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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 - (* / :

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 - (* / :

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 - (* / :

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 2 - 1.37 - 0.89 - 0.55 - 0.25 - 0 0.22 0.45 0.75 1.3 2111 mat :

Y 0.519 0.414 - 1.061 - 0.987 - 0.347 - 2.791 1.081 0.765 1.181 0.588 0.466 - 111 mat :

u 3 - 1 - 0.7 - 0.4 - 0.0625 - 0.125 0.25 0.75 1.3 1.5 101 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 7 - 5 - 3 - 1 - 0.7 - 0.4 - 0.0625 - 0.125 0.25 0.75 1.3 1.5 121 mat

Add 3 user knots ... at will modify the two 1rst ones at will at the end => 4 more knots than original 'u' vector.

u 12 el 3.5 :

u 13 el 4 :

u 14 el 5.5 :

normalise Bp(n,x) wrt 't' t u :

t transpose 7 - 5 - 3 - 1 - 0.7 - 0.4 - 0.0625 - 0.125 0.25 0.75 1.3 3.5 4 5.5 114 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. n x Bp t n el t n 1 + el t n 2 + el t n 3 + el 41 mat x b :

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 # # 131 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

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

s M 1 -^Y * :

Once 's' in hand, you can isolate 's' and modify at will ... !

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

vx 2 - 2.5 1.99 - range :

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 :

Spline Spline :

Complete plot of Lagrange Quadratic Spline

XY Spline X Y x cinterp 3 1 sys