<p> Next visit "singular end point" </p>

314b7bd0-5750-4f63-83d2-62596fcae9a1 30 3 5 decimal

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

f a b maxerr n SR h b a - n / : h Simpson 13 / h 2 / a f b f + (* h a k h * + f k 1 n 1 - sum * + (* 23 / h * a h 2 / - k h * + f k 1 n sum * + : ε h Simpson eval : dintp ε : ε abs maxerr > n 2 n * : h b a - n / : dint 13 / h 2 / a f b f + (* h a k h * + f k 1 n 1 - sum * + (* 23 / h * a h 2 / - k h * + f k 1 n sum * + eval : ε dint dintp - 2 / : dintp dint : 51 line while dint dint ε + : 61 line :

x f x sqrt exp :

подынтегральная функция

integrand

t0 1 time :

Σ f 02106 -^100 SR :

Σ 5.407528146177

====================================================== >=>=>

1 time t0 - 14 s *

Technical accuracy 5.40752818465627 ≡

109 -^50 7.5 1011 -^* (109 -^100 7.5 1011 -^* (23 mat 15 min * 15 min * 21 mat ≡

5.40752818465627 Σ - 3.8 108 -^*

work sheet option integral accuracy: 1000 f a b Int Smath 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^- (* : 32 / b a - (* w G w 01 int * 41 line eval :

t0 1 time :

f 02 Int 5.40752818465657

1 time t0 - 4 s *

5.40752818465627 f 02 Int - 31013 -^* -

a 0 : b 2 : 12 mat

Romberg kernel w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + 0 w ≤ (w 1 ≤ (& cases :

Romberg integrand w G w s f 12 w * 1 - (2^- (* :

Romberg rectangular/parabolic estimates w D b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + w diff 0 w ≤ (w 1 ≤ (& cases :

Integral acuracy => 1000 5.40752818465627 x f x a b int - 2.567 106 -^*

The function

x f

The Romberg kernel

x s

The Romberg parabolic estimates

x D

ΣSimpsonRichardson f 02106 -^100 SR : 5.4075281461769

ΣRomberg f 02 Int : 5.40752818465657

x f x a b int 5.40752561744737

Comapare integrators Technical accuracy 5.40752818465627 ≡ Mathcad Adaptive 5.40752818465632 ≡ Smath Romberg 5.40752818465657 ≡ Smath Integrator 1000 5.40752561744737 ≡ Smath Richardson 5.40752818461769 ≡ 51 mat

f a b n Simpson h b a - n / : 13 / h 2 / a f b f + (* h a k h * + f k 1 n 1 - sum * + (* 23 / h * a h 2 / - k h * + f k 1 n sum * + 21 line :

QED => Quod Erat Demonstrandum

QED Smath integrator is "Simpson" x f x 02 int f 021000 Simpson - 81015 -^* -

Technical accuracy

a b Clear 1

x f x sqrt exp :

a b m x f x a b int maple : 2 b sqrt exp 1 - b sqrt + (* a sqrt exp a sqrt * - a sqrt exp + (*

a 0 : b 2 : 12 mat

a b m 5.40752818465627

<p> Next visit "singular end point" </p>

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

a 0 : b 1 : 12 mat

b m 1 -

x m e 0 + 15 red b b m + 15 red 2 5 mat 2 1 sys

x f x 1 - x ln + (* :

b m 1 -

'x' not defined ... i.e: 
"singular end point" ln(0) x f x 0 e int #

"Romberg singular" ... option integral accuracy: 1000 f a b Int x f x a b int 32 / b a - (* w G w 01 int * ≡ (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^- (* : 32 / b a - (* w G w 1015 -^1 int * 41 line :

Smath native integrator x f x 1015 -^e int 1.84726388246312 -

Mathcad Adaptive 1.84726402473266 - ≡

Smath Romberg singular end point f a e Int 1.84726402472862 -

Didactic: Simgular end point Smath  adaptive integration.
Prohibitive in computation load, but may be useful in some
user application. f a b TOL Singular 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 : h b a - N 1 + / : j 2 N range x0 a j 1 - (h * + : x1 x0 h + : Y2 Y2 h x0 f x1 f + 2 / (* + : 31 line for 51 line while Y2 51 line eval :

t0 1 time :

x f x 1 - x ln + (* :

a 0 : b e : 12 mat

Σ f a b 0.000001 Singular :

Σ 1.8472638 -

1 time t0 - min 1.3