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

64aeb7dd-6b40-4720-a199-aad56cc3b9e0 10 4 5 decimal

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

Legendre  setup 6179 needs "local assignment" 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 11 line :

n x poly i 0 n range p i el i x L : for 1 p stack 21 line :

x y XY x y / - :

n 12 : j 13 : 12 mat

n x poly j el XY 2 1 sys

x P 0 x L 1 x L 2 x L 3 x L 4 x L 5 x L 61 sys :

x P

n 4 :

SYMBOLIC n x poly n 1 + el 35 x 2^* 6 - 7 x 2^* + (* + 8 /

Set n= [8,9], hover to see the [8, 9] Mathcad sanity.

t0 1 time :

Mathcad sanity [8, 9]

λ V Ω λ z L z coeffs maple eval : n order the collection n 1 Ω rows 2 range stack : ct 2 if λ odd ct 1 if λ even ct λ 2 mod 1 ≡ 21 if : output coeffs ordered wrt 'n' i n New ct el Ω i el : ct ct 2 + : 21 line for New 81 line 21 line :

n 24 : (order ≡ C n V : r C polyroots : 31 line

ODD 0315128 / 0115532 / - 0900964 / 0643532 / - 012155128 / 101 mat :

EVEN 35128 / 031532 / - 0346564 / 0300332 / - 06435128 / 91 mat :

n 24

Wr i 1 n range W i el 21 r i el (2^- (* n 1 + (2^n 1 + r i el L (2^* / eval : for W r augment 21 line :

Abramowitz: ~ 11½ decimals [1965, p. 916] Wr 11 mat 0.127938195344 0.064056892855 - 0.127938195344 0.064056892855 0.125837456411 0.191118867527 - 0.125837456411 0.191118867527 0.121670472581 0.315042679527 - 0.121670472581 0.315042679527 0.115505668891 0.433793507909 - 0.115505668891 0.433793507909 0.107444268488 0.545421471001 - 0.107444268488 0.545421471001 0.097618654219 0.648093652447 - 0.097618654219 0.648093652447 0.086190161258 0.740124190879 - 0.086190152823 0.740124190879 0.073346496324 0.820001986991 - 0.073346496324 0.820001986991 0.059298531685 0.88641552567 - 0.059298531685 0.88641552567 0.044277450368 0.938274553318 - 0.044277450368 0.938274553318 0.028531309002 0.974728555132 - 0.028531309002 0.974728555132 0.01234124198 0.995187220251 - 0.01234124198 0.995187220251 242 mat 11 mat

1 time t0 - 3 s *

roots j 1 r rows range p j el . : for j 1 r rows range zero j el 0 : for r zero p augment 31 line :

roots n x L 2 1 sys

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.

x f x sqrt exp :

x f

Нахождение нулей полиномов Лежандра

Penyelesaian Polinomial Legendre

Legendre Quadrature

weight, wi w wr 1 col :

abscissea, xi u wr 2 col :

m wr rows : 63

range of integration a 0 : b 2 : 12 mat

a b Quad i 1 m range y i el b a - 2 / (u i el * b a + 2 / (+ eval : for y b a - (2 / w i el y i el f * (i 1 m sum eval * 31 line :

sanity check the quadruplet Mathcad 11 Romberg 5.40752818465632 ≡ Technical accuracy 5.40752818465627 ≡ Smath Romberg 5.40752818465657 ≡ Smath Integrator 5.407526 ≡ 41 line

a b Quad 5.4075293

Legendre poly user application . . .

]]>

Legendre  setup 6179 needs "local assignment" 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 11 line :

n x poly i 0 n range p i el i x L : for 1 p stack 21 line :

x y XY x y / - :

zoom 0.875 :

x λ 11 - x ≤ (x 1 ≤ (& cases :

n 5 : j n 1 + : 12 mat

r n zoom x * poly j el x λ * 0 ≡ x 1 - 1 solve :

r 0.8038 - 0 0.8038 31 mat

roots j 1 r rows range p j el . : for j 1 r rows range zero j el 0 : for r zero p augment 31 line :

n x poly j el XY 2 1 sys roots n zoom x * poly j el x λ * 2 1 sys

roots 0.8038 - 0.00 . 0.8038 0.33 mat

SYMBOLIC ... export for foreign in-situ application x export n x poly n 1 + el (: x 157 x 2^* 10 - 9 x 2^* + (* + (* 8 /

Negate in local definition x Lagrange x r n el - (n 1 r rows product 11 line - :

x Lagrange x λ *

0.5 Lagrange 0.198

cpoy/paste ▼ x Lagrange x 160755856966161200000000000000 / - (- (x 0 - (* x 8037792258092991000000000000000 / - (* -

x Symbolic x 160755856966161200000000000000 / - (- (x 0 - (* x 8037792258092991000000000000000 / - (* - :

x Symbolic x λ * ComboBox

CWD C:\SmathFile CurrentDirectory :

x Lagrange x Symbolic (:

x Lagrange CWD Lagrange wfile 1

x MyLagrange CWD Lagrange rfile :

x MyLagrange x λ *

end of lecture