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

abc83e55-fbeb-499c-8062-f84451b71c8a 26 15 12 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  Respect some rules for ± ∞ [a, b] 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 :

1. Frechet ... integrate Romberg-Simpson 2. ........... integrate ODE dn_GearsBDF

3 appVersion 0.98.6179 ≡

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 - 20 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

12 Frechet 29807

sol sol rows row 122978612 mat

3. cspline Frechet ... integrate Romberg-Simpson

XY X 0201 range : i 1 X rows range f i el X i el f eval : for X f augment 31 line :

X XY 1 col : Y XY 2 col : 12 mat x s X Y x cinterp : 21 sys

x c 10 x ≤ (x 20 ≤ (& cases :

XY 0403165822467351434600055046635957238981574910741078111614125191346714437154211641117406184041940220401212 mat

x cspline_Frechet s 0 x 10 Int :

Confirm canvas plot cumulative

x s x c * x cspline_Frechet x c * 2 1 sys

4sure, acc option
50 ≤ acc  non-sense x f x 020 int #

puzzling ... s(x)
compatible integrate x s x 020 int 33165

s 02010 Int 33238

4. Frechet ... integrate Adaptive

α 1523043253 - 27354.7 - 43256.75 25388.3 - 40061 mat :

Fréchet: extreme value-II x α 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 + :

Smath  adaptive integration ... accommodates hard end points f a b TOL Singular Y1 b a - (a b + 2 / f * : Y2 b a - 6 / 2 a * b + 3 / f a 2 b * + 3 / f + (* : N 2 : Y1 Y2 - abs TOL ≥ Y1 Y2 : N 2 N * : Y2 0 : h b a - N 1 + / : j 2 N range x0 a j 1 - (h * + : x1 x0 h + : Y2 Y2 h x0 f x1 f + 2 / (* + : 31 line for 51 line while Y2 51 line eval :

a 0 : b 20 : 12 mat TOL 10 : 21 line

0 α f 403

TOL wisely wrt expected magnitude 21 mat 1 2.6 min * ≡ (33209 ≡ 21 mat

x f x α f :

t0 1 time :

f a b TOL Singular 33202

1 time t0 - min 0.4

mild example integration

α 1.1077 142.40751 1.01751 - 0.07235 13.58616 - 51 mat :

x α Φ x α 1 el α 2 el x α 4 el / - exp * + α 3 el x α 5 el / - exp * + (* :

end 1 :

x c 10 x ≤ (x end ≤ (& cases :

x Y D x α Φ :

init 0 :

sol init 0 end 100 x Y D dn_GearsBDF :

sol sol rows row 1 0.765 12 mat

x α Φ x 01 int 0.764852

x α Φ x c * sol 2 1 sys Int(f,a,b,n) ... Simpson/Romberg Kernel [Jean]

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  Respect some rules for ± ∞ [a, b] 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 :

The naive test function ... best => Int(f,a,b,n)

x f 1 x 1 - (2^- sqrt :

all dn_ODE solvers fail x Y D x f :

1 - 8 / 3 sqrt * 16 / π * + 0.307092424652189

sol 0021000 x Y D rkm9st :

good for plot ...otherwise 5 D

x f sol 2 1 sys

sol sol rows row 2 1.570793 12 mat

acc 10000 x f x 02 int 1.570796 ≡

π 2 / 1.5707963267949

t0 1 time :

f 022000 Int 1.57079632679491

1 time t0 - min 0.2

TOL 0.000001 :

profitable ... hyperslow f 02 TOL Singular 1.570796