hatch

b73c6a08-a9ba-4f2c-bb1c-01c443b9456c 68 4 5 decimal

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

About these 8 numeric Integrators . . .

appVersion(4)≡"0.98.6179.21440"

∫bisection . . . slow but productive, may fail.

∫adaptive . . . all difficult cases.

∫romberg . . . most difficult cases.

∫simplex . . . occasinally exact, never dramatically wrong .

∫simpson . . . if result varies lot [100, 1000, 10000]

sanity check using other integrators.

Σ(data) . . . FD data set, may be noisy, very good yardstick.

Mesh fine enough the difficult region(s).

Legendre(w,x) . . . Demo.

ODE solvers . . . Cumulative, often convenient.

]]>

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 :

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].
Lot more efficient tham MCD that does not understand tis code. # 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 2 : Y1 Y2 - abs TOL ≥ Y1 Y2 : N 2 N * : Y2 0 : 13 mat h b a - N / : j 1 N range x0 a j 1 - (h * + : x1 x0 h + : 12 mat Y2 Y2 h x0 f x1 f + 2 / (* + : 21 line for 31 line while Y2 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 ∫simplex 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 :

∫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 :

∫Romberg(f,a,b,n) Simpson/Romberg Kernel

Basic Romberg-Simpson ... integration acc from options f a b 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^- (* : 32 / b a - (* w G w 01 int * 31 line eval :

f a b n ∫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 
Integration accuracy from user command line 'n' f a b n ∫Romberg 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 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 :

...
...
--"--
0.5 min
0.4 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 ∫simplex f a b 11 ∫simplex f a b 12 ∫simplex 31 mat 0.904001 0.90389 0.903846 31 mat

1 time t0 - 1.1 s *

'N' iterations
precision N 100 : acc 107 -^: 21 sys

t0 1 time :

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

1 time t0 - 0.5 s *

TOL 0.000001 :

t0 1 time :

x f a b TOL ∫bisection 0.903858

hoover algo

1 time t0 - min 0.5

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 * :

x c 10 x ≤ (x 1 ≤ (& cases :

Ψmax 0.484

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 kg/sec ≡ x downstream/uptream ratio dimensionless ≡ 91 sys

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

x f x 0.528 1 int 0.179725

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 x c * 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 0.000001 :

t0 1 time :

x f a b TOL ∫bisection 0.179725

hoover algo

1 time t0 - min 0.8

b 11012 -^- :

t0 1 time :

f a b 100 0.000001 ∫Adaptive 0.179725

1 time t0 - 0.4 s *

t0 1 time :

f a b 20 ∫Romberg 0.179725

1 time t0 - 0.2 s *

t0 1 time :

Simplex f a b 11 ∫simplex 0.179725

1 time t0 - 0.4 s *

Sanity . . . Devil fnct

Export this section

]]> ∫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 ∫simplex 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 :

∫Adaptive(f,a,b,N,ε) Carlos

f a b N ε ∫Adaptive 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 41 line eval :

∫romberg(f,a,b,n) ... Simpson/Romberg Kernel

Basic Romberg-Simpson ... integration acc from options f a b 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^- (* : 32 / b a - (* w G w 01 int * 31 line eval :

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 
Integration accuracy from user command line 'n' f a b n ∫romberg 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 0.5 1 - ^ x - exp * π sqrt x sqrt erf * 2 1 sys

<= Wolfram Alpha

MCD 11 default adaptive ... 0.611991 x 0.5 1 -^x - exp * x 1015 -^0.1 int Don't even attempt ≡

Infinite  limit x f x 0.5 1 -^x - exp * :

'N' iterations
precision N 2000 : p 1017 -^: 21 sys

lower limit
upper limit a 1029 -^: b 0.1 : 21 sys

t0 1 time :

f a b N p ∫Adaptive 0.611991366111712

π sqrt b sqrt erf * 0.611991366111770

1 time t0 - min 0.0

t0 1 time :

Romberg converges with 'n' ... slow f a b 100000 ∫romberg 0.61199

1 time t0 - min 0.0

∫simplex Emeritus . . . ~ 11 D 0.4 s

]]>

t0 1 time :

f 0 b 12 ∫simplex 0.611991366116

1 time t0 - 0.4 s *

x f a b 0.01 ∫bisection Never result ? ≡

data

data 5 - 0.03846 4.95 - 0.03921 4.9 - 0.03998 4.85 - 0.04078 4.8 - 0.0416 4.75 - 0.04244 4.7 - 0.04331 4.65 - 0.0442 4.6 - 0.04513 4.55 - 0.04608 4.5 - 0.04706 4.45 - 0.04807 4.4 - 0.04912 4.35 - 0.05019 4.3 - 0.05131 4.25 - 0.05246 4.2 - 0.05365 4.15 - 0.05488 4.1 - 0.05615 4.05 - 0.05746 4 - 0.05882 3.95 - 0.06023 3.9 - 0.06169 3.85 - 0.0632 3.8 - 0.06477 3.75 - 0.06639 3.7 - 0.06807 3.65 - 0.06982 3.6 - 0.07163 3.55 - 0.07352 3.5 - 0.07547 3.45 - 0.0775 3.4 - 0.07962 3.35 - 0.08182 3.3 - 0.0841 3.25 - 0.08649 3.2 - 0.08897 3.15 - 0.09155 3.1 - 0.09425 3.05 - 0.09706 3 - 0.1 2.95 - 0.10307 2.9 - 0.10627 2.85 - 0.10962 2.8 - 0.11312 2.75 - 0.11679 2.7 - 0.12063 2.65 - 0.12465 2.6 - 0.12887 2.55 - 0.13329 2.5 - 0.13793 2.45 - 0.14281 2.4 - 0.14793 2.35 - 0.15332 2.3 - 0.15898 2.25 - 0.16495 2.2 - 0.17123 2.15 - 0.17786 2.1 - 0.18484 2.05 - 0.19222 2 - 0.2 1.95 - 0.20822 1.9 - 0.21692 1.85 - 0.22612 1.8 - 0.23585 1.75 - 0.24615 1.7 - 0.25707 1.65 - 0.26864 1.6 - 0.2809 1.55 - 0.2939 1.5 - 0.30769 1.45 - 0.32232 1.4 - 0.33784 1.35 - 0.3543 1.3 - 0.37175 1.25 - 0.39024 1.2 - 0.40984 1.15 - 0.43057 1.1 - 0.45249 1.05 - 0.47562 1 - 0.5 0.95 - 0.52562 0.9 - 0.55249 0.85 - 0.58055 0.8 - 0.60976 0.75 - 0.64 0.7 - 0.67114 0.65 - 0.70299 0.6 - 0.73529 0.55 - 0.76775 0.5 - 0.8 0.45 - 0.8316 0.4 - 0.86207 0.35 - 0.89087 0.3 - 0.91743 0.25 - 0.94118 0.2 - 0.96154 0.15 - 0.978 0.1 - 0.9901 0.05 - 0.99751 1.8 1014 -^* - 1 0.05 0.99751 0.1 0.9901 0.15 0.978 0.2 0.96154 0.25 0.94118 0.3 0.91743 0.35 0.89087 0.4 0.86207 0.45 0.8316 0.5 0.8 0.55 0.76775 0.6 0.73529 0.65 0.70299 0.7 0.67114 0.75 0.64 0.8 0.60976 0.85 0.58055 0.9 0.55249 0.95 0.52562 1 0.5 1.05 0.47562 1.1 0.45249 1.15 0.43057 1.2 0.40984 1.25 0.39024 1.3 0.37175 1.35 0.3543 1.4 0.33784 1.45 0.32232 1.5 0.30769 1.55 0.2939 1.6 0.2809 1.65 0.26864 1.7 0.25707 1.75 0.24615 1.8 0.23585 1.85 0.22612 1.9 0.21692 1.95 0.20822 2 0.2 2.05 0.19222 2.1 0.18484 2.15 0.17786 2.2 0.17123 2.25 0.16495 2.3 0.15898 2.35 0.15332 2.4 0.14793 2.45 0.14281 2.5 0.13793 2.55 0.13329 2.6 0.12887 2.65 0.12465 2.7 0.12063 2.75 0.11679 2.8 0.11312 2.85 0.10962 2.9 0.10627 2.95 0.10307 3 0.1 3.05 0.09706 3.1 0.09425 3.15 0.09155 3.2 0.08897 3.25 0.08649 3.3 0.0841 3.35 0.08182 3.4 0.07962 3.45 0.0775 3.5 0.07547 3.55 0.07352 3.6 0.07163 3.65 0.06982 3.7 0.06807 3.75 0.06639 3.8 0.06477 3.85 0.0632 3.9 0.06169 3.95 0.06023 4 0.05882 4.05 0.05746 4.1 0.05615 4.15 0.05488 4.2 0.05365 4.25 0.05246 4.3 0.05131 4.35 0.05019 4.4 0.04912 4.45 0.04807 4.5 0.04706 4.55 0.04608 4.6 0.04513 4.65 0.0442 4.7 0.04331 4.75 0.04244 4.8 0.0416 4.85 0.04078 4.9 0.03998 4.95 0.03921 5 0.03846 2012 mat :

Data set from foreign source

data rows 201

t0 1 time :

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

Finite differences Cumulative Integral

data Σ data 2 1 sys

data Σ data Σ rows row 5 2.747 12 mat

1 time t0 - 2.4 s *

∫legendre . . . Demo

]]> Legendre wr 63 ... 64

wr63 0.0494723666239310 0.0000000000000000 0.0494118330399182 0.0494521871161596 - 0.0494118330399182 0.0494521871161596 0.0492303804237476 0.0987833564469453 - 0.0492303804237476 0.0987833564469453 0.0489284528205120 0.1478727863578720 - 0.0489284528205120 0.1478727863578720 0.0485067890978838 0.1966003467915067 - 0.0485067890978838 0.1966003467915067 0.0479664211379951 0.2448467932459534 - 0.0479664211379951 0.2448467932459534 0.0473086713122689 0.2924940585862514 - 0.0473086713122689 0.2924940585862514 0.0465351492453837 0.3394255419745844 - 0.0465351492453837 0.3394255419745844 0.0456477478762926 0.3855263942122479 - 0.0456477478762926 0.3855263942122479 0.0446486388259414 0.4306837987951116 - 0.0446486388259414 0.4306837987951116 0.0435402670830276 0.4747872479948044 - 0.0435402670830276 0.4747872479948044 0.0423253450208158 0.5177288132900333 - 0.0423253450208158 0.5177288132900333 0.0410068457596664 0.5594034094862850 - 0.0410068457596664 0.5594034094862850 0.0395879958915441 0.5997090518776252 - 0.0395879958915441 0.5997090518776252 0.0380722675843496 0.6385471058213654 - 0.0380722675843496 0.6385471058213654 0.0364633700854573 0.6758225281149861 - 0.0364633700854573 0.6758225281149861 0.0347652406453559 0.7114440995848458 - 0.0347652406453559 0.7114440995848458 0.0329820348837793 0.7453246483178474 - 0.0329820348837793 0.7453246483178474 0.0311181166222198 0.7773812629903724 - 0.0311181166222198 0.7773812629903724 0.0291780472082805 0.8075354957734567 - 0.0291780472082805 0.8075354957734567 0.0271665743590979 0.8357135543195029 - 0.0271665743590979 0.8357135543195029 0.0250886205533450 0.8618464823641238 - 0.0250886205533450 0.8618464823641238 0.0229492710048899 0.8858703285078534 - 0.0229492710048899 0.8858703285078534 0.0207537612580391 0.9077263027785316 - 0.0207537612580391 0.9077263027785316 0.0185074644601613 0.9273609206218432 - 0.0185074644601613 0.9273609206218432 0.0162158784103383 0.9447261340410098 - 0.0162158784103383 0.9447261340410098 0.0138846126161156 0.9597794497589419 - 0.0138846126161156 0.9597794497589419 0.0115193760768800 0.9724840346975701 - 0.0115193760768800 0.9724840346975701 0.0091259686763267 0.9828088105937273 - 0.0091259686763267 0.9828088105937273 0.0067102917659601 0.9907285468921895 - 0.0067102917659601 0.9907285468921895 0.0042785083468638 0.9962240127779701 - 0.0042785083468638 0.9962240127779701 0.0018398745955771 0.9992829840291237 - 0.0018398745955771 0.9992829840291237 632 mat :

wr64 0.0486909570091397 0.0243502926634244 - 0.0486909570091397 0.0243502926634244 0.0485754674415034 0.0729931217877990 - 0.0485754674415034 0.0729931217877990 0.0483447622348030 0.1214628192961206 - 0.0483447622348030 0.1214628192961206 0.0479993885964583 0.1696444204239928 - 0.0479993885964583 0.1696444204239928 0.0475401657148303 0.2174236437400071 - 0.0475401657148303 0.2174236437400071 0.0469681828162100 0.2646871622087674 - 0.0469681828162100 0.2646871622087674 0.0462847965813144 0.3113228719902110 - 0.0462847965813144 0.3113228719902110 0.0454916279274181 0.3572201583376681 - 0.0454916279274181 0.3572201583376681 0.0445905581637566 0.4022701579639916 - 0.0445905581637566 0.4022701579639916 0.0435837245293235 0.4463660172534641 - 0.0435837245293235 0.4463660172534641 0.0424735151236536 0.4894031457070530 - 0.0424735151236536 0.4894031457070530 0.0412625632426235 0.5312794640198946 - 0.0412625632426235 0.5312794640198946 0.0399537411327203 0.5718956462026340 - 0.0399537411327203 0.5718956462026340 0.0385501531786156 0.6111553551723933 - 0.0385501531786156 0.6111553551723933 0.0370551285402400 0.6489654712546573 - 0.0370551285402400 0.6489654712546573 0.0354722132568824 0.6852363130542333 - 0.0354722132568824 0.6852363130542333 0.0338051618371416 0.7198818501716109 - 0.0338051618371416 0.7198818501716109 0.0320579283548516 0.7528199072605319 - 0.0320579283548516 0.7528199072605319 0.0302346570724025 0.7839723589433414 - 0.0302346570724025 0.7839723589433414 0.0283396726142595 0.8132653151227975 - 0.0283396726142595 0.8132653151227975 0.0263774697150547 0.8406292962525803 - 0.0263774697150547 0.8406292962525803 0.0243527025687109 0.8659993981540928 - 0.0243527025687109 0.8659993981540928 0.0222701738083833 0.8893154459951141 - 0.0222701738083833 0.8893154459951141 0.0201348231535302 0.9105221370785028 - 0.0201348231535302 0.9105221370785028 0.0179517157756973 0.9295691721319396 - 0.0179517157756973 0.9295691721319396 0.0157260304760247 0.9464113748584028 - 0.0157260304760247 0.9464113748584028 0.0134630478967186 0.9610087996520538 - 0.0134630478967186 0.9610087996520538 0.0111681394601311 0.9733268277899110 - 0.0111681394601311 0.9733268277899110 0.0088467598263639 0.9833362538846260 - 0.0088467598263639 0.9833362538846260 0.0065044579689784 0.9910133714767443 - 0.0065044579689784 0.9910133714767443 0.0041470332605625 0.9963401167719553 - 0.0041470332605625 0.9963401167719553 0.0017832807216964 0.9993050417357722 - 0.0017832807216964 0.9993050417357722 642 mat :

This page was compiled by Mike "Pomax" Kamermans, on June 5th, 2011, as a resource for the primer on Bezier curves, specifically the section that deals with arc length computation. You are free to do what you like with this data. Go make the world a better place than it was when you found it.

Int(f,a,b,n)

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 :

Σalgo(lo,mesh,hi,∫)

plug integral curve

x Clear 1

Cumulative integration algo style ... lo mesh hi ∫ Σalgo cumullative Integrat'n algo style Y 0 : data 0 : 12 mat x lo hi mesh range : data x ∫ vectorize augment : ux data 1 col : uy data 2 col : Y 1 el 0 : i 2 ux rows range Δx i el ux i el ux i 1 - el - : Y i el Y i 1 - el 12 / uy i 1 - el uy i el + (Δx i el * (* + : 21 line for ux Y augment 91 line :

select wr63/wr64

Test [Alvaro] x f 4 x 2^1 + / :

Σalgo(lo,mesh,hi,∫) ... poor candidate for this application

x f

w wr63 1 col : (weight, wi ≡ x wr63 2 col : (abscissea,xi ≡ a 0 : b 1 : 12 mat m wr63 rows : 41 line

y i 1 m range y i el b a - 2 / (x i el * b a + 2 / (+ : for y 21 line :

a 1 - : b 0 : 12 mat wr64 ≡ a 0 : b 1 : 12 mat wr63 ≡ 21 mat

t0 1 time :

Area b a - 2 / (w i el y i el f * (i 1 m sum eval * π - : 21015 -^*

1 time t0 - 2.5 s *

t0 1 time :

f a b 10000 Int π - 21014 -^*

1 time t0 - 34.7 s *

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 :

x Clear 1

x g 0 x * :

t0 1 time :

lo 0 : mesh 0.0005 : hi 1 : 13 mat ∫ x f eval : Σ lo mesh hi ∫ Σalgo : 31 sys

DISAMBIGUATE ... Σalgo(lo,mesh,hi,∫) is a cumulative export data set. In fact very fast as it calculates 2000 times in 9 s. The meshed Int(f,a,b,2000) calculates once in 7 s. Legendre is 64 mesh in 2.5 s

Slow as it plots all calculs

x f Σ x f x g 0 1 10 hatch 3 1 sys

1 time t0 - 8.7 s *

n Σ rows : 2001

Σ n 2 el π - 4.2 108 -^* -

superb but ...
less accurate [1000] x f x 01 int π - 1.5 1014 -^* -

Simpson incompatible f(x) ... X^Y not scalar wrt 'x'

α 1523043253 - 27354.7 - 43256.75 25388.3 - 40061 mat :

init 0 : end 12 : N 960 : 13 mat

Fréchet: extreme value-II x f α 1 el α 2 el x α 4 el - (/ (α 5 el^* α 2 el x α 4 el - (/ (α 3 el^- exp * α 6 el + :

x c 10 x ≤ (x end ≤ (& cases :

0 f 403

x Y D x f :

sol init 0 end N x Y D rkfixed :

x f x c * sol 2 1 sys

Result above max ...
Smath X^Y not scalar wrt 'x'
Rescued  via Romberg kernel x f x 012 int #

sol sol rows row 122980712 mat

f 012100 ∫Romberg 29808