dbc90b24-73d1-4f92-8141-44c44321c9da 32 6 9 decimal

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

Part 1: otherwise dressed from 'Carlos'=====================

N 6 :

i 1 N range f i el x i 1 -^: for

m n dot x f x g * x 1 - 1 int ≡

f transpose 1 x x 2^x 3^x 4^x 5^16 mat

o 1 el f 1 el :

<= pre-initialise

kernel algo: MUST BE pre-initialised o i 2 f rows range d 0 : j 1 i 1 - range a f i el o j el * x 1 - 1 int o j el o j el * x 1 - 1 int / 15 round : d d a o j el * - : 21 line for o i el f i el d + (: 31 line for :

Explain:

1. "Syntax incorrect" at the user level but the system of 'o' is evaluated at the kernel level, awaiting the indexed call(s) of oi. 2. The ratio of the two dot product needs be vectorised, Smath does not do it from normal code. The 'round(_,15)' has same vectorising effect, thus enabling the evaluation of the 'a(s)' coefficients, which coefficients are pure numeric. 3. The pest remains about extracting meaningful components for export and more usefull sub-use. However OK for QuickPlot. 4. Lagrange "roots style" export is compatible for Engineering works, mostly fitting sampled data. Think in term of Fourier spectrum from clean model.

Collect/Normalise function style

x o1 o 1 el :

x o2 o 2 el :

x o3 o 3 el :

x o4 o 4 el :

x o5 o 5 el :

x o6 o 6 el :

x o1 1

x o2 x

x o3 166666666666667 - 500000000000000 x 2^* + 500000000000000 /

x o4 244140625000000000000000000 x * 1500000000001 - 2500000000000 x 2^* + (* 1220703125166666666666667 - 500000000000000 x 2^* + (* - 610351562500000000000000000000000000000 /

Poly x o1 x o2 x o3 x o4 x o5 x o6 61 sys :

Poly

The ROBUST root(s) Solve scanner f x0 x1 TOL Solve ρ x f x x0 x1 solve : i 1 ρ rows range a ρ i el 0.1 ρ i el * - : b ρ i el 0.1 ρ i el * + : xn b a + (2 / : ε TOL : n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 61 line X i el xn : 21 line for X 31 line :

Export Lagrange roots style

x f x o6 :

TOL 1015 -^:

r f 1 - 1 TOL Solve : 0.90618 - 0.538469 - 5.51813 1015 -^* 0.538469 0.90618 51 mat

x Lagrange x r 1 el - (x r 2 el - (* x r 3 el - (* x r 4 el - (* x r 5 el - (* :

x Lagrange

Part 2: dressed tutor & export style 'Jean'===================

x f1 1 :

x f2 x :

x f3 x 2^:

x f4 x 3^:

x f5 x 4^:

x o1 x f1 :

x o1 1 :

a2 x o1 x f2 * x 1 - 1 int x o1 x o1 * x 1 - 1 int / 15 round :

The method exposed here is piece wise, more effort demanding but convenient for Engineering works ... visible.

x o2 x f2 a2 x o1 * - :

a2 0

x o2 x :

a3 x o1 x f3 * x 1 - 1 int x o1 x o1 * x 1 - 1 int / 15 round :

b3 x o2 x f3 * x 1 - 1 int x o2 x o2 * x 1 - 1 int / 15 round :

x o3 x f3 a3 x o1 * - b3 x o2 * - :

a3 0.333333

b3 0

x o3 x 2^13 / - :

a4 x o1 x f4 * x 1 - 1 int x o1 x o1 * x 1 - 1 int / 15 round :

b4 x o2 x f4 * x 1 - 1 int x o2 x o2 * x 1 - 1 int / 15 round :

c4 x o3 x f4 * x 1 - 1 int x o3 x o3 * x 1 - 1 int / 15 round :

x o4 x f4 a4 x o1 * - b4 x o2 * - c4 x o3 * - :

a4 0

b4 0.6

c4 0

x o4 x 3^35 / x * - :

a5 x o1 x f5 * x 1 - 1 int x o1 x o1 * x 1 - 1 int / 15 round :

b5 x o2 x f5 * x 1 - 1 int x o2 x o2 * x 1 - 1 int / 15 round :

c5 x o3 x f5 * x 1 - 1 int x o3 x o3 * x 1 - 1 int / 15 round :

d5 x o4 x f5 * x 1 - 1 int x o4 x o4 * x 1 - 1 int / 15 round :

x o5 x f5 a5 x o1 * - b5 x o2 * - c5 x o3 * - d5 x o4 * - :

a5 0.2

b5 0

c5 0.857143

d5 0

x o5 x 4^335 / + 67 / x 2^* - :

combine the terms 67 / 13 / - (* - 15 / - 335 /

5 first orthogonal functions

x o1 x o2 x o3 x o4 x o5 5 1 sys

NOT convenient symbolic x o1 x o2 x o3 x o4 x o5 51 sys 1 x 1 - 3 x 2^* + 3 / x 3 - 5 x 2^* + (* 5 / 3110 x 2^* - (* 35 x 4^* + 35 / 51 sys

x o1 x o2 x o3 x o4 x o5 51 sys 1 x x 2^13 / - x 3^35 / x * - x 4^335 / + 67 / x 2^* - 51 mat ≡

x o1 x f1 :

x o2 x f2 o1 f2 dot o1 o1 dot / x o1 * - :

x o3 x f3 o1 f3 dot o1 o1 dot / x o1 * - o2 f3 dot o2 o2 dot / x o2 * - :

x o4 x f4 o1 f4 dot o1 o1 dot / x o1 * - o2 f4 dot o2 o2 dot / x o2 * - o3 f4 dot o3 o3 dot / x o3 * - :

x o5 x f5 o1 f5 dot o1 o1 dot / x o1 * - o2 f5 dot o2 o2 dot / x o2 * - o3 f5 dot o3 o3 dot / x o3 * - o4 f5 dot o4 o4 dot / x o4 * - :