<p> plotG(vx,vy,a,b,c) </p>

d3ab7f05-29ab-4882-ae71-e63c11400ce0 31 3 5 decimal

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

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 :

1D simplex integration

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

Carlos added,
enables recursive N 2 k matrix ≡ M nr nc matrix ≡ 21 sys

M idx M transpose 1 col rows :

5 gencm1 idx 5

Carlos  'submat' 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 33 weights 0.043058311287478 0.090299823633157 - 0.054241071428571 0.008465608465608 - 41 mat

n s pts 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

14 pts 0.875 0.625 0.375 0.125 14 mat 0.8333 0.5 0.1667 13 mat 0.75 0.25 12 mat 0.5 11 mat 41 mat

17 pts 0.929 0.786 0.643 0.5 0.357 0.214 0.071 17 mat 0.917 0.75 0.583 0.417 0.25 0.083 16 mat 0.9 0.7 0.5 0.3 0.1 15 mat 0.875 0.625 0.375 0.125 14 mat 0.833 0.5 0.167 13 mat 0.75 0.25 12 mat 0.5 11 mat 71 mat

Project Demo

1. Some functions will take native scalar 'x' ... sineintegral example.

2. Otherwise: conform to simplex rule => index 'x' in the RHS member.

]]>

t0 1 time :

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 * + : 21 line for v 51 line :

x f 1 - (m^x 12 m * +^* 12 m * + (12 m * + (! (* / (m 020 sum eval :

s ≤ 40 upper limit in Σ ∫(f,n,s) n 1 : s 7 : 12 mat

comparative sanity wrt 'm' in f(x) f n s ∫ 0.48638537623533 11 mat

1 time t0 - 1.8 s *

t0 1 time :

acc 1000 x f x 01 int 0.486385376235322

x f

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

1 time t0 - 18.7 s *

t0 1 time :

setup 1D simplex n 1 : s 7 : 12 mat x f 0.5 x 1 el (0.5^- exp : 21 sys

2.4 # 01 int

seemingly close f n s ∫ 0.871

1 time t0 - 0.1 s *

x 1 el Gamma unknown ≡ x 1 el sqrt exp exact 2 ≡ 0.5 x 1 el (0.5^- exp ~ 3½ D ≡ 31 line

ε 1012 -^:

'x' not defined => Simpson
gets lost in Gamma(x). x Gamma x ε 1 int #

Pt n s pts : w n s 1 - weights : 21 line

x Φ x 1 el x : x f 21 line :

View the feed-in to simplex integration algorithm Pt w 12 mat 0.929 0.786 0.643 0.5 0.357 0.214 0.071 17 mat 0.917 0.75 0.583 0.417 0.25 0.083 16 mat 0.9 0.7 0.5 0.3 0.1 15 mat 0.875 0.625 0.375 0.125 14 mat 0.833 0.5 0.167 13 mat 0.75 0.25 12 mat 0.5 11 mat 71 mat 2.223 4.195 - 2.548 0.56 - 0.037 0 6.889 108 -^* 71 mat 12 mat

x Φ

Example for advanced integrators

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

x 1 x - ( * sqrt 1 1 - 2 x * + ( x 1 x - ( * sqrt * 2 x * 1 x - ( * / - ( 2 ^ + sqrt 2 1 sys

basket handle x f x 1 el 1 x 1 el - (* sqrt :

arc length integrand x arc 1 x f x diff (2^+ sqrt :

setup 1D simplex n 1 : s 11 : 12 mat x f x 1 el 1 x 1 el - (* sqrt : 21 sys

sanity confirmed f n s ∫ 0.393

setup 1D simplex n 1 : s 40 : 12 mat x f 11 - 2 x 1 el * + (x 1 el 1 x 1 el - (* sqrt * 2 x 1 el * 1 x 1 el - (* / - (2^+ sqrt : 21 sys

sanity confirmed f n s ∫ 1.574

reputed upper limit in Smath 6179 # # 40 sum

vy 17 weights : i 1 vy rows range vy rows / : 21 line

just to view something

i vy augment