<p> Legendre wr 63 ... 64 </p>

da398f07-fb6e-4d6b-8aee-7861933a01f4 37 4 5 decimal

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

Нахождение нулей полиномов Лежандра

Penyelesaian Polinomial Legendre

Legendre Quadrature

Insert:Numeric up-down

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

Polynomial order n < 12

n 12

Legendre  more profitable than Rodrigues n z L 1 - (k^2 n * 2 k * - (! (z n 2 k * -^* (* 2 n^k! (* n k - (! (n 2 k * - (! (* (* / (k 0 n 2 / Floor sum :

t0 1 time :

T 1 el n z L :

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

Optimiz NONE k 1 : k n ≤ k k 1 + : T k 1 + el T k el (z diff k / : for

Limits including polyroots 1. Rodrigues n=8 => 46 sec 2. L(n,x) n=12 => 3 sec

z D T :

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

r 0 D polyroots :

W Quadrature starts n [2] i 1 r rows range w i el 21 r i el (2^- (* n 1 + (2^n 1 + r i el L (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 n x L 2 1 sys

n 12

1 time t0 - 3 sec *

5 x L 0 ≡ x Algsys #

This document, delivers approx. 11 true decimals

Web reference rounded to 15 decimals is sanity check wrt Abramowitz.

W 0.249147045813334 0.125233408511383 - 0.249147045813334 0.125233408511383 0.233492536538249 0.367831498998169 - 0.233492536538249 0.367831498998169 0.203167426723504 0.58731795428675 - 0.203167426723504 0.58731795428675 0.160078328543607 0.769902674193765 - 0.160078328543607 0.769902674193765 0.106939325978222 0.904117256370389 - 0.106939325978222 0.904117256370389 0.047175336426294 0.981560634247238 - 0.047175336426294 0.981560634247238 122 mat

1 0.2491470458134028 -0.1252334085114689 2 0.2491470458134028 0.1252334085114689 3 0.2334925365383548 -0.3678314989981802 4 0.2334925365383548 0.3678314989981802 5 0.2031674267230659 -0.5873179542866175 6 0.2031674267230659 0.5873179542866175 7 0.1600783285433462 -0.7699026741943047 8 0.1600783285433462 0.7699026741943047 9 0.1069393259953184 -0.9041172563704749 10 0.1069393259953184 0.9041172563704749 11 0.0471753363865118 -0.9815606342467192 12 0.0471753363865118 0.9815606342467192

https://pomax.github.io/bezierinfo/legendre-gauss.html

<p> Legendre wr 63 ... 64 </p>

wr 0.0486909570091397 0.0243502926634244 - 0.0486909570091397 0.0243502926634244 0.0485754674415034 0.0729931217877990 - 0.0485754674415034 0.0729931217877990 0.0483447622348030 0.1214628192961206 - 0.0483447622348030 0.1214628192961206 0.0479993885964583 0.1696444204239928 - 0.0479993885964583 0.1696444204239928 0.0475401657148303 0.2174236437400071 - 0.0475401657148303 0.2174236437400071 0.0469681828162100 0.2646871622087674 - 0.0469681828162100 0.2646871622087674 0.0462847965813144 0.3113228719902110 - 0.0462847965813144 0.3113228719902110 0.0454916279274181 0.3572201583376681 - 0.0454916279274181 0.3572201583376681 0.0445905581637566 0.4022701579639916 - 0.0445905581637566 0.4022701579639916 0.0435837245293235 0.4463660172534641 - 0.0435837245293235 0.4463660172534641 0.0424735151236536 0.4894031457070530 - 0.0424735151236536 0.4894031457070530 0.0412625632426235 0.5312794640198946 - 0.0412625632426235 0.5312794640198946 0.0399537411327203 0.5718956462026340 - 0.0399537411327203 0.5718956462026340 0.0385501531786156 0.6111553551723933 - 0.0385501531786156 0.6111553551723933 0.0370551285402400 0.6489654712546573 - 0.0370551285402400 0.6489654712546573 0.0354722132568824 0.6852363130542333 - 0.0354722132568824 0.6852363130542333 0.0338051618371416 0.7198818501716109 - 0.0338051618371416 0.7198818501716109 0.0320579283548516 0.7528199072605319 - 0.0320579283548516 0.7528199072605319 0.0302346570724025 0.7839723589433414 - 0.0302346570724025 0.7839723589433414 0.0283396726142595 0.8132653151227975 - 0.0283396726142595 0.8132653151227975 0.0263774697150547 0.8406292962525803 - 0.0263774697150547 0.8406292962525803 0.0243527025687109 0.8659993981540928 - 0.0243527025687109 0.8659993981540928 0.0222701738083833 0.8893154459951141 - 0.0222701738083833 0.8893154459951141 0.0201348231535302 0.9105221370785028 - 0.0201348231535302 0.9105221370785028 0.0179517157756973 0.9295691721319396 - 0.0179517157756973 0.9295691721319396 0.0157260304760247 0.9464113748584028 - 0.0157260304760247 0.9464113748584028 0.0134630478967186 0.9610087996520538 - 0.0134630478967186 0.9610087996520538 0.0111681394601311 0.9733268277899110 - 0.0111681394601311 0.9733268277899110 0.0088467598263639 0.9833362538846260 - 0.0088467598263639 0.9833362538846260 0.0065044579689784 0.9910133714767443 - 0.0065044579689784 0.9910133714767443 0.0041470332605625 0.9963401167719553 - 0.0041470332605625 0.9963401167719553 0.0017832807216964 0.9993050417357722 - 0.0017832807216964 0.9993050417357722 642 mat :

wr 0.0494723666239310 0.0000000000000000 0.0494118330399182 0.0494521871161596 - 0.0494118330399182 0.0494521871161596 0.0492303804237476 0.0987833564469453 - 0.0492303804237476 0.0987833564469453 0.0489284528205120 0.1478727863578720 - 0.0489284528205120 0.1478727863578720 0.0485067890978838 0.1966003467915067 - 0.0485067890978838 0.1966003467915067 0.0479664211379951 0.2448467932459534 - 0.0479664211379951 0.2448467932459534 0.0473086713122689 0.2924940585862514 - 0.0473086713122689 0.2924940585862514 0.0465351492453837 0.3394255419745844 - 0.0465351492453837 0.3394255419745844 0.0456477478762926 0.3855263942122479 - 0.0456477478762926 0.3855263942122479 0.0446486388259414 0.4306837987951116 - 0.0446486388259414 0.4306837987951116 0.0435402670830276 0.4747872479948044 - 0.0435402670830276 0.4747872479948044 0.0423253450208158 0.5177288132900333 - 0.0423253450208158 0.5177288132900333 0.0410068457596664 0.5594034094862850 - 0.0410068457596664 0.5594034094862850 0.0395879958915441 0.5997090518776252 - 0.0395879958915441 0.5997090518776252 0.0380722675843496 0.6385471058213654 - 0.0380722675843496 0.6385471058213654 0.0364633700854573 0.6758225281149861 - 0.0364633700854573 0.6758225281149861 0.0347652406453559 0.7114440995848458 - 0.0347652406453559 0.7114440995848458 0.0329820348837793 0.7453246483178474 - 0.0329820348837793 0.7453246483178474 0.0311181166222198 0.7773812629903724 - 0.0311181166222198 0.7773812629903724 0.0291780472082805 0.8075354957734567 - 0.0291780472082805 0.8075354957734567 0.0271665743590979 0.8357135543195029 - 0.0271665743590979 0.8357135543195029 0.0250886205533450 0.8618464823641238 - 0.0250886205533450 0.8618464823641238 0.0229492710048899 0.8858703285078534 - 0.0229492710048899 0.8858703285078534 0.0207537612580391 0.9077263027785316 - 0.0207537612580391 0.9077263027785316 0.0185074644601613 0.9273609206218432 - 0.0185074644601613 0.9273609206218432 0.0162158784103383 0.9447261340410098 - 0.0162158784103383 0.9447261340410098 0.0138846126161156 0.9597794497589419 - 0.0138846126161156 0.9597794497589419 0.0115193760768800 0.9724840346975701 - 0.0115193760768800 0.9724840346975701 0.0091259686763267 0.9828088105937273 - 0.0091259686763267 0.9828088105937273 0.0067102917659601 0.9907285468921895 - 0.0067102917659601 0.9907285468921895 0.0042785083468638 0.9962240127779701 - 0.0042785083468638 0.9962240127779701 0.0018398745955771 0.9992829840291237 - 0.0018398745955771 0.9992829840291237 632 mat :

This page was compiled by Mike "Pomax" Kamermans, on June 5th, 2011, as a resource for the primer on Bezier curves, specifically the section that deals with arc length computation. You are free to do what you like with this data. Go make the world a better place than it was when you found it.

The extended Γ(x)

x G x Γ x Gamma : x 0 ≥ x Γ π x abs 1 + (* x abs 2 + Γ π x abs 1 + (* sin * / if 21 line :

This function is a recast from Gilberto E. Urroz Works for in-situ calculations. Tabulate wisely to escape singularities.

x G 0 2 1 sys

Legendre quadrature general use

L H N xd U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for U 41 line :

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

a 0.125 : b 4 : N 62 : 13 mat

U a b N xd :

data i 1 U rows range vy i el U i el G : 11 line for U vy augment 21 line :

a b c spec . 12 red 13 mat :

XY data a b c spec pts :

Low a 10 a 5 - 22 mat :

High b 10 b 5 - 22 mat :

data XY Low High 4 1 sys

Run Legendre Quadrature

weight, wi w wr 1 col :

abscissea, xi x wr 2 col :

NOT an integrator ! data 2 col Mean b a - (* 8.0655

m wr rows :

m 63

y i 1 m range y i el b a - 2 / (x i el * b a + 2 / (+ : for y 21 line :

b a - 2 / (w i el y i el * (i 1 m sum eval * 7.99

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

x f x G :

data Σ m 2 el 7.77

X data 1 col :

h X X length el X 1 el - data rows 1 - / : 116 /

1rst derivative  applied on interpolating functions x h D1 x 2 h * + f - x 2 h * - f + (- 12 h * / :

NOT compatible x D a x ≤ (x b ≤ (& x G x diff Not defined if :

x h D1 1.464 0 + 20 red 1 5 mat 2 1 sys

inflexion 1.464 :

NOT scalar x G x a b int #

NOT scalar x Gamma x a b int #