4d5445d3-24c3-4c7f-8c16-eb7ba0d9630d 12 4 5 decimal

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

x c0 10 x ≤ (x 1 ≤ (& cases : x c1 11 x ≤ (x 10 ≤ (& cases : 21 line

plot x sqrt x c0 * x sqrt x c1 * 1 1.5 1 0.5 22 mat 31 sys :

plot

This Smath Romberg . . .

. . . reconciliates the arclength of the magnitude

Power fnct m*x^n ... in the entire domain

m => magnitude, x_range(0..∞).

For the unary magnitude m = 1, higher accuracy

is obtained from maple code arc(n,x)... in the

entire domain [x = 0 ... x = ∞] & much faster.

Arc(a,n,x) reconciliates m*x^n in the domain

[x = 0.0001 ... x = ∞].

Ro mberg reconciliates the entire domain

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Smath integrator 1 m u n^* u diff (2^+ sqrt u 1 x int

Romberg low region [acc 10000] f ε x Int

x f 1 m x n^* x diff (2^+ sqrt :

m 1 : n 13 / : 12 mat

ε 1015 -^:

f a b Int x f x a b int 32 / b a - (* w G w 01 int * ≡ (memo ≡ 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 ε 1 int * 41 line :

Higher accuracy of the "unary" 1*x^n
recast maple scalar arclength x domain [0..∞] @ magnitude a ≡ 1 n x arc x 2^n x ln * exp (2^n 2^* + sqrt u 1 - n + (* n u 2^n u ln * exp (2^n 2^* + sqrt * / u ε x int n * + n / :

13 / 10 arc 10.6362946

n 12 / :

λ 0.00001 :

arclength [λ 0.00001 ..1], sanity 6 decimals n 1 arc n λ arc - 1.4757806

Recapitulate the project

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reconciliate the "m" low range
accuracy decreases  λ ~ 0.00001 a m n x Arc 1 m u n^* u diff (2^+ sqrt u a x int :

λ 0.0001 :

sanity λ 0.0001 λ 1.5 12 / 1 Arc 1.85297

Romberg reconciliates "entire domain"
------   [0.000000000001 ..∞]  ----- m 1.5 : n 12 / : x f 1 m x n^* x diff (2^+ sqrt : 14 mat

λ 0.000000000001 :

sanity 4½ decimals f λ 1 Int 1.86793

sanity 6 decimals[+] f 110 Int 9.6165