3D Simplex Maths

56740386-8105-462a-9874-3460034ab221 13 15 12 decimal

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

x y z f x z + exp 0.5 y 3^- (* :

3 appVersion 0.98.6179 ≡ Timing < 1 min ≡ 21 sys

a 0 : b 1 : 12 mat

12 min acc 50 x y z f z a b x - y - int y a b x - int x a b int

3D Simplex Maths

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

much faster as pre-assigned n 3 : s 4 : 12 mat pts n s Triple : 21 mat

Simplex  triple integration algorithm f n s ∫∫∫ v 0 : Pt pts : w n s 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 :

Simplex integration is on the 0 .. 1

]]>

x f x 1 el x 2 el x 3 el f :

t0 1 time :

sanity ... 0.1296229687487563 f 34 ∫∫∫ 0.129622968748757

1 time t0 - 0.1 s *

s > 7 ... check numerical instability
faster as pre_assigned n 2 : s 7 : 12 mat pts n s Double : 21 mat

Simplex integration algorithm f n s ∫∫ v 0 : Pt pts : 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 :

x f x 1 el exp 0.5 x 2 el (3^- (* :

t0 1 time :

sanity MCD ... 0.299449943475250 f 27 ∫∫ 0.29944994347527

1 time t0 - 0.2 s *

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 :

x f x - exp x sqrt / :

106 -^f 1000

ε 1012 -^: b 0.1 : 12 mat

x c 1 ε x ≤ (x b ≤ (& cases :

x f x c * b 50 b 50 - 2 2 mat 2 1 sys

Reserve Simplex [infinite limit and singular end point]. Explore for best 's' below. Next best => Adaptive 14 D's 2 min.

t0 1 time :

sanity 12 D f 0111 ∫ 1.49364826562379

1 time t0 - 0.4 s *

π sqrt 1 erf * 1.49364826562485

Explore f 019 ∫ f 0110 ∫ f 0111 ∫ f 0112 ∫ f 0113 ∫ 51 mat : 1.49364826611909 1.49364826565873 1.49364826562379 1.49364826563812 1.49364826566673 51 mat

x y z f x y + exp 0.5 z 3^- (* :

a 0 : b 1 : 12 mat

12 min acc 50 ! x y z f z a b x - y - int y a b x - int x a b int 0.12962296880572

Romberg

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 :

x f x sqrt :

set acc 10000 u f u 010 int 21.0792837853217

2010 sqrt * 3 / 21.0818510677892

f 01010000 Int 21.0818510677891