hatch

b73c6a08-a9ba-4f2c-bb1c-01c443b9456c 34 3 12 decimal

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

hatch

Erzeugt eine Matrix für die Darstellung vertikaler Linien zwischen den Funktionen f(x) und 
g(x) im Bereich von x1 bis x2 mit 2*n Linien. Generates a matrix for 2*n vertical lines (hatching) between functions f(x) and g(x)
in x-range from x1 to x2 f g x.1 x.2 N hatch dx x.2 x.1 - N / (: h x.1 x.1 f x.1 x.1 g 22 mat : x x.1 x.2 dx - x.1 dx + range h h x dx 2 / + x dx 2 / + g x dx 2 / + x dx 2 / + f x dx + x dx + f x dx + x dx + g 42 mat stack : for h 41 line :

∫bisection(f,a,b,TOL) ..

Iterative bisection ... prohibitive in time, didactic.
May rescue the projet if no other method is profitable. For less 
drastic functions ∫bisection timing drops [sin(x)² ½ timing]. f a b TOL ∫bisection Y1 b a - (a b + 2 / f * : Y2 b a - 6 / 2 a * b + 3 / f a 2 b * + 3 / f + (* : N 1 TOL / : Y1 Y2 - abs TOL ≥ Y1 Y2 : N 2 N * : Y2 0 : h b a - N 1 - / : j 1 N range x0 a j h * + : x1 x0 h + : 12 mat Y2 Y2 h x0 f x1 f + 2 / (* + : 21 line for 51 line while 2 Y2 * Y1 - 51 line eval :

∫simplex(f,a,b,s)

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

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^22 s * -^* d n + 2 i * - (d^i! (d n + i - (! (* / * : for w 31 line :

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 :

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 1D

Recast the simplex domain

Romberg-Simplex kernel 1D integrator  ... domain [a,b] f a b s ∫ 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 ∫ * : 31 line :

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

∫Adaptive(f,a,b,N,ε)

f a b N ε ∫Adaptive s 0 : sum 0 : i 1 : tol i el ε : aa i el a : h i el b a - (2 / : 51 line .... fa i el a f : fc i el a h i el + f : fb i el b f : s i el h i el fa i el 4 fc i el * + fb i el + (3 / * : L i el 1 : 51 line 13 mat above => initialise the while loop i 0 > fd aa i el h i el 2 / + f : fe aa i el 3 h i el * 2 / + f : 12 mat s1 h i el fa i el 4 fd * + fc i el + 6 / (* : s2 h i el fc i el 4 fe * + fb i el + 6 / (* : 12 mat v1 aa i el : v2 fa i el : v3 fc i el : v4 fb i el : v5 h i el : v6 ε : v7 s i el : v8 L i el : 18 mat i i 1 - : s1 s2 + v7 - abs v6 < sum sum s1 s2 + (+ : v8 N ≥ Solution Insufficient Number of Intervals : i i 1 + : aa i el v1 v5 + : fa i el v3 : fc i el fe : fb i el v4 : 14 mat h i el v5 2 / : tol i el v6 2 / : s i el s2 : L i el v8 1 + : 14 mat i i 1 + : aa i el v1 : fa i el v2 : fc i el fd : fb i el v3 : h i el h i 1 - el : s i el s1 : 16 mat L i el L i 1 - el : 61 line if if 51 line while sum 51 line eval :

X vx 121 pts 1 row transpose sort 1 el : vx reverse 21 line :

CWD C:\SmathFile CurrentDirectory :

X CWD Xsimplex wfile 1

X CWD Xsimplex rfile :

x f x - exp x sqrt / :

x c 10 x ≤ (x 1 ≤ (& cases :

M key FindIndex r M rows : c M cols : 12 mat i 1 : j 1 : 12 mat j c ≤ (M i j el key ≠ (* i r < i i 1 + : i 1 : j j 1 + : 21 line if while j c ≤ i j 21 mat 0 if 41 line :

ε 1012 -^:

split X X f vectorize augment :

x g 0 :

x f x c * x f x g ε 0.5 30 hatch split . 12 black plot 3 1 sys

idx X 0.5 FindIndex 1 el : 11

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

a 0.02381 : b 0.5 : 12 mat

t0 1 time :

...
...
--"--
5 min
1.4 s
0.4 s Wolfram 0.903858 ≡ Mathcad 0.903858 ≡ bisection 0.903858 ≡ adaptive 0.903858 ≡ simplex 0.903846 ≡ 51 sys

Simplex f a b 10 ∫ f a b 11 ∫ f a b 12 ∫ 31 mat 0.904001 0.90389 0.903846 31 mat

1 time t0 - 1.3 s *

'N' iterations
precision N 100 : acc 109 -^: 21 sys

t0 1 time :

Zeta(2) ... f a b N acc ∫Adaptive 0.903858

1 time t0 - 2 s *

TOL 0.0001 :

t0 1 time :

f a b TOL ∫bisection 0.903858

hoover algo

1 time t0 - min 5.6

'N' from inner algo iter N TOL * : 4

Sanity . . . ∫bisection . . . ∫adaptive

Air isentropic discharge

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k 1.4 : (Air Isentropic coefficient 20 °C ≡

Ψmax k k 1 - / sqrt 2 k 1 + / ln k k 1 - / * exp (2 k /^2 k 1 + / ln k k 1 - / * exp (k 1 + k /^- sqrt * :

Ψmax 0.484

x f 0 x ≤ (x 0.528 ≤ (& (Ψmax 0.528 x ≤ (x 1 ≤ (& k k 1 - / sqrt x 2 k /^x k 1 + k /^- sqrt * if if :

T 293 : (Air temperature 293 K ≡ ρo 1.293 : (Air kg/m^3 STP ≡ Ao 0.05 : (Area orifice m^2 ≡ Pv 900 : (Vessel Pressure kPa ≡ p2 101.325 : (Outlet backup P kPa ≡ ρv 1.293 Pv 101.325 / * : (Static Air ρv kg/m^3 ≡ ρv ρo Pv p2 / (1 k /^* : (Dynamic kg/m^3 ≡ x Q Ao 2 Pv * ρv * sqrt * x Ψ * : (MassFlow rate kg/sec ≡ x downstream/uptream P ratio dimensionless ≡ 91 sys

Isentropic Discharge Coefficient wrt p1/po where p1 =inlet P , po = outlet P p1/po => choke flow pressure ratio @ sonic velocity for Air @ 293 K ... p1/po = 0.528

0 0.484 0.528 0.484 2 2 mat x f 0.528 0.484 . 15 black 1 5 mat 3 1 sys

ρv 6.153

Qmax Ao 2 Pv * ρv * sqrt * :

Qmax 5.262169

x f k k 1 - / sqrt x 2 k /^x k 1 + k /^- sqrt * :

a 0.528 : b 1 : 12 mat

for 6 decimals TOL 0.0001 :

t0 1 time :

f a b TOL ∫bisection 0.179725

hoover algo

1 time t0 - min 7.8

'N' from inner algo iter N TOL * : 4

b 11012 -^- :

t0 1 time :

f a b 100 0.000001 ∫Adaptive 0.179725

1 time t0 - 0.6 s *