897f720d-1914-40ad-8316-d8b5e9b3f9f0 50 6 9 decimal

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The Gram-Schmidt orthogonal procedure

This project is Mathcad 8 Pro. NOT reproducible in Smath 5346. Up until now, I found no way to force Smath integrator understand the dot product algorithm. Alternately, construct the orthogonal set: terms by terms. Terms that are 0 are ignored.

m n dot x f x g * x 1 - 1 int :

x f3 x 2^:

x f4 x 3^:

x f5 x 4^:

x f1 1 :

x f2 x :

x o1 x f1 :

x o1 x f2 * x 1 - 1 int x o1 x o1 * x 1 - 1 int / 15 round 0

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

x o2 x :

1 - (x o1 x f3 * x 1 - 1 int x o1 x o1 * x 1 - 1 int / * 15 round x o1 * 0.333333 -

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

x o3 x 2^13 / - :

1 - (x o2 x f4 * x 1 - 1 int x o2 x o2 * x 1 - 1 int / * 15 round 0.6 -

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 o4 x 3^35 / x * - :

1 - (x o1 x f5 * x 1 - 1 int x o1 x o1 * x 1 - 1 int / * 15 round 0.2 -

1 - (x o3 x f5 * x 1 - 1 int x o3 x o3 * x 1 - 1 int / * 15 round 0.857143 -

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 * - ≡

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

57 x 4^* 23 x 2^* 1 - (* - (* 7 - 35 / x 4^103 x 2^* 1 - (* 7 + 35 / - ≡ x 4^335 / + 67 / x 2^* - ≡ 3110 x 2^* - (* 35 x 4^* + 35 / ≡ 715 x 4^* - (* 101 - 3 x 2^* + (* + 35 / - ≡

x collect x o1 x o2 x o3 x o4 x o5 51 sys :

x collect 1 x 1 - 3 x 2^* + 3 / x 3 - 5 x 2^* + (* 5 / 3110 x 2^* - (* 35 x 4^* + 35 / 51 sys

x collect

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

X_Y ... only 3 x arguments x ortho 0 x 2^13 / - x 3^35 / x * - x 4^335 / + 67 / x 2^* - 41 sys :

x mcd 1 x x 2^13 / - x 3^35 / x * - x 4^335 / + 67 / x 2^* - 51 sys :

x mcd 1 - 1 - 1 - 1 1 1 1 1 - 1 - 1 - 5 2 mat 2 1 sys

x Unit x : X_Y rejects the 4th argument in 'x' ≡

x ortho 1 - 1 - 1 - 1 1 1 1 1 - 1 - 1 - 5 2 mat 2 1 sys

vx vy a b c plotG a char ≡ b size ≡ c color ≡ 13 mat n vx length : plot vx 1 el vy 1 el a b c augment : i 2 n range plot plot vx i el vy i el a b c augment stack : for plot 51 line :

x f 1 - x ≤ (x 1 ≤ (& x 4^335 / + 67 / x 2^* - out range if :

data x 1 - 1 0.875 - range : i 1 x rows range Y i el x i el f : for x Y augment 31 line :

X data 1 col : Y data 2 col : XY X Y . 12 red plotG : 31 line

X Y x cinterp XY 2 1 sys

Summarise an application ...

1. You have sampled data, hope not too noisy. 2. From educated guess, polyfit 4th order. 3. Attempt to recognise the coefficients. 4. Declare the fit is Legendre "o5(x)". ________________________ Noisy sampled data would freak the coeficients but NOT beyond recognition. The f3(x), f4(x)... can be anything, but chek the "o(x)" system. From the knowledge that the fit is Legendre "o5(x)", you have in hand the complete set of more orthogonal functions for whatever sub-use.

setup the solver n X length : PolyOrder 4 : degree PolyOrder 1 + : j 0 degree range : 41 line

x φ 1 x x 2^x 3^x 4^51 mat :

Polyfit matrix j 1 degree range i 1 n range M i j el X i el φ j el : for for

Φ M transpose M * : d M transpose Y * : b Φ 1 -^eval d * : 31 line

n 17

Φ 170 6.375 0 4.283203 0 6.375 0 4.283203 0 6.375 0 4.283203 0 3.410065 0 4.283203 0 3.410065 0 4.283203 0 3.410065 0 2.942635 55 mat

Tabulate M = in the right margin to see the complete evolution ... M, Φ, d, b

recast the scalar model x fit b x φ * :

d b 12 mat 0.27606 0 0.285176 0 0.386854 51 mat 0.085714 0 0.857143 - 0151 mat 12 mat

x fit XY 2 1 sys

recognise coefficients 335 / 67 / 21 mat 0.085714286 0.857142857 21 mat

e1 1 el e 1 el e 1 el abs / : i 2 n range j 1 i 1 - range e i el e i el e1 j el e i el * (e1 j el * - : for e1 i el e i el e i el abs / : 21 line for e1 31 line