plot(data,a,b,c)

64aeb7dd-6b40-4720-a199-aad56cc3b9e0 33 4 5 decimal

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

plot(data,a,b,c)

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

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 :

4 appVersion 0.98.6179.21440 ≡

Legendre Polynomials & Quadrature

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

x y axes x y / - :

n 12 : j n 1 + : 12 mat

n x poly j el axes 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 :

maxorder 30 ≡ n 8 : (order ≡ C n V : r C polyroots : r r eval : 51 line

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

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

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 0.362683783378 0.183434642496 - 0.362683783378 0.183434642496 0.313706645878 0.525532409916 - 0.313706645878 0.525532409916 0.222381034453 0.796666477414 - 0.222381034453 0.796666477414 0.101228536291 0.960289856498 - 0.101228536291 0.960289856498 82 mat

1 time t0 - 1 s *

roots r r 0 * augment : roots roots . 14 black plot : 21 line

roots n x L 2 1 sys

r 0.1834 - 0.1834 0.5255 - 0.5255 0.7967 - 0.7967 0.9603 - 0.9603 81 mat

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

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.

Нахождение нулей полиномов Лежандра

Penyelesaian Polinomial Legendre

Legendre Quadrature

No matter that some numerical integrators reveal higher accuracy than Legendre. This type of quadrature is the core integrator in many standards and hyperfast. It allows traceability for QA approval.

weight, wi w wr 1 col :

abscissea, xi u wr 2 col :

m wr rows : 63

Legendre Quadrature ... algo style a b f 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 :

x f x sqrt exp :

t0 1 time :

02 f Quad 5.4075293

1 time t0 - 0.1 s *

x f

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

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 axes 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 r r 0 * augment : roots roots . 14 black plot : 21 line

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

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

Supplement stand alone export section . . .

]]> 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 :

Hermite spline parameters
.....
Experiment n = 0 ... -0.125 b 0.5 : c 1 - : n 0 : 31 mat

x λ 10 x ≤ (x 2 ≤ (& cases :

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

Not knowing more about other possible numerical integrators in lieu, Legendre Quadrature.

recast the spline function x f x b c spline :

x f x λ *

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

w wr 1 col : (weight, wi ≡ x wr 2 col : (abscissea,xi ≡ a 0 : b 2 : m wr rows : 13 mat 31 line

ArcLength 1 : Area 1 : 21 mat

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

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

Arc length quadrature [64] ArcLength b a - 2 / (w i el y i el arc * (i 1 m sum eval * : ArcLength 2.1832

Area quadrature [64] Area b a - 2 / (w i el y i el f * (i 1 m sum eval * : Area 1.5 1015 -^*

excat 'n = 0' x f x 02 int

Exercise for Legendre Quadrature

x q x sqrt 2 x 3 - x + (* + (* 2 / - : x λ 231 x - (* - 1 x - (2^+ (1 x - sqrt * 2 / - : x h x q x λ + 2 / : x λ 10 x ≤ (x 1 ≤ (& cases : 41 sys

w wr 1 col : (weight, wi ≡ x wr 2 col : (abscissea,xi ≡ a 0 : b 1 : m wr rows : 13 mat 31 line

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

x h x λ *

ε 0.000001 : ε h 1 ε - h 0.5 h 41 mat 1106 -^* 0.0005 - 0.0005 - 0.2652 - 41 mat

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

Legendre quadrature [64] Arc b a - 2 / (w i el y i el f * (i 1 m sum eval * : 1.226

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

Cannot calculate,
else error message.
Work with limits. x f x 01 int

Compatible Simpson x h x 01 int 0.2095 -

ε 0.000001 :

Undetermined
wrt 'ε' x f x ε 1 ε - int

acc 2.0214 1.3019 1.2364 31 sys 10010001000031 sys ≡

∫kernel(f,a,b,n)) ... Simpson/Romberg Kernel

Basic Romberg-Simpson ... integration acc from options f a b Int 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 * 31 line eval :

Simpson Integrator in the Romberg Transform Domain [0,1] <= [a,b] ...  
... integrates  Smath ainterp(X,Y,x), cinterp(X,Y,x), cubic(x) from  l_p_c_spline 
Integration accuracy from user command line 'n' f a b n ∫kernel w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : α 0 : β 1 : h β α - n / : 14 mat Simpson 13 / h 2 / α G β G + (* h α k h * + G k 1 n 1 - sum * + (* 23 / h * α h 2 / - k h * + G k 1 n sum * + eval : 32 / b a - (* Simpson * 41 line :

on low span   ►  ... small 'n' f ε 1 ε - 40 ∫kernel 1.2254

end of supplement

Sanity Legendre vs known exact . . .

]]> Legendre wr63 ... wr64

wr63 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 :

wr64 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 :

Test function x f 4 x 2^1 + / :

select wr63/wr64

x f $w wr64 1 col : (weight, wi ≡ x wr64 2 col : (abscissea,xi ≡ a 1 - : b 0 : 12 mat m wr rows : 41 line$ $a 1 - : b 0 : 12 mat wr64 ≡ a 0 : b 1 : 12 mat wr63 ≡ 21 mat$ $y i 1 m range y i el b a - 2 / (x i el * b a + 2 / (+ : for y 21 line :$ $t0 1 time :$ $Area b a - 2 / (w i el y i el f * (i 1 m sum eval * π - : 0$ $1 time t0 - 2.6 s *$ $x f x a b int π - 81015 -^*$ $x f x 1 - 0 int π x f x 01 int 31 mat 3.141592653589800 3.141592653589790 3.141592653589790 31 mat$ $u b 0 u 2 - ≤ 16 / 2 u + (3^* 2 - u ≤ (u 1 - ≤ (& 16 / 2 u + (3^* 23 / 1 u + (3^* - 1 - u ≤ (u 0 ≤ (& 23 / 1 u - (3^* - 16 / 2 u - (3^* + 0 u ≤ (u 1 ≤ (& 16 / 2 u - (3^* 1 u ≤ (u 2 ≤ (& 0 cases :$

The Gaussian b-spline

Gaussian basis 'b-spline'

2 - x ≤ x 2 ≤ & x b Undefined if

0 b 0.6667

known exact u b u 2 - 2 int 1.000000000000000

w wr63 1 col : (weight, wi ≡ x wr63 2 col : (abscissea,xi ≡ a 2 - : b 2 : 12 mat m wr rows : 41 line

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

Area b a - 2 / (w i el y i el b * (i 1 m sum eval * : 1.000001

More Gauss-Lobatto coeffs [100] from https://keisan.casio.com/exec/system/1280801905

end of supplement

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

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

x D x h x diff :

title_1 0.25 0.15 - Hermite chainette 12 blue 15 mat : title_2 0.175 2 Arc Integrand 12 blue 15 mat : 21 sys

x h 0.1 x D * title_1 3 1 sys x f x D title_2 3 1 sys

Sanity Hermite Chainette arc length . . .

]]>

L Δ N inc X 0 : i 1 N range X i 1 + el L Δ i * + eval : X 1 el L : 21 line for X 31 line :

x f x sqrt 2 x 3 - x + (* + (* 21 x - (2 x + (* - (1 x - sqrt * + 4 - / eval : Δ 0 0.001 1000 inc : data Δ Δ f vectorize augment : data data eval : 41 sys

sanity end coord's data data rows row 1012 mat

Riemann rectangles Finite Differences Arc length Finite differences derivative X 0 : Y 0 : 12 mat X data 1 col : Y data 2 col : 12 mat n X rows : deriv 1 el 0 : deriv n el 0 : i 2 n 1 - (range deriv i el 0.5 Y i 1 - el Y i el - X i 1 - el X i el - / Y i el Y i 1 - el - X i el X i 1 - el - / + (* eval : for deriv 1 el deriv 2 el : deriv n el deriv deriv rows 1 - el : Dp X deriv augment : Arc integrand .................... xx Dp 1 col : δx Dp 2 col : 12 mat i 1 δx rows range yy i el 1 δx i el (2^+ sqrt eval : 11 line for Arc xx yy augment : Arc length ....................... vx Arc 1 col : vy Arc 2 col : 12 mat S 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - eval : S i el S i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + eval : 21 line for Σ vx S augment : 191 line

Chainette & derivative

Dp data 2 1 sys

Arc integrand

Arc 1 1 sys

increase 0.001 canvas demand 0.625

user @ over  N=1000 user 0.626 1000 * : length Σ 2 col : 21 sys

length user el 0.749

Cumulative arc length

Σ 0.001 user * length user el + 40 red 1 5 mat 2 1 sys

@ this fineness of FD, it is reputed of somewhat greater accuracy than Legendre.

Arc length
@ user canvas Σ Σ rows row Σ 626 row 21 mat 1 1.228 12 mat 0.625 0.749 12 mat 21 mat