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

4d547b85-7682-4ffb-ac92-9e1ea43afb7a 50 6 12 decimal

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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 :

∫Adaptive(f,a,b,N,ε)

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 :

Smath/Maple Symbolic x sqrt x int maple 2 x 2 nthroot 3^* 3 /

x ln x int maple x 1 - x ln + (*

a 0 : b 2 : 21 line

x f x sqrt exp :

sanity check the quadruplet Mathcad 11 Romberg 5.40752818465632 ≡ Technical accuracy 5.40752818465627 ≡ Smath Romberg 5.40752818465657 ≡ Smath Integrator 5.407526 ≡ 41 line

0 f 1

2 f 4.11325

x f 0 0 f + 20 red 1 5 mat 2 1 sys

f a b 2000 Int 5.40752818465635

x f x sqrt :

Otherwise x sqrt x int 23 / x 32 /^* ≡

a 0 : b 2 : 21 line

exact from symbolic x m 23 / x 32 /^* :

X x sqrt x m - 0 ≡ x 12 solve : 1.500000000000000

x sqrt x m X X f + 200 black 1 5 mat 3 1 sys

Simpson [acc = 100]
 ...  Int(f,a,b,n)
True value symbolic x f x a b int f a b 2000 Int 2 m 31 mat 1.88553689854704 1.88561808316414 1.88561808316413 31 mat

x f x 1 - x ln + (* :

a 1015 -^: b e : 12 mat

x ln x 1 - x ln + ( * e 0 + 50 black 1 5 mat 3 1 sys

Simpson [acc 100]
Romberg kernel
Exact value x f x a b int f a b 5000 Int e 2^4 / - 31 mat 1.84724979780443 - 1.84726402473266 - 1.84726402473266 - 31 mat

X x ln x f - 0 ≡ x 3.75 4 solve : 3.857334825948710

u b 0 u 2 - ≤ 16 / 2 u + (3^* 2 - u ≤ (u 1 - ≤ (& 16 / 2 u + (3^* 23 / 1 u + (3^* - 1 - u ≤ (u 0 ≤ (& 23 / 1 u - (3^* - 16 / 2 u - (3^* + 0 u ≤ (u 1 ≤ (& 16 / 2 u - (3^* 1 u ≤ (u 2 ≤ (& 0 cases :

The Gaussian b-spline

Gaussian basis 'b-spline'

2 - x ≤ x 2 ≤ & x b Undefined if

exact acc 100
exact by design x b x 2 - 2 int 1.000000000000000

Torricelli unary-theoretical discharge

A 1 : r 0.025 : g 9.8067 : 13 mat

time vs level x t A r / 2 g / sqrt * 1 x - sqrt * :

0 t 18.063984

level vs time x L 0 t x t - :

1 L 18.063984

ratio level % 0.75 :

x c 10 x ≤ cases :

a 0 : b 1 : 12 mat

% t 9.03

x f x t :

x t x c * x L x c * % % t % 0 2 2 mat 3 1 sys

Simpson [100] 3½ D
Romberg sanity 6 D x f x a b int f a b 100 Int 21 mat 12.042138 12.042656 21 mat

Recast Romberg kernel x f x L :

Simpson [100] 3½ D
Romberg  sanity 6 D x f x a b int f a b 100 Int 21 mat 6.021847 6.021328 21 mat

'N' iterations
precision N 1000 : ε 109 -^: 21 sys

Sanity adaptive L a b N ε ∫Adaptive 6.021328

∫Adaptive(f,a,b,N,ε) Program algo style is a bit slower than open style, more convenient. Adapted from Carlos code [20180110]

Independent section below

]]> Int(f,a,b,n) Int(f,a,b,n) ... Simpson/Romberg Kernel

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 :

∫Adaptive(f,a,b,N,ε)

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 :

Logarithmic integrals; dilog & Dilog

ε 1014 -^:

t f t ln 1 t - / :

t q 1 t - ln t / :

f 1 ε + x n Int

x c 1 ε x ≤ (x e ≤ (& cases :

t0 1 time :

x d f 1 ε + x 10000 Int :

[200000, 8.4 min]... 1.64493406682487
[100, 0.3 min] ..... 1.64491089757522 ε d 1.64493406453749

t f t 1 ε + x int x c * x ln 1 x - / 1 1 1 2 - 2 2 mat 3 1 sys

1 time t0 - min 1

ε dilog 1.64493406684808 ≡

Zeta(2) 0 dlog 1.644934066848226 ≡

ε 1014 -^:

Infinite  limit x f x ln 1 x - / :

x f 1 1 1 5 - 2 2 mat 2 1 sys

'N' iterations
precision N 2000 : p 1016 -^: 21 sys

User inputs

lower limit
upper limit a ε : b 1 ε - : 21 sys

t0 1 time :

Sanity -π^2/6 [0 .. 1] .....   -1.644934066848226
Wolfram stalls @ 5 D ... -1.64493 f a b N p ∫Adaptive 1.64493406684793 -

1 time t0 - min 1.8

https://www.wolframalpha.com/input/?i=integrate+ln(x)%2F(1-x)+from+0+to1

t0 1 time :

f a b 100000 Int 1.64493406682488 -

1 time t0 - min 7.6

Comparative conclusion . . .

1. Both: Int(f,a,b,n) and ∫Adaptive(f,a,b,N,ε) support ± infinite limit(s) and singular end point.

2. Observe Wolfram Alpha hits the wall @ 5 decimals in this last demo example.

3. Only exp(sqrt(x)) has been found more accurate Int(f,a,b,n) Romberg-Simpson kernel.

4. at = accuracy, ∫Adaptive(f,a,b,N,ε) runs much faster . . . convince yourself from last demo.

5. You can plot the cumulative integration on canvas @ low [100] while accessing much higher

Engineering accuracy as needed in the document.

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.....
1.8 min
7.6 min Sanity 1.644934066848226 - ≡ f a b N p ∫Adaptive 1.64493406684793 - ≡ f a b 100000 Int 1.64493406682488 - ≡ 31 mat

Carlos => ∫Adaptive(f,a,b,N,ε)

Jean => Int(f,a,b,n)

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