Bisec= 31 Trisec= 32 Trisec+ = 14 Newton= 14 (X–1.234) * (X–5.678) * (X+12.345)[–8, –15]1E–10 1E–7 –12.345Bisec= 33 Trisec= 20 Trisec+ = 5 Newton= 6 Bisec= 35 Trisec= 30 Trisec+ = 13 Newton= 12 18

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Thank You!! 19

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