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

08858393-025d-4cba-9065-1df9d727d751 12 6 12 decimal

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

x sin x / x 0 ∞ int 34901405563.8022 -

π 2 /

it needs to be

x sin x / x ∞ - ∞ int #

x x 1 - 1 int 2.9606 1017 -^*

1017 -^

it correct, when

looks like 0

it needs to be just π

x 1 - x ln / x 1 ∞ int #

1 x / x 1 ∞ int #

1 x / x 0 ∞ int 333333333345.544

Nothing true, cuz this three ints diverges

x 1 - x ln / x 01 int #

this is ln(2)

x x 0100 int 5000

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 :

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

ε 1014 -^:

x f x 1 - x ε + ln / :

x f

'N' iterations
precision N 2000 : p 1013 -^: 21 sys

lower limit
upper limit a 0 : b 1 : 21 sys

Sanity ln(2)   0.693147180559945 f a b N p ∫Adaptive 0.693147180559504

x f x sin x / :

0 f #

x f x sin x ε + / :

x f

'N' iterations
precision N 2000 : p 1014 -^: 21 sys

∞

lower limit
upper limit a 0 : b 10 : 21 sys

Wolfram 1.65835 f a b N p ∫Adaptive 1.65835

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

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 5.40752818465635 ≡ Smath Integrator 5.407526 ≡ 41 line

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

Wolfram does not understand ... pure freak 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

Long range Romberg-Simpson Emeritus

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

x q 4 x 10 ≤ x x 10 ≥ cases :

x 016 0.1 range : XY x x q vectorize augment : data XY o 3 black plot : 31 sys

Numerical engine does not recognize > 15 D input

..... OK 
..... OK
from otherwise 9.999999999999999 q 10.000000000000001 q 5 q 31 mat 410431 mat

data

NOT exact [100] x q x 1016 int 77.94

x f x q :

f 1016100 Int 78.000000000000100