algo(lo,mesh,hi,∫)

5c60c9c7-0039-42c0-939e-8ac72caf0c5d 22 4 5 decimal

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

Style_12: Profitable/accurate FD [Finite Differences] ...

algo(lo,mesh,hi,∫)

lo mesh hi f algo data U lo hi mesh range : i 1 U rows range vy i el U i el f eval : for U vy augment 31 line : data data : data Σ FD Cumulative 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 : 31 line data Σ 21 line :

Σalgo(lo,mesh,hi,∫)

plug integral curve

x Clear 1

Cumulative integration algo style ... lo mesh hi ∫ Σalgo cumullative Integrat'n algo style Y 0 : data 0 : 12 mat x lo hi mesh range : data x ∫ vectorize augment : ux data 1 col : uy data 2 col : Y 1 el 0 : i 2 ux rows range Δx i el ux i el ux i 1 - el - : Y i el Y i 1 - el 12 / uy i 1 - el uy i el + (Δx i el * (* + : 21 line for ux Y augment 91 line :

Γ(x) [15 D] Alvaro

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

Hypergoeometric,
normalized N n p x PDF N 1 + Γ n 1 + Γ * N n + p - 1 + Γ * p 1 + Γ * x 1 + Γ x - n + 1 + Γ * N n + 1 + Γ * x N + p - 1 + Γ * x - p + 1 + Γ * / :

N 60 :

n 40 :

p 10 :

shift 0 :

x f N n p x PDF eval :

Continuous scalar PDF Hypergeometric.

N n 5 x shift - PDF N n 10 x shift - PDF N n 15 x shift - PDF 3 1 sys

option: Int. acc 100
native Smath Gamma(x)
rejected/crash Smath u CDF x f x 0 u int :

10.975 CDF 0.999

ε 1012 -^:

match the 'p' codes [5, 10, 15] p 10 :

t0 1 time :

Φ x f : lo 0 : mesh 0.025 : hi p 1 + ε - : 13 mat Σ lo mesh hi Φ algo : 31 line

u 0 p 0.1 range : CDF u u CDF vectorize augment : 21 mat

1 min [440 points]

N n 10 x shift - PDF Σ 2 1 sys

6.5 min [100 points]

N n 10 x shift - PDF CDF 2 1 sys

1 time t0 - min 0.8

n Σ rows : 440

Σ n row 10.975 0.999 12 mat

μ 0 : σ 2 : 12 mat ∫ x μ σ pdf.Normal eval : CDFN 10 - 9.99 - 10 ∫ Σalgo : 31 sys

t0 1 time :

∫ CDFN 2 1 sys

1 time t0 - 0.5 s *

CDFN rows 2001

Project notes . . .

FD is a Lagrangian method , ancestror of numerical CAS.

It ranges from very accurate to exact, especially suited

for applications otherwise cumbersome to numerical CAS.

In this demo, Hypergeometric is ideal Simpson but so

slow to useless @ higher # points.

Σalgo(lo,mesh,hi,∫) is the Emeritus for ∫ compatible.

for ∫ NOT compatible . . . alternate algo(lo,mesh,hi,Φ)

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SIMPLY DECEPTIVE ... !
1. CDF(u) pefect Simpson  case
2. algo considered exact
3. Σalgo considered exact u CDF vectorize 100 pts ~ 6 min ≡ algo 440 pts ~ 1 min ≡ Σalgo 2000 pts 0.5 s ≡ 31 mat

... attempting your function in Σalgo(-10,-9.99,10,∫) red indicateS NOT compatible

Σalgo(lo,mesh,hi,∫)

plug integral curve

x Clear 1

Cumulative integration algo style ... lo mesh hi ∫ Σalgo cumullative Integrat'n algo style Y 0 : data 0 : 12 mat x lo hi mesh range : data x ∫ vectorize augment : ux data 1 col : uy data 2 col : Y 1 el 0 : i 2 ux rows range Δx i el ux i el ux i 1 - el - : Y i el Y i 1 - el 12 / uy i 1 - el uy i el + (Δx i el * (* + : 21 line for ux Y augment 91 line :

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

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 :

∫Adaptive(f,a,b,N,ε) Carlos

f a b N ε ∫Adaptive 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 41 line eval :

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

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

x h x q x λ + 2 / :

ε 104 -^:

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

t0 1 time :

∫ x arc eval : x f x arc : arcLength ε ε 0.0001 + 1 ε - ∫ Σalgo : 31 sys

ε f 25

1 time t0 - min 2.3

n arcLength rows : 9999

3½ decimals certified arcLength n row 0.9999 1.2228 12 mat

Simpson kernel dependent upon 'n' f ε 1 ε - 100 Int 1.2226

2.5 x h * arcLength 2 1 sys

sanity Mathcad 1.2226
iter 100 ... plenty
'ε' canvas accuracy
'a' infinite limit
'b' infinite limit x f x arc eval : iter 100 : ε 0.00001 : a 0.0001 : b 0.9999 : 51 sys

f a b iter ε ∫Adaptive 1.2226

x f

The triplet integrators agree closely. For this example the verdict is: ∫Adaptive(#,#,#,#,#) Romberg-Simpson Int(f,a,b,n) renders best on long range of the integrand. Finite Differences algo(lo,mesh,hi,f) may be adopted as universal accurate.