7f6daa9e-a4fe-4774-9ec7-e537cded15e6 35 4 5 decimal

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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 :

User applied example ... adapted from [Mathcad 11]

k x B1 k 4 el x - (2^k 4 el k 2 el - (k 4 el k 3 el - (* / 11 line :

X 0 0.5 1 1.5 2 2.5 3481 mat :

Y 12 3.1 2.9 1.7 2.8 6281 mat :

u 1 - 0.75 1.25 1.75 2.75 3 3.25 71 mat :

k x B2 k 4 el k 1 el - (k 3 el x - (2^* - k 4 el k 3 el - (k 3 el k 2 el - (* k 3 el k 1 el - (* / 11 line :

k x B3 k 4 el k 1 el - (k 2 el x - (2^* k 4 el k 2 el - (k 3 el k 2 el - (* k 2 el k 1 el - (* / 11 line :

Add two auto knots on the LHS
there are 'n' intervals. i 1 t0 i el u i el 2 u i 1 + el u i el - (* - : t1 i el u i el u i 1 + el u i el - (- : 21 line for u t0 t1 u stack : 21 line

u transpose 4.5 - 2.75 - 1 - 0.75 1.25 1.75 2.75 3 3.25 19 mat

add/modify RHS u 9 el 3.5 : u 10 el 4 : u 11 el 5.5 : 31 sys

User adapt first & 2nd point u 1 el 1.5 - : u 2 el 10 - : 12 mat

t u : isolate  'u' for 't' ≡

t transpose 1.5 - 10 - 1 - 0.75 1.25 1.75 2.75 3 3.5 4 5.5 111 mat

Lagrange Quadratic Knots Generatrix k x b k 1 el x ≤ (x k 2 el ≤ (& k x B1 k x B2 + k x B3 + k 2 el x ≤ (x k 3 el ≤ (& k x B1 k x B2 + k 3 el x ≤ (x k 4 el ≤ (& k x B1 0 if if if 11 line :

Lagrange poly from Quad knots.
Observe the local definition. n x Bp j n 11 line : t j el t j 1 + el t j 2 + el t j 3 + el 41 mat x b 21 line :

x plot 1 x Bp 2 x Bp 3 x Bp 4 x Bp 5 x Bp 6 x Bp 7 x Bp 8 x Bp 9 x Bp 10 x Bp 11 x Bp 111 sys ≡

For the 11 knots, here are the 11 Lagrange polynomials

1 x Bp 2 x Bp 3 x Bp 4 x Bp 5 x Bp 6 x Bp 7 x Bp 8 x Bp 9 x Bp 10 x Bp 11 x Bp 11 1 sys

Quad tridiagonal matrix

Tridiag of QuadCoeff i 1 X rows range j 1 X rows range M i j el j X i el Bp : for for 21 line

s M 1 -^Y * :

The Quadratic tridiagonal matrix M 0.03 0.716 0.254 00000 0.003 0.425 0.571 000000 0.056 0.819 0.125 000000 0.125 0.792 0.083 000000 0.375 0.575 0.05 00000 0.042 0.508 0.45 0000000 0.667 0.333 00000000 0.75 88 mat

s 1.5125 - 0.292341 3.291238 3.094179 0.468444 5.406556 7.186887 2.666667 81 mat

<= user s 0.295 - 0.292 3.291 3.094 0.468 5.407 7.187 2.571 81 mat :

x L s j el j x Bp * (j 1 s rows sum 11 line :

1.5 - L 0.1597

6 L 0

compute 1 time :

vx 2 - 7 1.975 - range :

Original knots

u 1 - 0.75 1.25 1.75 2.75 3 3.25 71 mat :

v 000000071 mat :

Spline i 1 vx rows range vy i el vx i el L eval : for vx vy augment 21 line :

XY X Y . 10 black plotG :

knots u v . 10 black plotG :

Spline Spline :

t transpose 1.5 - 10 - 1 - 0.75 1.25 1.75 2.75 3 3.5 4 5.5 111 mat

to 0 t * vectorize :

The augmented/adapted set of user knots MyKnots t to o 10 blue plotG :

Complete plot of Lagrange Quadratic Spline Flat entrance/departure

XY Spline knots MyKnots 4 1 sys

1 time compute - 7 s *

User interpolation X Spline 1 col : Y Spline 2 col : 12 mat x Lagrange X Y x ainterp : x λ 12 - x ≤ (x 7 ≤ (& cases : 31 mat

x Lagrange x λ * 2 - 2 2 - 2 - 2 2 mat 2 1 sys

X 0 0.5 1 1.5 2 2.5 3481 mat :

x L s j el j x Bp * (j 1 s rows sum :

0 L 1.0361

x f x L :

h 0.001 :

x h f D1 x 2 h * - f x 2 h * + f - 4 h * / - :

x Deriv x h f D1 2 - x ≤ (x 6 ≤ (& cases :

D x 2 - 6 1.99 - range : i 1 x rows range D i el x i el Deriv eval : for x D augment 31 line :

Derivative of the Lagrange Knots Quadratic. Spline is quadratic, derivative is piecewise linear. Gradient is 2nd deriv of spline at support points 'X'

D X s augment Spline XY 4 1 sys

Stem fill plot for QuickPlot unicolor data STEM stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - (range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - (range V V stem j el stack : for V V data data rows row stack : 71 line :

Spline XY Spline STEM 5 1 sys

Spline rows 361

data Spline :

Finite differences derivative X data 1 col : Y data 2 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 - : 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

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 :

Finite differences Cumulative Integral

data Σ data 2 1 sys