<p> hatch(f,g,u,v,N) </p>

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Simpson-Richardson Integration

t0 1 time :

x f x sqrt exp :

подынтегральная функция

integrand

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 :

Σ f 02106 -^50 SR :

Σ 5.407528146177

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

1 time t0 - 9 sec *

5.40752818465627 Σ - 0.000000038

f 02109 -^50 SR 9 min * ≡ 5.407528184581 ≡ Romberg accuracy 3 sec 31013 -^* - ≡ SimpsonRichardson 9 min 81011 -^* ≡ 31 mat

Technical accuracy 5.40752818465627 ≡

x f x 02 int 5.407525617

work sheet 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 01 int * 41 line :

t0 1 time :

f 02 Int 5.40752818465657

1 time t0 - 3 sec *

5.40752818465627 f 02 Int - 31013 -^* -

General comment: If you just want to use the Smath operator, set the work sheet option > 1000. That will satisfy most, if not all, Engineering works. In this examle, Romberg Int(f,a,b) is about twice nore accurate than SR(f,a,b,maxerr,n) at ~ ½ time.

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 - 8.357681 1015 -^* -

X 12331 mat :

Y 121 - 31 mat :

x f X Y x cinterp :

<p> hatch(f,g,u,v,N) </p>

Erzeugt eine Matrix für die Darstellung vertikaler Linien zwischen den Funktionen f(x) und 
g(x) im Bereich von x1 bis x2 mit 2*n Linien. Generates a matrix for 2*n vertical lines (hatching) 
between functions f(x) and g(x) in x-range from x1 to x2 f g u v N hatch dx v u - N / (: h u u f u u g 22 mat : x u v dx - u dx + range h h x dx 2 / + x dx 2 / + g x dx 2 / + x dx 2 / + f x dx + x dx + f x dx + x dx + g 42 mat stack : for h 41 line :

x g 0 :

H x f x g 0.5 325 hatch :

X Y augment x f H 3 1 sys

f 0.5 31000 Simpson 2.708333

x f x 0.5 3 int 2.708333