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

81b6c9b9-2820-4587-9fbe-f132675ecd40 3 4 5 decimal

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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 
Integration accuracy from user command line 'n' ... NUMERIC 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 / : 15 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 :

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 + :

end 12 :

x c 10 x ≤ (x end ≤ (& cases :

0 f 403

x Y D x f :

init 0 :

sol init 0 end 100 x Y D dn_GearsBDF :

in-situ Simpson acc

x Frechet f 0 x 10 Int :

t0 1 time :

x f x c * sol 2 1 sys x f x c * x Frechet x c * 2 1 sys

1 time t0 - 2 s *

data X 0121 range : i 1 X rows range F i el X i el Frechet : for X F augment 31 line :

data 0014702186035727411494517110621429724388826339927642102855611292451229807132 mat

Result is above max 
allowed positive number x f x 0 end int #

12 Frechet 29807

sol sol rows row 122978612 mat

ROMBERG RECURSIVE

f a b n S AlgoRomberg h 1 el b a - : R 11 el h 1 el 2 / a f b f + (* : 12 mat k 2 n range h k el b a - 2 k 1 -^/ : R k 1 el 12 / R k 1 - 1 el h k 1 - el a 2 i * 1 - (h k el * + f i 12 k 2 -^sum (* + (* : 21 line for j 2 n range A j 1 - : B 4 j 1 -^: C B 1 - : 13 mat k j n range R k j el B R k A el * R k 1 - A el - C / : for 21 line for R n n el 41 line :

x Frechet1 f 0 x 6 S AlgoRomberg :

t0 1 time :

x f x c * sol 2 1 sys x f x c * x Frechet1 x c * 2 1 sys

1 time t0 - 17 s *

i 113 range F1 i el i 1 - Frechet1 : for

X1 0121 range :

data1 X1 F1 augment :

data1 0014702186035727411494517111621429724389826339927642102855611292451229807132 mat

Result is above max 
allowed positive number x f x 0 end int #

12 Frechet1 29807

sol sol rows row 122978612 mat

ADAPTIVE RECURSIVE

Adaptive recursive quadrature based on the Simpson Rule for the calculation of integrals

f a b N ε ∫AdapRecursive I.0 b a - 6 / a f 4 a b + 2 / f * + b f + (* : ε 10 ε * : f a b N ε I.0 Recursive eval 31 line eval :

f a b N ε I.0 Recursive N 0 ≡ I.0 Number of Iterations exceeded : c a b + (2 / : I.1 c a - 6 / a f 4 a c + 2 / f * + c f + (* : I.2 b c - 6 / c f 4 c b + 2 / f * + b f + (* : I.0 I.1 - I.2 - abs ε > I.0 f a c N 1 - ε 2 / I.1 Recursive f c b N 1 - ε 2 / I.2 Recursive + : I.0 I.1 I.2 + : if 41 line if I.0 21 line :

ε 0.9 :

x Frechet2 f 0 x 6 ε ∫AdapRecursive :

t0 1 time :

x f x c * sol 2 1 sys x f x c * x Frechet2 x c * 2 1 sys

1 time t0 - 5 s *

data X 0121 range : i 1 X rows range F i el X i el Frechet2 : for X F augment 31 line :

data 0014702186035726411494517111621429724389826339927642102855611292461229807132 mat

Result is above max 
allowed positive number x f x 0 end int #

12 Frechet2 29807

sol sol rows row 122978612 mat