Legendre wr 63 ... 64

652d7d2c-24c0-40f5-8ffd-2600365c00ca 61 4 5 decimal

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

Part: 1

Part:1 int acc 10000 => 1.4 min Part:2 int acc 100 ...

Test function x f x sqrt exp :

x f

Lecture:

Quadratures are generally obsolete [Legendre is the most common].

They can be used piecewise ... This document illustrates Romberg

integrator. The rough plot on range [0 .. 2] does not evidence the

killer region wrt "Simpson" integrator. From educated practice, if

Romberg does not improve, you must devise other approach.

Romberg has two virtues:

1. it reduces the integration range 0 .. 1 , thus more points for its

terminal Simpson integrator ∫G(w)

2. The integrand G(w) is now of a better form for Simpson Integrate.

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sanity check the quadruplet ... acc 10000 Mathcad 11 Romberg 5.40752818465632 ≡ Technical accuracy 5.40752818465627 ≡ Smath Romberg 5.40752818465638 ≡ Smath Integrator 10000 5.4075281 ≡ 41 line

a 0 : b 2 : 12 mat w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + : w G w s f 12 w * 1 - (2^- (* : 31 line

x s f 1 2 x * 1 - ( 2 ^ - ( x G 3 1 sys

Explain Romberg:

1. The Romberg integrand is the product of the transform

by the parabola. It reduces the original [a, b] space to

[0, 1]. The final integration is dimensionned 3/2(b-a).

2. wrt to Smath integrator, it has the virtue of improving

the integration ... 10000 accuracy is now on smaller span.

3. The other virtue results from transforming the original

integrand toward a form more appropriate for the Smath

native Simpson integrator.

=========================================

Do not conclude Romberg will cope with steep region(s)

of any function. Some Engineering standards require

only 4 decimals. Here, we attempt to be more Scientific.

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Confusing linear look of the test f(x)

x f

f a b Int x f x a b int 32 / b a - (* w G w 01 int * ≡ memo ≡ 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 :

x f x 02 int 5.4075281

f 02 Int 5.40752818465638

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

weight, wi w wr 1 col :

abscissea, xi x wr 2 col :

m wr rows : 64

a 0 : b 2 : 21 line

x f x sqrt exp :

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 f * (i 1 m sum eval * 5.40753

x Φ x :

x Φ

a sense of scale/accuracy 1. "fm = fermi = 10e-15" 2. classical diameter electron ...... 5.635882 fm ...... In this example, the native Smath 7 true decimals is now 13½

x Φ x 04 int 8

Part: 2

An Engineering function

Copy this section, paste in blank sheet, set integral accuracy 100.

By default, function definition is symbolic.
f(x) is  already the symbolic expansion of its
normalised rational fraction generatrix. x f x x β 3 el x * β 2 el + β 4 el x 2^* + (* β 1 el + (* 1 x x β 7 el x * β 6 el + β 8 el x 2^* + (* β 5 el + (* + / :

β 4.69292878185 2.485645088668 0.005049502793 - 0.008399478307 - 5.19337552684 0.698811411628 0.043875966805 - 0.000673223801 - 81 mat :

X 12 :

x c 10 x ≤ (x 15 ≤ (& cases :

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 datum x g 0 :

X X f + 15 black 1 5 mat x f x c * x f x g 0 X 20 hatch 3 1 sys

Note:

You can test Romberg => not to avail.

f(x) is very accurate but only an approximation.

For sure, it does not look linear compared to test.

However, Smath integrator is convenient.

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x f x 0 X int 25.6122

Σ CumulativeIntegration U 0 X 0.1 range : i 1 U rows range Cumul i el x f x U 1 el U i el int eval : for U Cumul augment 41 line :

enable @ integral accuracy 100