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

cf7a2705-8708-455b-b499-929de959fd6a 14 4 5 decimal

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Integration Trapeze ...

Basic Romberg-Simpson ... integration acc from options f a b 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^- (* : 32 / b a - (* w G w 01 int * 31 line eval :

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' f a b n ∫kernel 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 :

Test function x f x sqrt exp :

∫kernel

∫trap

∫adaptive

∫simplex

sanity check the quintuplet Mathcad 11 Adaptive 5.40752818465632 ≡ Technical  accuracy 5.40752818465627 ≡ Smath ∫kernel 5.40752818465635 ≡ Smath Simpson 5.407581 ≡ Smath ∫trap Demo 5.40758178 ≡ 51 line

t0 1 time :

f 023000 ∫kernel 5.40752818465635

1 time t0 - min 0.2

algo style similar  ... ∫kernel f a b n ∫trap b a - 2 n * / a f b f + (* b a - n / a k b a - n / * + f k 1 n 1 - sum * + :

t0 1 time :

∫ x f 02200000 ∫trap : 5.407528178

1 time t0 - min 4

Simpson 10000 5.407528103

∫kernel 5.40752818465635

x f 1 1 sys

t0 1 time :

acc 10000 t f t 02 int 5.407528103

1 time t0 - min 0.3

NOTES ... 1. ∫trap is a pure curisity Edu. 2. ∫kernel is not panacea ... will not integrate ∞ limit, singular end point. 3. ∞ limit, singular end point, via Carlos ∫adaptive. 4. For this demo,∫simplex warrants ~ 11 decimals ...∫simplex ... offered below. ...∫simplex solves nicely ∞ limit/singular end point.

x f x erf :

13 minutes ∫ f 1017 -^2500000 ∫trap : 1.43678843916568

Exact ... t erf vectorize t 02 int 1.43678843916719

f 022000 ∫kernel 1.43678843916719

Simplex Maths [1,2,3]

vx vy a b c plotG a char ≡ b size ≡ c color ≡ 13 mat n vx length : plot vx 1 el vy 1 el a b c augment : i 2 n range plot plot vx i el vy i el a b c augment stack : for plot 51 line :

k gencm1 N 2 k matrix : i 1 k range N 1 i el i 1 - : N 2 i el k i - : 21 line for N 31 line :

nr nc v genm M nr nc matrix : i 1 nr range j 1 nc range M i j el v : for for M 31 line :

5 gencm1 012344321025 mat

330 genm 00000000033 mat

M idx M transpose 1 col rows :

5 gencm1 idx 5

M 1 n 1 + (range 2 w range el M 1 n 1 + (2 w submatrix ≡

330 genm idx 3

without submatrix(...) n k gencm n 1 ≡ k gencm1 M n 1 + 10 genm : i 0 k range N n 1 - k 1 + (i - gencm : w N idx 1 - : O 1 w 1 + i genm : N O N stack : M M N augment : 51 line for w M idx 0 - : M 1 n 1 + (range 2 w range el 41 line if 11 line :

15 gencm 012344321025 mat

Only true for n=2 and any 'k'

24 gencm 11 mat 000001111222334012340123012010432103210210100315 mat 11 mat

n s weights d 2 s * 1 + : i 0 s range w i 1 + el 1 - (i^22 s * -^* d n + 2 i * - (d^i! (d n + i - (! (* / * : for w 31 line :

sanity 15 D 32 weights 0.050793650793651 0.096428571428571 - 0.044444444444444 31 mat

n s Single d 2 s * 1 + : i 0 s range M n s 1 + (i - gencm : M M 2 n 1 + 1 M idx submatrix : Pt i el 2 M * 1 + d n + 2 i * - / : 31 line for Pt 31 line :

Specific for n=1

n s 1 + (i - gencm

13 Single 0.8333 0.5 0.1667 13 mat 0.75 0.25 12 mat 0.5 11 mat 31 mat

n s Double d 2 s * 1 + : i 0 s range M n s i - gencm : M M 2 n 1 + 1 M idx submatrix : Pt i el 2 M * 1 + d n + 2 i * - / : 31 line for Pt 31 line :

Specific for n=2

n s i - gencm

show 27 Double : simplex show 1 el transpose : vx simplex 1 col : vy simplex 2 col : 41 line

23 Double 0.143 0.429 0.714 0.143 0.429 0.143 0.714 0.429 0.143 0.429 0.143 0.143 26 mat 0.2 0.6 0.2 0.6 0.2 0.2 23 mat 0.333 0.333 21 mat 31 mat

S vx vy o 5 red plotG :

simplex 0 0 0 1 1 1 1 0 0 0 5 2 mat S 0.2 1 0.2 0 2 2 mat 5 1 sys

The doctored triple integration

23 gencm 000011122301230120103210210100310 mat

n s Triple d 2 s * 1 + : v 11113 mat transpose : d 2 s * 1 + : i 0 s range M n s i - gencm : M M 2 n 1 + 1 M idx submatrix : Pt i el 2 M * 1 + v augment : Pt s 1 + el v : scale i 1 + el 1 d n + 2 i * - / : 51 line for Pt scale * vectorize 51 line :

Specific for n=3

Test function x f x sqrt exp :

n 1 : s s : 12 mat

f a b s ∫ f n s ∫ v 0 : Pt n s Single : w n s 1 - weights : i 1 w rows range P Pt i el : v v w i el P j col f j 1 P idx sum * + : 21 line for v 51 line : recast the simplex domain [a, b] u b a - (34 / 2 x 1 el * 1 - (* 14 / 2 x 1 el * 1 - (3^* - 12 / + (* a + : x f u f 12 x 1 el * 1 - (2^- (* : simplex 32 / b a - (* f 1 s ∫ * : simplex 41 line 31 line :

t0 1 time :

f 0210 ∫ 5.407528184649

1 time t0 - 0.3 s *

sanity check the sixtuplet ... Mathcad 11 Adaptive 5.40752818465632 ≡ Technical  accuracy 5.40752818465627 ≡ Smath ∫kernel 5.40752818465635 ≡ Smath Simpson 5.407581 ≡ Smath ∫trap Demo 5.40758178 ≡ Smath ∫simplex 5.407528184649 ≡ 61 line

Explore f 029 ∫ f 0210 ∫ f 0211 ∫ f 0212 ∫ f 0213 ∫ 51 mat : 5.40752818453319 5.40752818464906 5.40752818462219 5.40752818466499 5.40752818463718 51 mat

jmG: 20230517