Legendre wr 63 ... 64

335786a9-8857-44e8-8c3d-243ece8edbd4 58 6 9 decimal

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

Нахождение нулей полиномов Лежандра

Penyelesaian Polinomial Legendre

Legendre Quadrature

Степень полинома Pangkat polinomial Polynomial order n 5 :

n u L 1 - (k^2 n * 2 k * - (! (u n 2 k * -^* (* 2 n^k! (* n k - (! (n 2 k * - (! (* (* / (k 0 n 2 / Floor sum :

Формула Родрига Formula Rodrigues Rodrigues formula z n P 12 n^n! (* / z 2^1 - (n^z n diff * :

T 1 el z n P :

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

practical limit n < 9 k 1 : k n ≤ k k 1 + : T k 1 + el T k el (z diff k / : for

z D T :

Коэффициенты полинома Лежандра Koefisien polinomial Legendre Legendre polynomial coefficients 0 D 0158 / 0354 / - 0638 / 61 mat

0 x L 1 x L 2 x L 3 x L 4 x L 5 x L 61 sys 1 x 1 - 3 x 2^* + 2 / x 3 - 5 x 2^* + (* 2 / 35 x 2^* 6 - 7 x 2^* + (* + 8 / x 157 x 2^* 10 - 9 x 2^* + (* + (* 8 / 61 sys

roots of Legendre polynomials
--------- order 'n' -------- 0 D polyroots 0 0.538469 - 0.538469 0.90618 - 0.90618 51 mat

r 0 D polyroots :

ii 1 r rows range p ii el . : for

ii 1 r rows range zero ii el 0 : for

collect x 0 P x 1 P x 2 P x 3 P x 4 P x 5 P r zero p augment 71 sys :

collect

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

0. prepare the "wr", n rows x 2 cols 1. Select/copy numbers in web site, one at time ... paste in Smath. 2. very easy: points repeat.

n 15 ≡

wr 0.1527533871307258 0.0765265211334973 - 0.1527533871307258 0.0765265211334973 0.1491729864726037 0.2277858511416451 - 0.1491729864726037 0.2277858511416451 0.1420961093183820 0.3737060887154195 - 0.1420961093183820 0.3737060887154195 0.1316886384491766 0.5108670019508271 - 0.1316886384491766 0.5108670019508271 0.1181945319615184 0.6360536807265150 - 0.1181945319615184 0.6360536807265150 0.1019301198172404 0.7463319064601508 - 0.1019301198172404 0.7463319064601508 0.0832767415767048 0.8391169718222188 - 0.0832767415767048 0.8391169718222188 0.0626720483341091 0.9122344282513259 - 0.0626720483341091 0.9122344282513259 0.0406014298003869 0.9639719272779138 - 0.0406014298003869 0.9639719272779138 0.0176140071391521 0.9931285991850949 - 0.0176140071391521 0.9931285991850949 202 mat :

0.2025782419255613 0.0000000000000000 0.1984314853271116 0.2011940939974345 - 0.1984314853271116 0.2011940939974345 0.1861610000155622 0.3941513470775634 - 0.1861610000155622 0.3941513470775634 0.1662692058169939 0.5709721726085388 - 0.1662692058169939 0.5709721726085388 0.1395706779261543 0.7244177313601701 - 0.1395706779261543 0.7244177313601701 0.1071592204671719 0.8482065834104272 - 0.1071592204671719 0.8482065834104272 0.0703660474881081 0.9372733924007060 - 0.0703660474881081 0.9372733924007060 0.0307532419961173 0.9879925180204854 - 0.0307532419961173 0.9879925180204854 152 mat

Legendre wr 63 ... 64

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.

Legendre-Gauss Quadrature,Arbitrary interval

Hermite spline project

x b c spline c x 1 n +^* 2 c * b + (x 2 n +^* - c b + (x 3 n +^* + :

The working experimental constrained "spline (x,v)... n = user" x v spline 0 x ≤ (x 1 ≤ (& v 2 el x 1 n +^* 2 v 2 el * v 1 el + (x 2 n +^* - v 2 el v 1 el + (x 3 n +^* + out range if :

v 0.5 1 - 21 mat :

n 0.5 - :

x Spline x v spline 1 x - v spline + 2 / :

x q x sqrt 2 x 3 - x + (* + (* 2 / - :

Unary User UnitSystem

x λ 231 x - (* - 1 x - (2^+ (1 x - sqrt * 2 / - :

UnitGraphArcLength 1 :

x h x q x λ + 2 / :

UnitGraphChainetteArea 1 :

x h x sqrt 2 x 3 - x + (* + (* 21 x - (2 x + (* - (1 x - sqrt * + 4 / -

w wr 1 col : weight, wi ≡ x wr 2 col : abscissea,xi ≡ 21 line

More Quadratures---------- Mathcad 11.2a Quadratures: 1. Legendre 2. Radau 3. Hermite 4. Laguerre 5. Lobato 6. Jacobi 7. ChebyShev

a 0 : b 1 : m wr rows : 31 line

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

Define the integrand x f 1 x h x diff (2^+ sqrt :

x Spline x h 2 1 sys

Legendre quadrature [64] ArcLength b a - 2 / (w i el y i el f * (i 1 m sum eval * : 1.2258

Arc length FD 1000 pts 1.221 ≡

Area b a - 2 / (w i el y i el h * (i 1 m sum eval * : 0.2095 -

The Hermite "chaînette" ... user experimental arc length

x q x λ x h 3 1 sys

f a b Int x f x a b int 32 / b a - (* w G w 01 int * ≡ memo Romberg ≡ w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : 32 / b a - (* w G w 01 int * 41 line :

w wr 1 col : x wr 2 col : m wr rows : 31 line

a 0 : b 1 : m wr rows : 31 line

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

x f 1 x h x diff (2^+ sqrt : integrand ≡

No matter how much maths you will attempt: you will never know the exact arc length of the chaînette. Just be happy of what the best of maths can answer, from being close to asymptotes. Romberg is comfortably convergent.

Q b a - 2 / (w i el y i el f * (i 1 m sum eval * :

Experiment.. a 0.0000001 : b 0.9999999 : 21 line

Romberg 1 1.2226 2 1.2295 3 1.2317 4 1.2323 42 mat :

Smath 1 1.2239 2 1.2446 3 1.3019 4 1.4821 42 mat :

f a b Int Romberg x f x a b int Smath Q Legendre 32 mat 1.2323 Romberg 1.4821 Smath 1.2258 Legendre 32 mat

Romberg Smath 2 1 sys

Smath

Romberg

Arc length FD 1000 pts 1.221 ≡

a 0.0001 : b 0.9999 : 21 line 1.2226 1.2239 1.2258 31 mat 12 mat

a 0.00001 : b 0.99999 : 21 line 1.2295 1.2446 1.2258 31 mat 12 mat

a 0.000001 : b 0.999999 : 21 line 1.2317 1.3019 1.2258 31 mat 12 mat

a 0.0000001 : b 0.9999999 : 21 line 1.2323 1.4821 1.2258 31 mat 12 mat

x h x f x h x diff 2 1 sys

Hermite chaînette

chaînette integrand

a 0 : b 1 : x f x h : 31 line

Area of the chaînette

x f x a b int f a b Int b a - 2 / (w i el y i el f * (i 1 m sum eval * 31 mat 0.209522901734104 - 0.20952380952382 - 0.209524211792767 - 31 mat

From above ....  before assigning 'x' to Legendre x h x 12 /^2 x 3 - x + (* + (* 21 x - (2 x + (* - (1 x - (12 /^* + 4 / - ≡

u Φ u 12 /^2 u 3 - u + (* + (* 21 u - (2 u + (* - (1 u - (12 /^* + 4 / - :

u Int u Φ u int maple :

u Int u 32 /^15 u 2^* - 710 - 9 u * + (* + (* 151 u - (2^* 71091 u - (* - (* + (1 u - (32 /^* + 210 /

hatch

Erzeugt eine Matrix für die Darstellung vertikaler Linien zwischen den Funktionen f(x) und 
g(x) im Bereich von x1 bis x2 mit 2*n Linien. Generates a matrix for 2*N vertical lines (hatching) 
between functions f(x) and g(x) in x-range from x1 to x2 f g x.1 x.2 N hatch dx x.2 x.1 - N / (: h x.1 x.1 f x.1 x.1 g 22 mat : x x.1 x.2 dx - x.1 dx + range h h x dx 2 / + x dx 2 / + g x dx 2 / + x dx 2 / + f x dx + x dx + f x dx + x dx + g 42 mat stack : for h 41 line :

hatch x g 0 :

Hermite chaînette and its area

x Φ x Int f g 0 1 20 hatch 3 1 sys

Area of the Hermite chaînette b Int a Int - 0.20952380952381 -