DeBoor Colibri Triplet

17b4eb89-9c30-4532-bd79-71f794c146b2 11 4 5 decimal

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The 3 most common Cumulative Integrators . . . [1] FD ... [2] RK's ... [3] ∫f(x)

[1] Finite Differences is very accurate as some kind of yardstick for most, if not all applications.

[2] must be tested vs FD to insure long range convergence.

[3] works well, but not for all demand s ... this particular parametric f(x,α) ... unexplained failure.

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α 1523043253 - 27354.7 - 43256.75 25388.3 - 40061 mat :

Fréchet: extreme value-II x α f α 1 el α 2 el x α 4 el - (/ (α 5 el^* α 2 el x α 4 el - (/ (α 3 el^- exp * α 6 el + :

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

data U 013 0.1 range : i 1 U rows range vy i el U i el α f eval : for U vy augment 31 line :

Finite differences Cumulative Integral

data Σ data 2 1 sys

x Y D x α f : sol 0013130 x Y D rkfixed : 21 sys

sol data Σ 2 1 sys

Simpson fails unexpectedly ? Σ U 013 0.1 range : i 1 U rows range Cumul i el x α f x U 1 el U i el int eval : for U Cumul augment 31 line :

Σ #

ε 1015 -^:

x integral x ln x int maple : x 1 - x ln + (*

Σ U ε 5 0.1 range : i 1 U rows range Cumul i el x ln x U 1 el U i el int eval : for U Cumul augment 31 line :

Σ x integral 2 1 sys

acc 1000 ≡

DeBoor Colibri Triplet

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

data 3 1.5 0 3.05 1.5019 0.0012 3.1 1.5075 0.005 3.15 1.5169 0.0112 3.2 1.53 0.02 3.25 1.5469 0.0312 3.3 1.5675 0.045 3.35 1.5919 0.0612 3.4 1.62 0.08 3.45 1.6519 0.1013 3.5 1.6875 0.125 3.55 1.7269 0.1512 3.6 1.77 0.18 3.65 1.8169 0.2112 3.7 1.8675 0.245 3.75 1.9219 0.2812 3.8 1.98 0.32 3.85 2.0419 0.3612 3.9 2.1075 0.405 3.95 2.1769 0.4512 4 2.25 0.5 4.05 2.3256 0.5475 4.1 2.4025 0.59 4.15 2.4806 0.6275 4.2 2.56 0.66 4.25 2.6406 0.6875 4.3 2.7225 0.71 4.35 2.8056 0.7275 4.4 2.89 0.74 4.45 2.9756 0.7475 4.5 3.0625 0.75 4.55 3.1506 0.7475 4.6 3.24 0.74 4.65 3.3306 0.7275 4.7 3.4225 0.71 4.75 3.5156 0.6875 4.8 3.61 0.66 4.85 3.7056 0.6275 4.9 3.8025 0.59 4.95 3.9006 0.5475 54 0.5 5.05 4.1 0.4563 5.1 4.2 0.425 5.15 4.3 0.4063 5.2 4.4 0.4 5.25 4.5 0.4062 5.3 4.6 0.425 5.35 4.7 0.4562 5.4 4.8 0.5 5.45 4.9 0.5562 5.5 5 0.625 5.55 5.1 0.7062 5.6 5.2 0.8 5.65 5.3 0.9062 5.7 5.4 1.025 5.75 5.5 1.1562 5.8 5.6 1.3 5.85 5.7 1.4562 5.9 5.8 1.625 5.95 5.9 1.8062 662 6.05 6.0937 2.1925 6.1 6.175 2.37 6.15 6.2437 2.5325 6.2 6.3 2.68 6.25 6.3437 2.8125 6.3 6.375 2.93 6.35 6.3937 3.0325 6.4 6.4 3.12 6.45 6.3938 3.1925 6.5 6.375 3.25 6.55 6.3438 3.2925 6.6 6.3 3.32 6.65 6.2438 3.3325 6.7 6.175 3.33 6.75 6.0938 3.3125 6.8 6 3.28 6.85 5.8938 3.2325 6.9 5.775 3.17 6.95 5.6438 3.0925 7 5.5 3 7.05 5.355 2.905 7.1 5.22 2.82 7.15 5.095 2.745 7.2 4.98 2.68 7.25 4.875 2.625 7.3 4.78 2.58 7.35 4.695 2.545 7.4 4.62 2.52 7.45 4.555 2.505 7.5 4.5 2.5 7.55 4.455 2.505 7.6 4.42 2.52 7.65 4.395 2.545 7.7 4.38 2.58 7.75 4.375 2.625 7.8 4.38 2.68 7.85 4.395 2.745 7.9 4.42 2.82 7.95 4.455 2.905 8 4.5 3 8.05 4.5419 3.1 8.1 4.5675 3.2 8.15 4.5769 3.3 8.2 4.57 3.4 8.25 4.5469 3.5 8.3 4.5075 3.6 8.35 4.4519 3.7 8.4 4.38 3.8 8.45 4.2919 3.9 8.5 4.1875 4 8.55 4.0669 4.1 8.6 3.93 4.2 8.65 3.7769 4.3 8.7 3.6075 4.4 8.75 3.4219 4.5 8.8 3.22 4.6 8.85 3.0019 4.7 8.9 2.7675 4.8 8.95 2.5169 4.9 9 2.25 5 9.05 1.9819 5.0975 9.1 1.7275 5.19 9.15 1.4869 5.2775 9.2 1.26 5.36 9.25 1.0469 5.4375 9.3 0.8475 5.51 9.35 0.6619 5.5775 9.4 0.49 5.64 9.45 0.3319 5.6975 9.5 0.1875 5.75 9.55 0.0569 5.7975 9.6 0.06 - 5.84 9.65 0.1631 - 5.8775 9.7 0.2525 - 5.91 9.75 0.3281 - 5.9375 9.8 0.39 - 5.96 9.85 0.4381 - 5.9775 9.9 0.4725 - 5.99 9.95 0.4931 - 5.9975 10 0.5 - 61413 mat :

Cntrl 1.5 0 1.5 03150744254 0.5 - 6 0.5 - 692 mat :

DeBoor data 2 col data 3 col augment data 1 col data 2 col augment data 1 col data 3 col augment 31 sys :

XY axes system x y XY x y / - :

DeBoor XY 2 1 sys

control polygon X 1.5 35745 0.5 - 71 mat : Y 010424671 mat : 12 mat

Points X Y . 14 black plotG :

De Boor parametric order 2 wrt Smath ... kind of Convex Hull

X Y augment DeBoor 1 el Points 3 1 sys Calculate components Sx,Sy

Finite differences derivative X data 1 col : Y data 3 col : 12 mat n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - (range deriv i el 0.5 Y i 1 - el Y i el - X i 1 - el X i el - (/ Y i el Y i 1 - el - X i el X i 1 - el - (/ + (* : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows 1 - el : D X deriv augment : Arc integrand .................... xx D 1 col : δx D 2 col : 12 mat i 1 δx rows range yy i el 1 δx i el (2^+ sqrt eval : 11 line for Arc xx yy augment : Arc length ....................... vx Arc 1 col : vy Arc 2 col : 12 mat S 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - abs : S i el S i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for Σ vx S augment : 181 line

D Arc Σ

Each component Σx, Σy is calculated individually ... collected in their respective vector below.

Sx, Sy

Σx 0 0.05 0.1002 0.1508 0.2021 0.2543 0.3077 0.3626 0.4191 0.4774 0.5378 0.6003 0.6651 0.7324 0.8023 0.8748 0.95 1.0282 1.1092 1.1932 1.2802 1.3698 1.461 1.5533 1.6465 1.7409 1.8363 1.9327 2.0303 2.1289 2.2286 2.3294 2.4312 2.5342 2.6383 2.7434 2.8497 2.957 3.0655 3.175 3.2857 3.3973 3.5091 3.6209 3.7327 3.8445 3.9563 4.0681 4.1799 4.2917 4.4035 4.5153 4.6271 4.7389 4.8507 4.9625 5.0743 5.1861 5.2979 5.4097 5.5215 5.6305 5.7314 5.8216 5.9017 5.9725 6.0352 6.0914 6.1433 6.1937 6.2456 6.3018 6.3645 6.4353 6.5155 6.6057 6.7065 6.8183 6.9415 7.0761 7.2225 7.3753 7.5239 7.6632 7.7932 7.9141 8.0259 8.1289 8.2233 8.3094 8.3875 8.4583 8.5225 8.5809 8.635 8.6862 8.7365 8.7877 8.8417 8.9002 8.9644 9.0306 9.0913 9.1448 9.1955 9.2483 9.3077 9.3769 9.4581 9.5525 9.661 9.7842 9.9223 10.0757 10.2446 10.429 10.6291 10.845 11.0767 11.3243 11.5878 11.86 12.126 12.3785 12.6175 12.8431 13.0554 13.2542 13.4399 13.6123 13.7716 13.9179 14.0514 14.1723 14.2808 14.3773 14.4624 14.5369 14.602 14.6593 14.7134 1411 mat :

Σy 0 0.05 0.1001 0.1504 0.2009 0.2519 0.3035 0.3557 0.4087 0.4626 0.5174 0.5733 0.6304 0.6887 0.7483 0.8094 0.8719 0.9359 1.0016 1.0688 1.1378 1.2072 1.2745 1.3386 1.3996 1.458 1.5139 1.5678 1.6201 1.6711 1.7215 1.7715 1.8218 1.8729 1.9251 1.979 2.035 2.0933 2.1544 2.2185 2.2858 2.3534 2.4161 2.4723 2.5242 2.5746 2.6265 2.6827 2.7454 2.8162 2.8964 2.9866 3.0874 3.1992 3.3224 3.457 3.6033 3.7615 3.9315 4.1135 4.3076 4.5071 4.6987 4.8759 5.0388 5.1875 5.3222 5.443 5.5505 5.6449 5.727 5.7979 5.8593 5.9136 5.9644 6.0159 6.0722 6.1365 6.211 6.2971 6.3958 6.5021 6.6051 6.6994 6.7855 6.8637 6.9345 6.9986 7.0571 7.1111 7.1624 7.2126 7.2638 7.3179 7.3764 7.4405 7.5113 7.5895 7.6755 7.7699 7.8729 7.9825 8.0943 8.2061 8.3179 8.4297 8.5415 8.6533 8.7651 8.8769 8.9887 9.1005 9.2123 9.3241 9.4359 9.5477 9.6595 9.7713 9.8831 9.9949 10.1068 10.2174 10.3248 10.4278 10.5264 10.6207 10.7109 10.7969 10.8789 10.9571 11.0314 11.1021 11.1694 11.2335 11.2946 11.3529 11.4089 11.4628 11.515 11.5661 11.6166 1411 mat :

vX data 1 col : i 1 vX rows range Σxy i el Σx i el 2^Σy i el 2^+ sqrt : for Arclength vX Σxy augment : 31 line

X Arclength 1 col : Y Arclength 2 col : x Φ X Y x ainterp : 31 line

P 2 π * : λ P P Φ augment : 21 line

interpolate P Φ 8.0026

Arclength XY λ 3 1 sys