f2492a9a-2200-401f-ae26-d7dc7978c4b0 20 6 9 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 :

Part 1: Lagrange Quadratic basis experimental demo

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

Lagrange Quadratic Generatrix 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 :

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 :

Experimental knots.
Play knots, watch graphs. k 3 3.5 4.25 5.75 14 mat transpose :

Piecewise Quadratic segments x Q k 1 el x ≤ (x k 2 el ≤ (& k x B1 k x B2 + k x B3 + junction if k 2 el x ≤ (x k 3 el ≤ (& k x B1 k x B2 + junction if k 3 el x ≤ (x k 4 el ≤ (& k x B1 N/A if 31 mat :

knots k 1 el 0.12 black k 2 el 0.12 black k 3 el 0.12 black k 4 el 0.12 black 45 mat :

There are "Lagrange cubic knots". Not offered in Smath style. Quad knots: adapted from Robert A. Mathsoft Collaboratory.

Lagrange basis interpolating funct's

k x B1 k x B2 k x B3 knots 4 1 sys

Lagrange Quadratic "Generatrix"

x b k 1 el 0 . 12 black k 2 el 0 . 12 black k 3 el 0 . 12 black k 4 el 0 . 12 black 4 5 mat 2 1 sys

Piecewise Quadratic segments

x Q 1 el x Q 2 el x Q 3 el k 1 el 0 . 12 black k 2 el 0 . 12 black k 3 el 0 . 12 black k 4 el 0 . 12 black 4 5 mat 4 1 sys

b(x)+knots

Q(x)+knots

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 :

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 :

u 11 el 5.5 :

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 5.5 111 mat

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

A 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

A 1.512 - 0.292 3.291 3.094 0.468 5.407 7.187 2.667 81 mat

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

<= user

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

0 L 1.0361

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 :

t transpose 1.5 - 10 - 1 - 0.75 1.25 1.75 2.75 3 3.5 4 5.5 111 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 - 27 sec *

2 - x ≤ x 7 ≤ & x L out user range if

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 :

There are 8 pairs of data, thus 8 linear deriv segments. Deriv(Spline) is an image.