dilog(x0 ... Dilog(x)

8603dfac-fa78-4f8c-966d-4171d43c6bfa 7 9 12 decimal

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Numerical Integrators

]]> dilog(x0 ... Dilog(x)

Set integral accuracy 100 ε 1014 -^: x dilog t ln 1 t - / t 1 ε + x int : x Dilog 1 t - ln t / t ε x int - : 31 line

x dilog x Dilog 1 - π 2 ^ 6 / 2 π 2 ^ 6 / 2 2 mat 3 1 sys Int(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 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) 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 :

Logarithmic integrals; dilog & Dilog

∞

t f t ln 1 t - / :

t q 1 t - ln t / :

ε 1014 -^:

x c 1 ε x ≤ (x e ≤ (& cases :

t0 1 time :

x d f 1 ε + x 100 Int :

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

x d x c * x f 1 1 1 2 - 2 2 mat 3 1 sys

1 time t0 - min 1.9

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 1.0

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

t0 1 time :

f a b 20000 Int 1.64493406627038 -

1 time t0 - min 1.6