<p> Hermite interpolation </p>

259ac2f7-98e3-46ac-b152-ea8b10d17b20 8 4 5 decimal

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

Parametric Rkadapt RKF45 ... wrt 'β'  ►  D(t,X,β)) β 0.5 0.5 21 mat : t X β D X 2 el β 1 el X 2 el * X 1 el (2^1 - (* - β 2 el (2^X 1 el * - 21 mat : t.max 20 : N 2000 : init 0 β 1 el - 21 mat : t.0 init 1 el : 14 mat res init t.0 t.max N t X β D Rkadapt : X res 1 col : Y res 2 col : Z res 3 col : 13 mat IntegralPath Y Z augment : 61 sys

IntegralPath init 1 el init 2 el + 300 red 1 5 mat 2 1 sys

data n Decimate collect every 'n' pairs range 1 data rows n range : k 1 : i range u k el data i 1 el : v k el data i 2 el : k k 1 + : 31 line for u v augment 31 line 31 line :

Hermite cubic is happy of sparse data... van IntegralPath 30 Decimate eval :

van rows 69

van

user original polygon Xo van 1 col : Yo van 2 col : 21 sys

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 :

sub 1 :

subspaces is '1' for interpolation

New Xo sub subspace Yo sub subspace augment : New Xo 1 el Yo 1 el 12 mat New stack : 21 line

isolate New New :

X New 1 col : Y New 2 col : 21 line

populate segments t 01 0.1 range :

<p> Hermite interpolation </p>

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 :

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

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

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XoYo Xo Yo augment : XoYo XoYo o 5 black plot : 21 mat

Ω XoYo :

region of interpolation j 47 :

X Ω j 1 el Ω j 1 + 1 el Ω j 2 + 1 el Ω j 3 + 1 el 41 mat eval : 0.4398 - 0.8204 - 1.2006 - 1.533 - 41 mat

Y Ω j 2 el Ω j 1 + 2 el Ω j 2 + 2 el Ω j 3 + 2 el 41 mat eval : 1.2697 - 1.3366 - 1.2569 - 1.0115 - 41 mat

x sol X Y x ainterp :

or select 'U' somewhere in 'X' U X Mean : 0.9985 -

XY X Y augment : support XY . 14 black plot : 21 sys

U U sol + 40 blue 1 5 mat contour XoYo support 4 1 sys

U U sol 12 mat 0.9985 - 1.3215 - 12 mat