2e8a8eb0-4c06-4135-a17f-a8b55eec5b69 41 4 5 decimal

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

Нахождение нулей полиномов Лежандра

Penyelesaian Polinomial Legendre

Legendre Quadrature

t0 1 time :

Rodrigues z n P 12 n^n! (* / z 2^1 - (n^(z n diff * :

n 2 / Floor Floor ≡

n 2 / mwfloor

n 4 :

Legendre  Smath 6179 bug ... unexplained x n L 12 n^/ 1 - (k^2 n * 2 k * - (! (* k! (n k - (! (* n 2 k * - (! (* / x n 2 k * -^* k 0 n 2 / Floor sum * :

k! (n k - (! * n 2 k * - (! *

x n 2 k * - (^

The expansion is OK for plotting but NOT for recovering the last 'n' polynomial. The incorrect recovered last polynomial forces Smath to crash.

last 'n' polynomial Ω 1 el x n L :

Wrong last poly, crashes Smath 6179 Ω 35 x 2^* 6 - 7 x 2^* + (* + 8 / 11 mat

===================== Crashes below ===============

Optimiz symbolic k 1 : k n ≤ k k 1 + : Ω k 1 + el Ω k el (z diff k / : for

z D Ω :

Коэффициенты полинома Лежандра Koefisien polinomial Legendre Legendre polynomial coefficients 0 D explode to see coefficients ≡

r 0 D polyroots :

===================== Rodrigues OK below ====================

Legendre maths [coeffs, roots, wr]

t0 1 time :

Степень полинома Pangkat polinomial Polynomial order n 4 :

Legendre Σ  crashes Smath 6179 12 n^/ 1 - (k^2 n * 2 k * - (! (* k! (n k - (! (* n 2 k * - (! (* / x n 2 k * -^* k 0 n 2 / Floor sum *

Rodrigues formula n ≤ 8 x n P 12 n^n! (* / x 2^1 - (n^x n diff * (:

Legendre last polynomial T 1 el x n P :

Rodrigues

x 0 P x 1 P x 2 P x 3 P x 4 P x 5 P 6 1 sys

visible expansion   n ≤ 4 k 1 : k n ≤ k k 1 + : T k 1 + el T k el (x diff k / : for

x D T :

Coefficients 0 D 0.375 0 3.75 - 0 4.375 51 mat

r 0 D polyroots : 0.339981 - 0.339981 0.861136 - 0.861136 41 mat

timing sec n 4 ≡ (1.5 ≡ n 8 ≡ (56 ≡ n 12 ≡ (N/A ≡ 31 mat ≡

wr Quadrature starts n [2] i 1 r rows range w i el 21 r i el (2^- (* n 1 + (2^r i el n 1 + P (2^(* / : for w r augment 31 line :

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

R x n P 2 1 sys

n 4

wr 0.652145154862547 0.339981043584856 - 0.652145154862547 0.339981043584856 0.347854845137449 0.861136311594052 - 0.347854845137449 0.861136311594052 42 mat

1 time t0 - 1.5 s *