<p> Simplex Maths [1,2,3] </p>

5fdeb0b2-92cb-4969-badd-bdeea3e6709b 11 4 5 decimal

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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

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

t0 1 time :

much faster 
pre-assigned
SYMBOLIC n 3 : s 6 : 12 mat pts n s Triple : 21 mat

Not having otherwise here, the yard stick is Mathcad 11

4 appVersion 0.98.6179.21440 ≡

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

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

MCD 11 ... 0.129622968787563 f n s ∫∫∫ 0.129622968787599

1 time t0 - 2 s *

simplex is that much faster
... ± 4^-15 vs MCD 11 1260 * 2 / 360

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

◄ simplex ... MCD 11  ► 0.129622968787563 0.129622968787559

simplex depends upon 's' ... Whatsoever, always expect 10 ... 12 decimals You can experiment 's' convergent/divergent [5▲ , 6, 7▼]... 's 6' best. From this last remark, you don't need Mathcad anymore !