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

a310b871-3176-4788-8cae-89397f8a3aad 20 4 5 decimal

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PROJECT: ITER [Cadarache]

Created : jmGiraud

Dated: 2018_08_03

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unfamiliar Copper Resistivity .......................................................... a 1.454 10 17 - ^ * : b 4.437 : c 5.786 10 7 - ^ * : d 3.292 : f 7.649 : g 5.671 10 10 - ^ * : 2 3 mat T Cu g a T b ^ * 1 c T d ^ * 50 f ^ 1 T / ( f ^ * - exp * + / + : you may want X_Y plot if you miss ranging y, will crash Smath 2 1 line Cu resistivity in high induction magnetic field 5 1 line

Cu resitivity in high magnetic field [Model done for ITER Cadarache]

0 x Cu 2 1 sys x Cu 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 :

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

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 :

Data Σ Y 0 : 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 :

L H N xd U 0 : dx H L - N / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

x Γ x Lanczos :

x α β PDF 1 α Γ β α^* / x α 1 - (^* x β / - exp * :

ε 1012 -^:

α 0.75 :

β 250 :

x f x α β PDF eval :

t0 1 time :

enough 'n' on that difficult  long range n 100 :

last value cdf f ε 1000 n Int : 0.99

1 time t0 - 1 s *

x α β PDF

L 0.1 : H 1000 : N 1000 : 13 mat U L H N xd : 21 mat

Data Y 0 : j 1 U rows range Y j el U j el α β PDF eval : for U Y augment 31 line :

Σcumul Data Σ :

100 x α β PDF * Σcumul 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