4c9034db-b17c-4908-b7eb-42fa9536de36 69 3 5 decimal
&[DATE] &[TIME] - &[FILENAME]
&[PAGENUM] / &[COUNT]
 <p> Simplex Utilities </p>

5 gencm1 0 1 2 3 4 4 3 2 1 0 2 5 mat 0 0 0 0 0 0 0 0 0 3 3 mat

Carlos added, enables recursive

N 2 k matrix ≡ M nr nc matrix ≡ 2 1 sys 5

Carlos 'submat'

M 1 n 1 + ( range 2 w range el M 1 n 1 + ( 2 w submatrix ≡ 3

without submatrix(...)

n k gencm n 1 ≡ k gencm1 M n 1 + 1 0 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 : 5 1 line for w M idx 0 - : M 1 n 1 + ( range 2 w range el 4 1 line if 1 1 line : 0 1 2 3 4 4 3 2 1 0 2 5 mat

Only true for n=2 and any 'k'

 0 0 0 0 0 1 1 1 1 2 2 2 3 3 4 0 1 2 3 4 0 1 2 3 0 1 2 0 1 0 4 3 2 1 0 3 2 1 0 2 1 0 1 0 0 3 15 mat 1 1 mat

The "simplex method" of integration [1D, 2D, 3D]
The method reproduces Mathcad 8 Pro.
0. Utilities are common to [1D, 2D, 3D]
1. weights(n,s) ... ∫(f,n,s) ... common to [1D, 2D, 3D]
2. pts(n,s) are specific to each dimension
1D pts(n,s)
2D pts(n,s)
3D pts(n,s)
4. 1D, 2D, 3D exemplified in separate work sheet.
Simplex method integrates on the normalized numerical
range [0 ... 1]. Romberg is one of those scaled method.
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The Simplex Compendium: 1. Maths Simplex Smath 2. Maths Simplex Int_1D 3. Maths Simplex Int_2D 4. Maths Simplex Int_3D 5. Maths Simplex MCD

jmGiraud: 20180101

Credit & thanks to Carlos for fixing the recursive.

Simplex multi-dimension integration algorithm

f n s ∫ v 0 : Pt n s 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 * + : 2 1 line for v 5 1 line :

weights(n,s)is universal for any dimension

n s weights d 2 s * 1 + : i 0 s range w i 1 + el 1 - ( i ^ 2 2 s * - ^ * d n + 2 i * - ( d ^ i ! ( d n + i - ( ! ( * / * : for w 3 1 line :

sanity 15 D

3 3 weights 0.043058311287478 0.090299823633157 - 0.054241071428571 0.008465608465608 - 4 1 mat

Specific for 1D

 0.875 0.625 0.375 0.125 1 4 mat 0.8333 0.5 0.1667 1 3 mat 0.75 0.25 1 2 mat 0.5 1 1 mat 4 1 mat 0.929 0.786 0.643 0.5 0.357 0.214 0.071 1 7 mat 0.917 0.75 0.583 0.417 0.25 0.083 1 6 mat 0.9 0.7 0.5 0.3 0.1 1 5 mat 0.875 0.625 0.375 0.125 1 4 mat 0.833 0.5 0.167 1 3 mat 0.75 0.25 1 2 mat 0.5 1 1 mat 7 1 mat

Specific for 2D

 0.143 0.429 0.714 0.143 0.429 0.143 0.714 0.429 0.143 0.429 0.143 0.143 2 6 mat 0.2 0.6 0.2 0.6 0.2 0.2 2 3 mat 0.333 0.333 2 1 mat 3 1 mat 0.2 0.6 0.2 0.6 0.2 0.2 2 3 mat 0.333 0.333 2 1 mat 2 1 mat 0.111 0.333 0.556 0.778 0.111 0.333 0.556 0.111 0.333 0.111 0.778 0.556 0.333 0.111 0.556 0.333 0.111 0.333 0.111 0.111 2 10 mat 0.143 0.429 0.714 0.143 0.429 0.143 0.714 0.429 0.143 0.429 0.143 0.143 2 6 mat 0.2 0.6 0.2 0.6 0.2 0.2 2 3 mat 0.333 0.333 2 1 mat 4 1 mat 0.091 0.273 0.455 0.636 0.818 0.091 0.273 0.455 0.636 0.091 0.273 0.455 0.091 0.273 0.091 0.818 0.636 0.455 0.273 0.091 0.636 0.455 0.273 0.091 0.455 0.273 0.091 0.273 0.091 0.091 2 15 mat 0.111 0.333 0.556 0.778 0.111 0.333 0.556 0.111 0.333 0.111 0.778 0.556 0.333 0.111 0.556 0.333 0.111 0.333 0.111 0.111 2 10 mat 0.143 0.429 0.714 0.143 0.429 0.143 0.714 0.429 0.143 0.429 0.143 0.143 2 6 mat 0.2 0.6 0.2 0.6 0.2 0.2 2 3 mat 0.333 0.333 2 1 mat 5 1 mat

Specific for 3D

 0.125 0.125 0.125 0.375 0.375 0.625 0.125 0.125 0.375 0.125 0.125 0.375 0.625 0.125 0.375 0.125 0.125 0.375 0.125 0.125 0.625 0.375 0.125 0.375 0.125 0.125 0.375 0.125 0.125 0.125 3 10 mat 0.167 0.167 0.5 0.167 0.167 0.5 0.167 0.167 0.5 0.167 0.167 0.167 3 4 mat 0.25 0.25 0.25 3 1 mat 3 1 mat 0.1 0.1 0.1 0.1 0.3 0.3 0.3 0.5 0.5 0.7 0.1 0.1 0.1 0.3 0.3 0.5 0.1 0.1 0.3 0.1 0.1 0.3 0.5 0.7 0.1 0.3 0.5 0.1 0.3 0.1 0.1 0.3 0.5 0.1 0.3 0.1 0.1 0.3 0.1 0.1 0.7 0.5 0.3 0.1 0.5 0.3 0.1 0.3 0.1 0.1 0.5 0.3 0.1 0.3 0.1 0.1 0.3 0.1 0.1 0.1 3 20 mat 0.125 0.125 0.125 0.375 0.375 0.625 0.125 0.125 0.375 0.125 0.125 0.375 0.625 0.125 0.375 0.125 0.125 0.375 0.125 0.125 0.625 0.375 0.125 0.375 0.125 0.125 0.375 0.125 0.125 0.125 3 10 mat 0.167 0.167 0.5 0.167 0.167 0.5 0.167 0.167 0.5 0.167 0.167 0.167 3 4 mat 0.25 0.25 0.25 3 1 mat 4 1 mat

Scaling 1D integration

x f x a b int scale scale x * f x 0 b scale / int * 2 1 mat x f scale x * f 2 1 sys iVBORw0KGgoAAAANSUhEUgAAABgAAAAjCAYAAACOysqWAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAADgSURBVEhL7ZFLDoMwDAVRxUkqTlZxLNSToZ4EQTWoH0Mcx4Gwy2K6ATNv1Kadp8Xk1tpoN4Iq0G8EVaDfCKpAvxFcL+junX5YiGZ4DuqDIlDAzxUV/aP/C1aUl07x+e5PULLiu34jKPlfjK8xFECJCrkeNoISFXI9bARwpmK/HgLBmYr9eggEcKRCWw+q4EiFth5UAeRUxNZDVJBTEVsPUQF4Kqz1YAo8FdZ6MAVgVaTWQ1JgVaTWQ1IAWoVnPbgEWoVnPbgEICu868EtkBXe9eAWABU56yFLQEXO+naeljeOnwVKRFNzTwAAAABJRU5ErkJggg==

Working the extended range

where =>

Simplex integration algorithm

f n s ∫ v 0 : Pt n s 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 * + : 2 1 line for v 5 1 line :

Recast simplex extended domain

The executable ranged simplex

x f 0.5 u ( 0.5 ^ - exp 1 2 x 1 el * 1 - ( 2 ^ - ( * :
Simplex 1 D conclusion
This test function is not numerically exactly integrable.
Advanced Adaptive warrants ~ 10 D [±]. Simplex
converges, calculates fast compared to Simpson 10000.
Simpson is reasonably accurate, ~ 70 times slower.
At high 's', the Σ in ∫(f,n,s) becomes unstable.
]]>

..sanity Adaptive=> 1.95866158051

3 2 / b a - ( * f n s ∫ * 1.95866158055 0.25 s *

==================================================================================

Simpson acc => 10000 ... close accuracy

0.5 u ( 0.5 ^ - exp u 0 4 int 1.95866195909 21 s *

Next => visit Simplex [1D, 2D, 3D]