XY

22fd2807-0f2d-4dcc-b4ab-39558392fdd4 56 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

XY

XY 3 1.5 3.05 1.502 3.1 1.508 3.15 1.517 3.2 1.53 3.25 1.547 3.3 1.568 3.35 1.592 3.4 1.62 3.45 1.652 3.5 1.688 3.55 1.727 3.6 1.77 3.65 1.817 3.7 1.868 3.75 1.922 3.8 1.98 3.85 2.042 3.9 2.108 3.95 2.177 4 2.25 4.05 2.326 4.1 2.402 4.15 2.481 4.2 2.56 4.25 2.641 4.3 2.722 4.35 2.806 4.4 2.89 4.45 2.976 4.5 3.062 4.55 3.151 4.6 3.24 4.65 3.331 4.7 3.422 4.75 3.516 4.8 3.61 4.85 3.706 4.9 3.802 4.95 3.901 54 5.05 4.1 5.1 4.2 5.15 4.3 5.2 4.4 5.25 4.5 5.3 4.6 5.35 4.7 5.4 4.8 5.45 4.9 5.5 5 5.55 5.1 5.6 5.2 5.65 5.3 5.7 5.4 5.75 5.5 5.8 5.6 5.85 5.7 5.9 5.8 5.95 5.9 66 6.05 6.094 6.1 6.175 6.15 6.244 6.2 6.3 6.25 6.344 6.3 6.375 6.35 6.394 6.4 6.4 6.45 6.394 6.5 6.375 6.55 6.344 6.6 6.3 6.65 6.244 6.7 6.175 6.75 6.094 6.8 6 6.85 5.894 6.9 5.775 6.95 5.644 7 5.5 7.05 5.355 7.1 5.22 7.15 5.095 7.2 4.98 7.25 4.875 7.3 4.78 7.35 4.695 7.4 4.62 7.45 4.555 7.5 4.5 7.55 4.455 7.6 4.42 7.65 4.395 7.7 4.38 7.75 4.375 7.8 4.38 7.85 4.395 7.9 4.42 7.95 4.455 8 4.5 8.05 4.542 8.1 4.567 8.15 4.577 8.2 4.57 8.25 4.547 8.3 4.508 8.35 4.452 8.4 4.38 8.45 4.292 8.5 4.188 8.55 4.067 8.6 3.93 8.65 3.777 8.7 3.608 8.75 3.422 8.8 3.22 8.85 3.002 8.9 2.768 8.95 2.517 9 2.25 9.05 1.982 9.1 1.728 9.15 1.487 9.2 1.26 9.25 1.047 9.3 0.848 9.35 0.662 9.4 0.49 9.45 0.332 9.5 0.188 9.55 0.057 9.6 0.06 - 9.65 0.163 - 9.7 0.253 - 9.75 0.328 - 9.8 0.39 - 9.85 0.438 - 9.9 0.473 - 9.95 0.493 - 10 0.5 - 1412 mat :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

XY rows 141

X XY 1 col : Y XY 2 col : 12 mat x s X Y x cinterp : x c 1 X Min x ≤ (x X Max ≤ (& cases : 31 mat

◄ In this example, though entrance/departure suggest linear ainterp(x), cinterp(x) does same fit. However, both are strictly limited.

Example 1 ... strickly limited

x s x c * X Min 4 X Min 4 - 2 2 mat X Max 4 X Max 4 - 2 2 mat 3 1 sys

data 0 0.00794 0.1 0.012 0.2 0.022 0.3 0.048 0.4 0.167 0.5 1 0.6 0.167 0.7 0.048 0.8 0.022 0.9 0.012 1 0.00794 112 mat :

X data 1 col : Y data 2 col : 12 mat support data . 14 black plot : 4 appVersion 0.98.6179.21440 ≡ 31 mat

x Akima X Y x ainterp : x cubic X Y x cinterp : x λ 10 x ≤ (x 1 ≤ (& cases : 31 sys

Example 2 ... speaks by itself

x Akima x λ * x cubic x λ * support 3 1 sys

Enveloppe 15.75 18.47 11.85 9.66 9.8 6.49 7.25 3.21 5.35 1.605 3.6 0.52 62 mat :

vx Enveloppe 1 col : vy Enveloppe 2 col : 12 mat x Env vx vy x ainterp : x env vx vy x cinterp : x c 1 2.4165 x ≤ (x 18 ≤ (& cases : E Enveloppe . 14 black plot : 51 mat

x Env x 03 solve 2.4165

Example 3 ... speaks by itself

x Env x c * x env x c * E 3 1 sys

XY 01 0.5 21 3.1 1.5 2.9 2 1.7 2.5 2.8 3642516171112 mat :

Force the departure [1..9,10,11] XY XY 111 range 12 range el :

vx XY 1 col : vy XY 2 col : 12 mat x Lagrange vx vy x ainterp : x lagrange vx vy x cinterp : x c 11 - x ≤ (x 8 ≤ (& cases : L XY . 14 black plot : 51 mat

Example 4 ... speaks by itself

x Lagrange x c * x lagrange x c * 6 3 6 1 - 2 2 mat L 4 1 sys

The phenomenon is known to depart "flat". ainterp(x) best faithfull entrance/departure.

In applied mathematics, an Akima spline is a type of non-smoothing spline that gives good fits to curves where the second derivative is rapidly varying. The Akima spline was published by Hiroshi Akima in 1970.

XY 1 0.5 1.5 1.25 1.75 3 2.25 3.25 3.5 34 0.75 5 0.5 6 0.25 82 mat :

X XY 1 col : Y XY 2 col : 12 mat x Akima X Y x ainterp : x Cubic X Y x cinterp : x c 1 0.75 x ≤ (x 7 ≤ (& cases : Ω XY . 14 black plot : 51 mat

Example 5 ... speaks by itself

x Akima x c * x Cubic x c * Ω 3 1 sys

user 2 :

Example 5 ... rescue entrance

x Akima x c * 1 user 5 user 2 2 mat Ω 1 0.5 o 40 red 1 5 mat 4 1 sys

x Akima x c * user - x 15 solve 1.5975 3.7253 21 mat

In this demo, you may have to rescue the entrance from a 3 points iterpolation. Hermite cubic 3 points offered below.

Hermite cubic entrance rescueing Akima

XY 1 0.5 1.5 1.25 1.75 3 2.25 3.25 3.5 34 0.75 5 0.5 6 0.25 82 mat :

XoYo XY 13 range 12 range el :

Xo XoYo 1 col : Yo XoYo 2 col : 12 mat

XoYo 1 0.5 1.5 1.25 1.75 332 mat

r 2 : X Xo r RepEnd : Y Yo r RepEnd : 31 sys

populate reasonably for export t 01 0.025 range :

Hermite interpolation Utilities

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

z n RepEnd m z rows : i 1 n range w i el z 1 el : for i n 1 + (m n + (range w i el z i n - el : for i m n + 1 + (m 2 n * + (range w i el z m el : for w 51 line :

data Strip nest stripped in 'R' u data 1 col : v data 2 col : k 0 : i 1 data rows 1 - range v i 1 + el v i el ≤ (v i 1 + el v i el ≥ (¤ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

M Unwrap V M 1 col : i 2 M cols range V V M i col stack : for V 31 line :

x c 10 x ≤ (x 1 ≤ (& cases :

Hermite interpolate t P 16 / t 3^- 3 t 2^* + 2 t * - 3 t 3^* 6 t 2^* - 3 t * - 6 + 3 t 3^* - 3 t 2^* + 6 t * + t 3^t - 41 mat * :

Hermite basis cubics 0...1

x P 1 el x c * x P 2 el x c * x P 3 el x c * x P 4 el x c * 4 1 sys

t j XP t P 1 el X j el * t P 2 el X j 1 + el * + t P 3 el X j 2 + el * + t P 4 el X j 3 + el * + : t j YP t P 1 el Y j el * t P 2 el Y j 1 + el * + t P 3 el Y j 2 + el * + t P 4 el Y j 3 + el * + : 21 sys

U j 1 X rows r 1 + (- (range : ct 1 : j j i 1 t rows range Hx i ct el t i el j XP eval : for ct ct 1 + : 21 line for Hx 41 line :

V j 1 X rows r 1 + (- (range : ct 1 : j j i 1 t rows range Hy i ct el t i el j YP eval : for ct ct 1 + : 21 line for Hy 41 line :

collect the project xy U Unwrap V Unwrap augment : contour xy Strip : contour contour t rows contour rows t rows 2 - (- (range 12 range el : 31 sys

contour 11 mat 1 0.5 1.0095 0.5086 1.0194 0.5176 1.0295 0.5271 1.0399 0.5371 1.0505 0.5476 1.0615 0.5586 1.0726 0.57 1.084 0.582 1.0956 0.5945 1.1074 0.6074 1.1194 0.6209 1.1316 0.6349 1.144 0.6494 1.1565 0.6644 1.1692 0.6799 1.182 0.696 1.1949 0.7126 1.208 0.7297 1.2211 0.7474 1.2344 0.7656 1.2477 0.7844 1.2611 0.8037 1.2745 0.8236 1.288 0.844 1.3015 0.865 1.315 0.8865 1.3286 0.9087 1.3421 0.9314 1.3556 0.9547 1.3691 0.9785 1.3826 1.003 1.396 1.028 1.4093 1.0536 1.4226 1.0799 1.4358 1.1067 1.4489 1.1341 1.4618 1.1622 1.4747 1.1908 1.4874 1.2201 1.5 1.25 1.5093 1.293 1.5184 1.3366 1.5274 1.3807 1.5362 1.4254 1.5449 1.4705 1.5534 1.516 1.5618 1.5618 1.57 1.608 1.578 1.6545 1.5859 1.7012 1.5937 1.7481 1.6012 1.7951 1.6087 1.8423 1.6159 1.8895 1.623 1.9368 1.63 1.984 1.6368 2.0312 1.6434 2.0782 1.6499 2.1252 1.6562 2.1719 1.6624 2.2184 1.6684 2.2646 1.6743 2.3105 1.68 2.356 1.6855 2.4011 1.6909 2.4458 1.6962 2.49 1.7012 2.5336 1.7062 2.5767 1.7109 2.6191 1.7155 2.6609 1.72 2.702 1.7243 2.7423 1.7284 2.7819 1.7324 2.8206 1.7362 2.8584 1.7399 2.8953 1.7434 2.9312 1.7468 2.9661 1.75 3812 mat 11 mat

a b c spec . 12 black 13 mat : data contour a b c spec pts : XoYo XY + 10 black plot : 31 line

vx contour 1 col : vy contour 2 col : 12 mat x f vx vy x ainterp : x λH 1 Xo 1 el x ≤ (x Xo 2 el ≤ (& cases : x λA 1 XY 21 el x ≤ (x XY 81 el ≤ (& cases : x Akima XY 1 col XY 2 col x ainterp : 51 mat

contour contour rows row 1.75 312 mat

L 0.65 : H 1.5 : N 20 : 13 mat xH L H L H L - N / (+ range : 21 mat Lo 1.5 : Ho 7 : N# 100 : 13 mat xA Lo Ho Lo Ho Lo - N# / (+ range : 21 mat 21 sys

blue Hermite entrance ... red Akima suite

x f x λH * x Akima x λA * XoYo 3 1 sys

i 1 xH rows range Hd i el xH i el f eval : for i 1 xA rows range Ad i el xA i el Akima eval : for 21 line

Global xH xA stack Hd Ad stack augment :

Duplicate @ junction ... ignored.

Global L 1 L 0.5 - 2 2 mat XoYo 3 1 sys

Refit and solve ... ►

solve4U  ▼ U 2 :

X Global 1 col : Y Global 2 col : 12 mat x refit X Y x ainterp : x λ 1 0.65 x ≤ (x 7 ≤ (& cases : 31 mat

x refit U - 0 ≡ x 0.65 8 solve 1.597 3.725 21 mat

Duplicate @ junction ... ignored.

0.65 U 6.5 U 2 2 mat x refit x λ * XoYo 3 1 sys

NOT sanity checked x refit x 16 int 8.5

data

data 0.65 0.319 0.692 0.327 0.735 0.339 0.778 0.355 0.82 0.374 0.862 0.398 0.905 0.425 0.948 0.456 0.99 0.491 1.032 0.53 1.075 0.573 1.118 0.619 1.16 0.669 1.202 0.722 1.245 0.781 1.288 0.843 1.33 0.911 1.372 0.985 1.415 1.065 1.458 1.153 1.5 1.25 1.555 1.628 1.61 2.113 1.665 2.584 1.72 2.918 1.775 3.03 1.83 3.086 1.885 3.132 1.94 3.167 1.995 3.195 2.05 3.216 2.105 3.23 2.16 3.24 2.215 3.247 2.27 3.252 2.325 3.256 2.38 3.26 2.435 3.265 2.49 3.268 2.545 3.271 2.6 3.273 2.655 3.275 2.71 3.274 2.765 3.272 2.82 3.269 2.875 3.264 2.93 3.256 2.985 3.246 3.04 3.234 3.095 3.218 3.15 3.2 3.205 3.179 3.26 3.154 3.315 3.125 3.37 3.093 3.425 3.056 3.48 3.016 3.535 2.945 3.59 2.762 3.645 2.49 3.7 2.162 3.755 1.808 3.81 1.46 3.865 1.15 3.92 0.91 3.975 0.771 4.03 0.742 4.085 0.729 4.14 0.715 4.195 0.701 4.25 0.688 4.305 0.674 4.36 0.66 4.415 0.646 4.47 0.632 4.525 0.619 4.58 0.605 4.635 0.591 4.69 0.578 4.745 0.564 4.8 0.55 4.855 0.536 4.91 0.522 4.965 0.509 5.02 0.495 5.075 0.481 5.13 0.468 5.185 0.454 5.24 0.44 5.295 0.426 5.35 0.412 5.405 0.399 5.46 0.385 5.515 0.371 5.57 0.358 5.625 0.344 5.68 0.33 5.735 0.316 5.79 0.302 5.845 0.289 5.9 0.275 5.955 0.261 6.01 0.248 6.065 0.234 6.12 0.22 6.175 0.206 6.23 0.193 6.285 0.179 6.34 0.165 6.395 0.151 6.45 0.138 6.505 0.124 6.56 0.11 6.615 0.096 6.67 0.083 6.725 0.069 6.78 0.055 6.835 0.041 6.89 0.028 6.945 0.014 701212 mat :

data rows 121

Scalar derivative of a data set

data Deriv data Derivative BackSubstitution X data 1 col : Y data 2 col : 12 mat n data 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 el : X 1 deriv * augment 91 line :

D data Deriv : Xd D 1 col : Yd D 2 col : 12 mat x d Xd Yd x ainterp : x λd 1 0.65 x ≤ (x 6 ≤ (& cases : 41 sys

data x d x λd * 2.67 2 2.67 2 - 2 2 mat 3 1 sys

Implicit solve x d x 23 solve 2.6691

D Max D Min 21 mat 8.6909 6.3818 - 21 mat