ac6d4494-845f-428d-886a-f50172e97733 26 4 5 decimal

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

Gramm-Schmidt orthogonal polynomials

N 6 :

Lagrange roots iterated product x r i el - (i 13 product x r 1 el - (x r 2 el - (* x r 3 el - (*

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

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 o(x)=o 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 o o :

Poly x o 1 el x o 2 el x o 3 el x o 4 el x o 5 el x o 6 el 61 sys :

Poly

Expand Symbolic

x o 6 el x 745058059692385669643357 - 1562500000 x 4^* + (* 827846696120510009375001 - 15625000 x 2^* + (* - (* 1164153218269350000000000 /

Isolate ...

Ω 1 el x 745058059692385669643357 - 1562500000 x 4^* + (* 827846696120510009375001 - 15625000 x 2^* + (* - (* 1164153218269350000000000 / :

μ 16 range Ω μ 1 + el Ω μ el (x diff μ / : for x d Ω : r 0 d polyroots : 31 line

◄ from Smath Handbook

Possibly more decimals ... depends upon accuracy of transit.

0 d 0 0.238099 0 1.111117 - 01071 mat

polyroots terminator [v] ▼  can't be '0' r 0 0.238099 0 1.111117 - 0161 mat polyroots : 0 0.538473 - 0.538473 0.906181 - 0.906181 51 mat

polyroots terminator can't be '0' R 0 0.238099 0 1.111117 - 01071 mat polyroots : #

R i 1 r rows range p i el . : for i 1 r rows range zero i el 0 : for r zero p augment 31 line :

R x o 6 el 2 1 sys

setup local definition x Lagrange x r j el - (j 1 r rows product 11 line :

R x Lagrange 2 1 sys

x f 3 x 3^* 2 x * - :

x f

02 - 0341 mat polyroots 0 0.816496580927726 - 0.816496580927726 31 mat

only 5 decimals x f 0 ≡ x 1 - 1 solve 0.816497086825068 - 0 0.816496311989528 31 mat

x f 0 ≡ x solve maple 06 sqrt 3 / 6 sqrt 3 / - 31 mat

6 sqrt 3 / 0.816496580927727

The maple companion is willing ... much less gears

x f x 745058059692385669643357 - 1562500000 x 4^* + (* 827846696120510009375001 - 15625000 x 2^* + (* - (* 1164153218269350000000000 / :

roots order is immaterial
in this real roots example.
In case of complex roots,
set Lagrange Π comlex top. x f 0 ≡ x solve maple 0 0.9062 0.9062 - 0.5385 0.5385 - 51 mat

# # # # product

roots of a random polynomial

r Clear 1

x f 1 - x + (1 - 2 x * + (* x * 14 x * + (* 4 x * 3 + (* 32 / :

r x f 0 ≡ x solve maple : 1 0.5 0 0.25 - 0.75 - 51 mat

Lagrange roots style x L x r n el - (n 1 r rows product :

R i 1 r rows range p i el . : for i 1 r rows range zero i el 0 : for r zero p augment 31 line :

R x L 2 1 sys

x f x 745058059692385669643357 - 1562500000 x 4^* + (* 827846696120510009375001 - 15625000 x 2^* + (* - (* 1164153218269350000000000 / :

r 0 0.906181463330827 0.906181463330827 - 0.538472081329767 0.538472081329767 - 51 mat :

x Lagrange x r n el - (n 1 r rows product :

x Lagrange

r x f 0 ≡ x solve maple :

x f 0 ≡ x 1 - 1 solve 0.906181339783194 - 0.538472152848052 - 0 0.538472069445396 0.906181419346945 51 mat