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

53c45d26-a522-4379-ab7d-c16c938509ee 5 15 12 decimal

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

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 :

a 0 : b 2 : 21 line

x f x sqrt exp :

t0 1 time :

sanity check the quadruplet Mathcad 11 Romberg 5.40752818465632 ≡ Technical accuracy 5.40752818465627 ≡ Romberg-Simpson [2000] 5.40752818465635 ≡ Smath Integrator 5.407526 ≡ 41 line

0 f 1

2 f 4.11325

x f 0 0 f + 20 red 1 5 mat 2 1 sys

f a b 2000 Int 5.40752818465635

1 time t0 - 11 s *

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 Iter Recursive N 0 ≡ Iter 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 + (* : Iter I.1 - I.2 - abs ε > Iter f a c N 1 - ε 2 / I.1 Recursive f c b N 1 - ε 2 / I.2 Recursive + : Iter I.1 I.2 + : if 41 line if Iter 21 line :

Function x f x sqrt exp :

Number of iterations N 1000 :

Precision ε 0.0000000000001 :

Lower Limit a 0 :

t0 1 time :

Upper Limit b 2 :

f a b N ε ∫AdapRecursive 5.407528184656300

1 time t0 - 89 s *

Function x f x sqrt exp :

Number of iterations Iter 1000 :

Precision ε 0.000000000001 :

Lower Limit a 0 :

t0 1 time :

Upper Limit b 2 :

f a b 10000 ε Iter Recursive 5.407528184656300

1 time t0 - 86 s *

x f

sanity check the quadruplet Mathcad 11 Romberg 5.40752818465632 ≡ Technical accuracy 5.40752818465627 ≡ Romberg-Simpson [2000] 5.40752818465635 ≡ Smath Integrator 5.407526 ≡ 41 line

x sqrt exp x a b int 5.40729869243192