plot(data,char,size,clr)

17b4eb89-9c30-4532-bd79-71f794c146b2 98 4 5 decimal

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

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 :

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 :

The 10 most common Cumulative Integrators . . .

]]>

Style_1: Finites differences data set

end 12 :

α 1523043253 - 27354.7 - 43256.75 25388.3 - 40061 mat :

x c 10 x ≤ (x end ≤ (& cases :

Fréchet: extreme value-II x f α 1 el α 2 el x α 4 el - (/ (α 5 el^* α 2 el x α 4 el - (/ (α 3 el^- exp * α 6 el + :

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 end 0.1 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 data Σ 2 1 sys

Style_2: ODE ... experiment 17 solvers from Smath library

x Y D x f : sol 005100 x Y D dn_GearsBDF : 21 sys

u sol 101 row 1 el : v sol 101 row 2 el : 21 mat 51710921 mat

sol data Σ u v + 20 black 1 5 mat 3 1 sys

Style_3: Simpson compatible discrete

ε 1015 -^:

x integral x ln x int maple : x 1 - x ln + (*

Σ U ε 5 0.5 range : i 1 U rows range Cumul i el x ln x U 1 el U i el int eval : for U Cumul augment 31 line :

Ln Σ o 5 blue plot :

setacc 1000 ≡

Ln x integral x ln 3 1 sys

Style_4: Romberg-Simpson compatible

α 1523043253 - 27354.7 - 43256.75 25388.3 - 40061 mat :

Fréchet: extreme value-II x f α 1 el α 2 el x α 4 el - (/ (α 5 el^* α 2 el x α 4 el - (/ (α 3 el^- exp * α 6 el + :

x y^y x ln * exp ≡

recast Frechet

x F α 1 el α 5 el α 2 el x α 4 el - (/ ln * exp * α 3 el α 2 el x α 4 el - (/ ln * exp - exp * α 6 el + :

x f x c * x F 2 1 sys

Native Simpson fails miserably.
Will crash the freaked plot ! NFG U 05 0.25 range : j 1 U rows range Cumul j el x F x U 1 el U j el int eval : for U Cumul Re augment 31 line :

recast for the suite x f x F :

Rescued Frechet  from  [  Int(f,a,b,n)  ] Σ U 0 end 0.25 range : j 1 U rows range Cumul j el f 0 U j el 10 Int eval : for U Cumul augment 31 line :

5 x f * x c * Σ 2 1 sys

evaluate range in-situ f 51210 Int eval 12697

evaluate in-situ from low ... user f 0810 Int eval 26339

Style_5: Σalgo(lo,mesh,h,∫) algo style ... as applicable ........ splines(x), ProductLog(x), min(x) aren't 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 :

∫ x f eval :

CDF 0 0.125 end ∫ Σalgo :

5 ∫ * x c * CDF 2 1 sys

μ 0 : σ 2 : 12 mat ∫ x μ σ pdf.Normal eval : 21 sys

CDF 10 - 9.9 - 10 ∫ Σalgo :

∫ CDF 2 1 sys

Style_6: Riemann rectangles FD exact example

XY 004 0.75 8 0.75 12 0.2 42 mat :

X XY 1 col : Y XY 2 col : 12 mat x f X Y x linterp : x t D x f : sol 0012120 x t D dn_GearsBDF : 41 sys

sol sol rows row 12.000 6.399 12 mat

Cumulative Σ over magnified XY

X 4 Y * augment sol 2 1 sys

populate sub-divide polygon sub 16 :

Xo XY 1 col : Yo XY 2 col : 21 sys

X Xo : Y Yo : 21 sys

sub-spaces

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

data sub subspace j 1 data rows 1 - (range Vsub j el i 1 sub range vx i el data j el i sub / data 1 j + el data j el - (* + : for vx 21 line eval : 11 line for v Vsub 1 el : j 2 Vsub rows range v v Vsub j el stack : for v 31 line 21 line :

Construct the sub-spaced polygon

XY X Y augment : origin XY + 15 blue plot : 21 line

New Xo sub subspace Yo sub subspace augment : New Xo 1 el Yo 1 el 12 mat New stack : 21 line

isolate New New :

X New 1 col : Y New 2 col : 12 mat SUB New . 12 black plot : 21 line

Discnotinuous equally sub-spaced ... populated for FD cumulative

origin XY SUB 3 1 sys

data Σ FD Cumulative Integration vx data 1 col : vy data 2 col : 12 mat Y 0 : Y 1 el 0 : 12 mat 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 51 line :

New New Σ 2 1 sys

4 0.75 * 2 / 4 0.75 * + (0.75 0.2 - (4 * 2 / (+ 0.2 4 * (+ 6.400000000000000

n New Σ rows : 49

New Σ n row 12.000000000000000 6.400000000000000 12 mat

Style_7: Multiple ordinary Simpson ... unpredictable accuracy

Depends upon option Simpson 'acc' # f Σ ε 1012 -^: U ε 5 0.05 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 :

Assign user as function

x q 1 x 2^1 + / :

x f 0.25 x * :

x f Σ x ln Σ x q Σ 3 1 sys

Σ U 2 - 2 1.9 - range : i 1 U rows range Cumul i el x erf vectorize x U 1 el U i el int eval : for U Cumul augment 31 line :

Σ 2 1 sys

The Emeritus champion . . . as applicable

]]>

t0 1 time :

Style_8: Σalgo(lo,mesh,hi,∫)... hyper-accurate/fast 1503 [5 s]

∫ x ln eval : lo 1012 -^: mesh 0.01 : hi 5 : 13 mat Σalgo_1 lo mesh hi ∫ Σalgo : 31 sys

∫ 0.25 x * eval : lo 0 : mesh 0.01 : hi 5 : 13 mat Σalgo_2 lo mesh hi ∫ Σalgo : 31 sys

∫ 1 x 2^1 + / eval : lo 0 : mesh 0.01 : hi 5 : 13 mat Σalgo_3 lo mesh hi ∫ Σalgo : 31 sys

1 time t0 - 5 s *

collect Σalgo_1 Σalgo_2 Σalgo_3 31 sys :

collect

Style_9: Σ(data) ... rescue => Σalgo(lo,mesh,hi,∫)

min(,) undocumented x f 2110 x - (* 260 x - 2 / 5128 x - (* 4 / 31 mat min :

x f x bound 412004164 - 22 mat 802008064 - 22 mat 41 sys

x bound 1080 x - (* 7 / :

x f x bound 41 200 41 64 - 2 2 mat 80 200 80 64 - 2 2 mat 4 1 sys

accepted
acc 50 x f x bound - (x 16.923 - 185 int 6911.5

x f x bound - 0 ≡ x 32 - 0 solve x f x bound - 0 ≡ x 160200 solve 21 mat 16.923 - 18521 mat

L H N Φ i 1 N range ux i el L i N / H L - (* + : uy i el ux i el f ux i el bound - (eval : 21 line for ux uy augment 21 line :

data 16.923 - 185400 Φ :

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 Σ

n data Σ rows :

data Σ n row 185 6911.4 12 mat

Style_10: Adaptive on canvas ... user point value

∫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 0.5 1 - ^ x - exp * π sqrt x sqrt erf * 2 1 sys

eval(,) gains 1 min
.....
N 100 plenty
'ε' canvas accuracy
.....
'a' infinite limit
'b' user end canvas plot x f x 0.5 1 -^x - exp * eval : iter 100 : ε 0.01 : a 1015 -^: b e : 51 sys

x Canvas f a x iter ε ∫Adaptive :

x c 1 a x ≤ (x e ≤ (& cases :

x λ 1 a x ≤ (x 5 ≤ (& cases :

x 0.5 1 -^x - exp * π sqrt x sqrt erf vectorize * - 0 ≡ x 0.25 0.5 solve 0.384

t0 1 time :

Adapt Discrete cumulative on that range ux a 5 0.1 range : j 1 ux rows range uy j el f a ux j el iter ε ∫Adaptive eval : for ux uy augment 41 line :

isolate < timing Adapt Adapt :

vx Adapt 1 col : vy Adapt 2 col : 12 mat x spline vx vy x cinterp : 21 sys

1 time t0 - min 0.6

Adapt rows 51

Scalar Canvas(x)*c(x) plots on each canvas pixel: higher timing

x f x Canvas x c * x spline x λ * e e spline + 40 black 1 5 mat 4 1 sys

REMEMBER ... The Rieman rectangles FD ranges from very accurate to exact. It is almost insensible to noisy data sets. ODE could be the first companion to experiment. Watch ± ∞ limit/singular generally a good candidate, very fast. Sanity check ODE vs Adaptive.

1 time t0 - min 2.8

Style_11: Reluctant Hypergeometric ...

Gamma(x) Lanzcos Davide

z Lanczos Davde Carpi 20200514 ε 11015 -^* : p 57.156235665862923517 59.597960355475491248 - 14.136097974741747174 0.49191381609762019978 - .33994649984811888699 104 -^* .46523628927048575665 104 -^* .98374475304879564677 - (104 -^* .15808870322491248884 103 -^* .21026444172410488319 - (103 -^* .21743961811521264320 103 -^* .16431810653676389022 - (103 -^* .84418223983852743293 104 -^* .26190838401581408670 - (104 -^* .36899182659531622704 105 -^* 141 mat : g 4.7421875 : z Re 0.5 < y π π z * sin 1 z - Lanczos * / : z z 1 - : x 0.99999999999999709182 : n 1 p length range x x p n el z n + / + : for t z g + 0.5 + : y 2 π * sqrt t z 0.5 +^* t - exp * x * : 51 line if y Im abs ε ≤ y Re y if 61 line :

g=4.7421875, n=15

x Lanczos

x 2.5 i * :

2.5 Lanczos 1.32934038817912

2.5 Gamma 1.32934227535731

Lanczos damned slow ... does NOT plot f1(x) x Γ x Gamma :

Hypergeometric vanishes
for  'u' > p + 1 -ε N n p u PDF N 1 + Γ n 1 + Γ * N n + p - 1 + Γ * p 1 + Γ * u 1 + Γ u - n + 1 + Γ * N n + 1 + Γ * u N + p - 1 + Γ * u - p + 1 + Γ * / :

N 60 :

n 40 :

shift 0 :

'p' PDF(N,n,p,u)

x f1 N n 5 x shift - PDF eval : x f2 N n 10 x PDF eval : x f3 N n 15 x PDF eval : 31 sys

5.999 f1 8.3832 107 -^*

ε 1012 -^:

x f1 x f2 x f3 0 0 + 100 black 1 5 mat 4 1 sys

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

Cumulative starts @ PDF(N,n,p,'0') data U 0 p 1 + ε - (0.025 range : i 1 U rows range vy i el N n p U i el PDF 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 :

end data rows : 440

user 175 :

∫end data Σ end row : 10.975 0.999 12 mat

data Unest v data 1 el : j 2 data rows range v v data j el stack : for v 31 line :

Hy data user row data Σ user row 21 mat Unest : 4.35 0.255 4.35 0.597 22 mat

Finite Differences Cumulative Integration

data data Σ p 1 + 0 + 40 black 1 5 mat Hy 1 1 el Hy 1 2 el + 20 black 1 5 mat Hy 2 1 el Hy 2 2 el + 20 black 1 5 mat 5 1 sys 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

Normalized Hypergeometric ... PDF, CDF

u x N n p Clear 1

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 pdf 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 pdf x 0 u int :

10.975 CDF 0.999