7f6daa9e-a4fe-4774-9ec7-e537cded15e6 18 4 5 decimal

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

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 :

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

Part 2: User applied example

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

u 9 el 3.5 :

u 10 el 4.15 :

u 11 el 5 :

u 1 el 1.5 - :

u 2 el 10 - :

User adapt first & last point

t u : isolate  'u' for 't' ≡

t transpose 1.5 - 10 - 1 - 0.75 1.25 1.75 2.75 3 3.5 4.15 5111 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 :

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 0 x Bp 11 x Bp 11 1 sys

Quad tridiagonal matrix

r 1 X rows range :

Tridiag of QuadCoeff i r j r 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 0000000 0.03 0.713 88 mat

s 1.5125 - 0.292341 3.291238 3.094179 0.468444 5.406556 7.186887 2.499928 81 mat

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

<= user

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 1.99 - 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 4.15 5111 mat

i 1 t rows range to i el 0 t i el * : for

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

Complete plot of Lagrange Quadratic Spline

XY Spline knots MyKnots 4 1 sys

1 time compute - 19 s *

The remaining of the Lagrange stuff Spline Lagrange[Quadratic Knots].sm

Quad knots <=> flat entrance/departure