Simplex Utilities

3f828ae4-edc8-4bee-910f-b60ff92710d4 84 3 12 decimal

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Simplex Utilities

k gencm1 N 2 k matrix : i 1 k range N 1 i el i 1 - : N 2 i el k i - : 21 line for N 31 line :

nr nc v genm M nr nc matrix : i 1 nr range j 1 nc range M i j el v : for for M 31 line :

5 gencm1 012344321025 mat

330 genm 00000000033 mat

Carlos added,
enables recursive N 2 k matrix ≡ M nr nc matrix ≡ 21 sys

M idx M transpose 1 col rows :

5 gencm1 idx 5

Carlos  'submat' M 1 n 1 + (range 2 w range el M 1 n 1 + (2 w submatrix ≡

330 genm idx 3

without submatrix(...) n k gencm n 1 ≡ k gencm1 M n 1 + 10 genm : i 0 k range N n 1 - k 1 + (i - gencm : w N idx 1 - : O 1 w 1 + i genm : N O N stack : M M N augment : 51 line for w M idx 0 - : M 1 n 1 + (range 2 w range el 41 line if 11 line :

15 gencm 012344321025 mat

Only true for n=2 and any 'k'

24 gencm 11 mat 000001111222334012340123012010432103210210100315 mat 11 mat

weights(n,s)is universal for any dimension n s weights d 2 s * 1 + : i 0 s range w i 1 + el 1 - (i^22 s * -^* d n + 2 i * - (d^i! (d n + i - (! (* / * : for w 31 line :

Simplex integration algorithm f n s ∫ v 0 : Pt n s pts : w n s 1 - weights : i 1 w rows range P Pt i el : v v w i el P j col f j 1 P idx sum * + : 21 line for v 51 line :

n s pts d 2 s * 1 + : i 0 s range M n s 1 + (i - gencm : M M 2 n 1 + 1 M idx submatrix : Pt i el 2 M * 1 + d n + 2 i * - / : 31 line for Pt 31 line :

Specific for 1D

Recast the simplex domain

Simplex 1D integrator  ... domain [a,b] ... 's' convergence f a b s ∫ u b a - (34 / 2 x 1 el * 1 - (* 14 / 2 x 1 el * 1 - (3^* - 12 / + (* a + : x f u f 12 x 1 el * 1 - (2^- (* : simplex 32 / b a - (* f 1 s ∫ * : simplex 41 line :

∫Adaptive(f,a,b,N,ε)

f a b N ε ∫Adaptive s 0 : sum 0 : i 1 : tol i el ε : aa i el a : h i el b a - (2 / : 51 line .... fa i el a f : fc i el a h i el + f : fb i el b f : s i el h i el fa i el 4 fc i el * + fb i el + (3 / * : L i el 1 : 51 line 13 mat above => initialise the while loop i 0 > fd aa i el h i el 2 / + f : fe aa i el 3 h i el * 2 / + f : 12 mat s1 h i el fa i el 4 fd * + fc i el + 6 / (* : s2 h i el fc i el 4 fe * + fb i el + 6 / (* : 12 mat v1 aa i el : v2 fa i el : v3 fc i el : v4 fb i el : v5 h i el : v6 ε : v7 s i el : v8 L i el : 18 mat i i 1 - : s1 s2 + v7 - abs v6 < sum sum s1 s2 + (+ : v8 N ≥ Solution Insufficient Number of Intervals : i i 1 + : aa i el v1 v5 + : fa i el v3 : fc i el fe : fb i el v4 : 14 mat h i el v5 2 / : tol i el v6 2 / : s i el s2 : L i el v8 1 + : 14 mat i i 1 + : aa i el v1 : fa i el v2 : fc i el fd : fb i el v3 : h i el h i 1 - el : s i el s1 : 16 mat L i el L i 1 - el : 61 line if if 51 line while sum 51 line eval :

1 x - ln - x / 1 10 1 1 - 2 2 mat 2 1 sys

Test function x f 1 x - ln - x / :

3 appVersion 0.98.6179 ≡ WS_timing 13 min ≡ 21 mat

'N' iterations
precision N 1000 : acc 1015 -^: 21 sys

User inputs

lower limit
upper limit a 1015 -^: b 11015 -^- : 21 sys

Zeta(2) 1.644934066848226

t0 1 time :

Zeta(2) ... f a b N acc ∫Adaptive 1.64493406684842

1 time t0 - 60 s *

x f x - exp x sqrt / :

a 1012 -^:

x c 1 a x ≤ (x 1 ≤ (& cases :

Simplex needs be scanned for best 's'

x f x c * 0.1 25 0.1 10 - 2 2 mat 2 1 sys

whatever [acc] ... can't integrate
Adaptive MCD ... 1.49364826562485
[Wolfram α]    ... 1.49364826562485 x f x a 1 int 1668.071

t0 1 time :

sanity 12 D f 0111 ∫ 1.493648265624

π sqrt 1 erf * 1.49364826562485

1 time t0 - 0.4 s *

Explore i 714 range ∫ i 7 - el f 01 i ∫ : for ∫ 21 line :

Explore 1.49364827281499 1.49364826611909 1.49364826565873 1.49364826562379 1.49364826563812 1.49364826566673 1.49364826557079 71 mat

Simplex reverses convergence as 's' increases. Once you have determined the direction wrt f(x) use the reported best 's' ... here 's=11'

Search: 's' reverse convergence s k 6 : s k 15 range f 01 s ∫ f 01 s 1 - ∫ ≥ index k el s : k k 1 + : 21 line k if for s k 1 + : 31 line :

Optimized best simplex 's' s 11

sanity ...  1.49364826562485 ... 12 D f 0111 ∫ 1.49364826562379

π sqrt 1 erf * f 0111 ∫ - 11012 -^*

Γ(x) [15 D] Alvaro Original

Γ.κ 0.99999999999999709182 57.156235665862923517 59.597960355475491248 - 14.136097974741747174 0.49191381609762019978 - 0.33994649984811888699 104 -^* 0.46523628927048575665 104 -^* 0.98374475304879564677 104 -^* - 0.15808870322491248884 103 -^* 0.21026444172410488319 103 -^* - 0.21743961811521264320 103 -^* 0.16431810653676389022 103 -^* - 0.84418223983852743293 104 -^* 0.26190838401581408670 104 -^* - 0.36899182659531622704 105 -^* 151 mat :

ζ Γ z ζ 1 - : zh z 0.5 + : zg zh 607128 / + : zp zg 0.5 zh *^: 14 mat 2 π * sqrt Γ.κ 1 el Γ.κ κ 1 + el z κ + / κ 114 sum + (* zp 2^* e zg -^* eval 21 line :

Valid for Re(ζ)>0

x c 10 x ≤ (x 5 ≤ (& cases :

Test function x f x Γ :

'N' iterations
precision N 1000 : acc 106 -^: 21 sys

x Γ x c * 1 5 1 5 - 2 2 mat 2 1 sys

User inputs

lower limit
upper limit a 1012 -^: b 1 : 21 sys

27.3843 Wolfram ≡

t0 1 time :

f a b N acc ∫Adaptive 27.3843

1 time t0 - 19 s *

About these two models ∫-integrators . . .

1. Both shapes will possibly result from Engineering works.

2. Both integrators are user range [a .. b].

3. Both model test functions have exact values for comparison

wrt the applicable ∫-integrators.

4. Observe the hyperfast Simplex, escapes ± ∞/singularities.

5. Simplex is 5 decimals or better from all examples explored.

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simplex x - exp x sqrt /

adaptive 1 x - ln - x /

Example for advanced integrators . . .

This arc length has no verdict except if possibly measured otherwise

]]> x 1 x - ( * sqrt 1 1 - 2 x * + ( x 1 x - ( * sqrt * 2 x * 1 x - ( * / - ( 2 ^ + sqrt 2 1 sys

basket handle x f x 1 el 1 x 1 el - (* sqrt :

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

setup 1D simplex n 1 : s 11 : 12 mat x f x 1 el 1 x 1 el - (* sqrt : 21 sys

sanity confirmed f n s ∫ 0.393

t0 1 time :

setup 1D simplex n 1 : s 40 : 12 mat x f 11 - 2 x 1 el * + (x 1 el 1 x 1 el - (* sqrt * 2 x 1 el * 1 x 1 el - (* / - (2^+ sqrt : 21 sys

MCD sanity  1.571 f n s ∫ 1.574

1 time t0 - 2.9 s *

x f x sqrt exp :

In this example, we don't know the best ∫ and whereas Simplex oscillates @ low 's' we are confined to manual search, reputed to start around 7/8. Here, from our freaked "Technical accuracy", we obtain ~ 10 Decimals. At least, Simplex will never be disatrous.

Search: 's' reverse convergence s cnt 8 : s cnt 15 range f 02 s ∫ f 02 s 1 - ∫ ≤ index cnt el s : cnt cnt 1 + : 21 line cnt if for s cnt 1 - : 31 line :

5.40752818465627 Maple Technical accuracy ≡

7

x f 2 10 2 2 - 2 2 mat 2 1 sys

s 11

f 027 ∫ f 028 ∫ f 029 ∫ f 0210 ∫ f 0211 ∫ 51 mat 5.40752817493085 5.40752818330281 5.40752818453319 5.40752818464906 5.40752818462219 51 mat

Simpson-Romberg integration . . .

' n' reconciliates Simpson acc, as local command line.

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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 f a b n 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^- (* : 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 :

sanity check the sextuplets Mathcad 11 Adaptiv 5.40752818465632 ≡ Technical accuracy 5.40752818465627 ≡ Simpson Transf'd.. 5.40752818465635 ≡ Smath Simpson 7½ D 5.4075281 ≡ Simplex 10½ D .... 5.40752818462 ≡ 51 line

x f x sqrt exp :

t0 1 time :

transformed domain [0,1] <= Simpson [a,b] f 022000 Int 5.40752818465635

1 time t0 - 10.3 s *

Adaptive not best 6½

infinite limit x f x 0.5 1 -^x - exp * :

'N' iterations
_ precision _iter 5000 : TOL 1015 -^: 21 sys

x f 1 1 1 1 - 2 2 mat 2 1 sys

lower limit 
upper limit a 1014 -^: b 1 : 21 sys

t0 1 time :

sanity Wolfram ... 1.49364826562485 f a b iter TOL ∫Adaptive 1.49364806562498

1 time t0 - min 2.6

π sqrt 1 erf * 1.49364826562485

Resume reconciliation . . .

1. The 4 ∫ are on the user range [a .. b] reduced [0 .. 1].

2. Ordinary Simpson still valid for trivial cases.

3. ∫_Simplex oscillates, thus difficult to implement,

perfect for arc length "basket handle" type demo.

4. ∫Adaptive more universal, specific to some fnct.

5. Int(f,a,b,n) Romberg-Simpson: improved Simpson,

sometimes has no improving effect.

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Legendre Quadratures . . .

Legendre Quadratures as applicable to specified Standards.

1. Area & ArcLength "UnitGraphic", scale as per project.

2. Longer Legendre tables exist, unable to collect.

https://keisan.casio.com/exec/system/1280624821

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f(x) as  integrand x f 1 x ln (2^+ sqrt :

x arc x f x diff :

arc length component x arc x ln 1 x ln 2^+ sqrt x * /

arc length integrand x f 1 x arc (2^+ sqrt :

Red imitates Venturi throat

x f 1 x ln ( 2 ^ + sqrt e 2 e 2 - 2 2 mat 3 1 sys

Apply Simplex ... unknown sanity
seemingly => converges @ higher 's' f 0 e 11 ∫ 9.926

Legendre wr 63 ... 64

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 :

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 :

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.

a 1015 -^: b e : m wr rows : 31 line

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

wrt unary graphic ArcUnit 1 : AreaUnit 1 : 21 sys

Legendre is reasonably accurate and fast. f a b Quad i 1 m range y i el b a - 2 / (λ 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 1 x ln (2^+ sqrt :

t0 1 time :

f a b Quad AreaUnit 3.57

arc length integrand x f 1 x arc (2^+ sqrt :

f a b Quad ArcUnit 9.74

1 time t0 - 0.4 s *

α 14 / : β 3 : 12 mat

xo 0.125 - :

Throat 0.5 :

x Nozzle 1 x xo - (α^(β^/ :

x c 10 x ≤ (x Throat ≤ (& cases :

X Throat Nozzle : 1.423

Pipe "ID" 0 Nozzle 4.757

x Nozzle x c * x Nozzle x c * - Throat X Throat X - 2 2 mat 0 0 Nozzle + 100 black 1 5 mat 0 0 Nozzle - + 100 black 1 5 mat 5 1 sys

Finite Differences [FD] . . .

FD is known to under-estimate integration.

Generally, it under-estimates little bit.

FD is the sole method for noisy data set,

and considered immune to noise.

Plot to see FD cumulative integration .

Other FD algorithms not explored.

]]> x 1 x - ( * sqrt 1 1 - 2 x * + ( x 1 x - ( * sqrt * 2 x * 1 x - ( * / - ( 2 ^ + sqrt 2 1 sys

basket handle x q x 1 x - (* sqrt :

formal arc length integrand x arc 1 x q x diff (2^+ sqrt :

arc length integrand
from symbolic expansion. x f 11 - 2 x * + (x 1 x - (* sqrt * 2 x * 1 x - (* / - (2^+ sqrt :

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 :

data U 0.001 0.999 0.002 range : i 1 U rows range vy i el U i el f eval : for U vy augment 31 line :

Finite differences Cumulative Integral

data Σ data 2 1 sys

Long calculate
Not contested 0.0001 0.9999 0.0002 range (1.55 ≡ MCDadaptive 1.57 ≡ 21 mat

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

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 :

Trace-view cumulative @ reduced accuracy
as counterpart of ordinary integrator. Σ x f x sqrt exp : U 02 0.1 range : i 1 U rows range Cumul i el f 0 U i el 100 Int eval : for U Cumul augment 41 line :

Σ . 12 black plot Σ 2 1 sys

Appendix . . . Singular escapes Division by zero

∫singular(f,#,#,TOL) . . . profitable piece-wise

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Iterative integration ... prohibitive in time, unexplored.
May rescue the projet if no other method is profitable. f a b TOL ∫singular Y1 b a - (a b + 2 / f * : Y2 b a - 6 / 2 a * b + 3 / f a 2 b * + 3 / f + (* : N 1 TOL / : Y1 Y2 - abs TOL ≥ Y1 Y2 : N 2 N * : Y2 0 : h b a - N 1 - / : j 1 N range x0 a j h * + : x1 x0 h + : 12 mat Y2 Y2 h x0 f x1 f + 2 / (* + : 21 line for 51 line while 2 Y2 * Y1 - 51 line eval :

f # # TOL ∫singular

t0 1 time :

a 109 -^: b 0.1 : c 0.25 : d 0.5 : 14 mat

x f x ln :

TOL 0.001 :

Does not support TOL 0.0001 ∫1 f a b TOL ∫singular : 0.3302 -

∫2 f b c TOL ∫singular : 0.2663 -

∫3 f c d TOL ∫singular : 0.25 -

x c 1 a x ≤ (x 1.25 ≤ (& cases :

∫4 f d 1 TOL ∫singular : 0.1534 -

hoover to see inside

x ln x c * b 1 b 6 - 2 2 mat c 1 c 4 - 2 2 mat d 1 d 2 - 2 2 mat 1 1 1 2 - 2 2 mat 5 1 sys

'N' as left in the kernel iter N TOL * : 4

Σ ∫1 ∫2 + ∫3 + ∫4 + : 1 -

1 time t0 - min 1

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

∫singular(f,a,b,TOL) .. Disabled for Export/Experiment

'n' intervals in range from start '0' n range interval j 1 n 1 + (range h j el range - range j * + n / : for h 21 line :

X 161 interval :

Iterative integration ... prohibitive in time, unexplored.
May rescue the projet if no other method is profitable. f a b TOL ∫singular Y1 b a - (a b + 2 / f * : Y2 b a - 6 / 2 a * b + 3 / f a 2 b * + 3 / f + (* : N 1 TOL / : Y1 Y2 - abs TOL ≥ Y1 Y2 : N 2 N * : Y2 0 : h b a - N 1 - / : j 1 N range x0 a j h * + : x1 x0 h + : 12 mat Y2 Y2 h x0 f x1 f + 2 / (* + : 21 line for 51 line while 2 Y2 * Y1 - 51 line eval :

X 0 0.0625 0.125 0.1875 0.25 0.3125 0.375 0.4375 0.5 0.5625 0.625 0.6875 0.75 0.8125 0.875 0.9375 1171 mat

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 :

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 :

x f x - exp x sqrt / :

x c 10 x ≤ (x 1 ≤ (& cases :

π sqrt 1 erf * 1.49364826562485

j 2 X rows range :

U X j el :

ε 1012 -^:

split U U f vectorize augment :

x g 0 :

x f x c * x f x g ε 0.3125 15 hatch split . 12 black plot 3 1 sys

TOL 0.001 :

t0 1 time :

Σ i 26 range u i 1 - el X i el : ∫ i 1 - el f X i 1 - el X i el TOL ∫singular : 21 line for u ∫ 3 round vectorize augment 21 line :

Σ Σ :

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

1 time t0 - min 10

MCDadaptive [0.001] 1.012 Σ 2 col sum 1.01

Σ 0.0625 0.488 0.125 0.189 0.1875 0.136 0.25 0.108 0.3125 0.089 52 mat

Trapeze-Rectangle . . . Romberg kernel

' n' reconciliates acc as local command line.

Trapeze => Poncelet . . . Rectangle => Riemann

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∫Trapeze in the Romberg Transform Domain [0,1] <= [a,b] ... f a b n Trapeze 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 Trapeze h α G β G + 2 / 1 α k h * + G k 1 n 1 - sum * + (* eval : 32 / b a - (* Trapeze * 41 line :

x f x sin 2^:

f 0240000 Trapeze 1.189200623

∫Rectangle in the Romberg Transform Domain [0,1] <= [a,b] ... f a b n Rectangle 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 Riemann h α k h * + G k 1 n sum * eval : 32 / b a - (* Riemann * 41 line :

Most robust integrator for Smath . . .

Carlos => ∫Adaptive(f,a,b,N,ε) . . . manage limits as needed

]]> ∫Adaptive(f,a,b,N,ε)

f a b N ε ∫Adaptive s 0 : sum 0 : i 1 : tol i el ε : aa i el a : h i el b a - (2 / : 51 line .... fa i el a f : fc i el a h i el + f : fb i el b f : s i el h i el fa i el 4 fc i el * + fb i el + (3 / * : L i el 1 : 51 line 13 mat above => initialise the while loop i 0 > fd aa i el h i el 2 / + f : fe aa i el 3 h i el * 2 / + f : 12 mat s1 h i el fa i el 4 fd * + fc i el + 6 / (* : s2 h i el fc i el 4 fe * + fb i el + 6 / (* : 12 mat v1 aa i el : v2 fa i el : v3 fc i el : v4 fb i el : v5 h i el : v6 ε : v7 s i el : v8 L i el : 18 mat i i 1 - : s1 s2 + v7 - abs v6 < sum sum s1 s2 + (+ : v8 N ≥ Solution Insufficient Number of Intervals : i i 1 + : aa i el v1 v5 + : fa i el v3 : fc i el fe : fb i el v4 : 14 mat h i el v5 2 / : tol i el v6 2 / : s i el s2 : L i el v8 1 + : 14 mat i i 1 + : aa i el v1 : fa i el v2 : fc i el fd : fb i el v3 : h i el h i 1 - el : s i el s1 : 16 mat L i el L i 1 - el : 61 line if if 51 line while sum 51 line eval :

infinite limit x f x - exp x sqrt / :

'N' iterations
_ precision _iter 5000 : TOL 1015 -^: 21 sys

lower limit 
upper limit a 1014 -^: b 0.3125 : 21 sys

t0 1 time :

sanity Wolfram , MCD ...  1.011725 f a b iter TOL ∫Adaptive 1.011725

1 time t0 - min 2.4

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infinite limit x f x - exp x sqrt / :

'N' iterations
_ precision _iter 5000 : TOL 1015 -^: 21 sys

lower limit 
upper limit a 1014 -^: b 1 : 21 sys

t0 1 time :

sanity Wolfram ... 1.49364826562485 f a b iter TOL ∫Adaptive 1.49364806562499

1 time t0 - min 2.6

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x 1 x - ( * sqrt 1 1 - 2 x * + ( x 1 x - ( * sqrt * 2 x * 1 x - ( * / - ( 2 ^ + sqrt 2 1 sys

arc length integrand
from symbolic expansion. x f 11 - 2 x * + (x 1 x - (* sqrt * 2 x * 1 x - (* / - (2^+ sqrt :

'N' iterations
_ precision _iter 5000 : TOL 1012 -^: 21 sys

lower limit 
upper limit a 1014 -^: b 11014 -^- : 21 sys

t0 1 time :

sanity MCD... 1.570796 f a b iter TOL ∫Adaptive 1.570796

1 time t0 - min 1.6