0abe4372-9a21-4721-9f73-cd85909a11f3 28 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 :

Define a Lagrange quadratic piecewise

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

1. The spline does not joint exactly but closely the data set, it does not mean smoothing. 2. There are 1 less basic knot than there are X's 3. the 'A' spline coefficients has 'n' rows as there are 'n' rows X. 4. This spline does not support d'(x)operator. 5. Most interesting to adjust the "fuites" at either or both ends. Specific to Smath:-------------------------- 1. The Bolean must be specified completely 2. Local operator j:=|n Point of interest:-------------------------- Any "knot poly" may be used to explore/fit asymmetric stastistical data set.

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 :

k 3 3.5 4 5.75 14 mat transpose :

x b Lagrange Quad Generatrix 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 21 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 :

v 000000071 mat :

Lagrange basis interpolating functions

k x B1 k x B2 k x B3 3 1 sys

Lagrange quadratic "Generatrix"

x b

4.5 b 0.3968

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

Add two auto-knots on the LHS, there are n intervals

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 :

u 11 el 5.75 :

u 1 el 1.5 - :

u 2 el 10 - :

User adapt first & last point

t u :

t transpose 1.5 - 10 - 1 - 0.75 1.25 1.75 2.75 3 3.5 4 5.75 111 mat

k x b specify Bolean completely 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 21 line :

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 0 x Bp 11 x Bp 111 sys ≡

Lagrange poly from quad knots n x Bp knots poly 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 21 line :

j 8 :

t j el t j 1 + el t j 2 + el t j 3 + el 41 mat 3 3.5 4 5.75 41 mat

Select/Isolate/Export uv u 03 0.05 range : i 1 u rows range vx i el u i el : vy i el 4 u i el Bp : 21 line for vx vy augment 21 line 21 line :

UV uv 1 col uv 2 col . 10 black plotG :

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

A specfied knot-poly

UV uv 2 1 sys

Install & solve the tridiagonal matrix

r 1 X rows range :

Tridiagonal i r j r M i j el j X i el Bp : for for 21 line

A M 1 -^Y * :

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 0.778 88 mat

A 1.512 - 0.292 3.291 3.094 0.468 5.407 7.187 2.571 81 mat

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

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

0 L 1.0361

vx 2 - 7 1.99 - range :

<= n=2 populate for more points at will

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

v 000000071 mat :

Spline i 1 vx rows range u i el vx i el : v i el vx i el L : 21 line for u v augment 21 line :

XY X Y . 10 black plotG :

knots u v . 10 black plotG :

t transpose 1.5 - 10 - 1 - 0.75 1.25 1.75 2.75 3 3.5 4 5.75 111 mat

t0 00000000000111 mat transpose :

MyKnots t t0 o 10 blue plotG :

XY Spline knots MyKnots 4 1 sys

data Deriv data Derivative BackSubstitution X data 1 col : Y data 2 col : 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 el : X deriv augment 101 line :

Spline Deriv