hatch region

625147d3-7c8d-4396-9800-56df7a104108 37 3 5 auto

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hatch region

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 x.1 x.2 N hatch dx x.2 x.1 - N / (: h x.1 x.1 f x.1 x.1 g 22 mat : x x.1 x.2 dx - x.1 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 :

Simpson Integrator in the Transformed Domain [0,1] <= [a,b]

The Smath "normalised" integrator companion ... enables integration of the 'ainterp, cinterp' 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 / : 13 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 * 51 line :

sanity check the sextuplets Mathcad 11 Adaptiv 5.40752818465632 ≡ Technical accuracy 5.40752818465627 ≡ Simpson Transf'd.. 5.40752818465635 ≡ Smath Simpson 7½ D 5.4075281 ≡ Simplex 10½ D .... 5.40752818462 ≡ MCD Romberg TOL -7 5.40752809219025 ≡ 61 line

x f x sqrt exp :

x g 0 :

transformed domain [0,1] <= Simpson [a,b] f 0210000 Int 5.40752818465635

region x f x g 0210 hatch :

x f

The naive test fnct to integrate

x f region 2 1 sys

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 * + eval 21 line :

Most remarkable Simpson Transformed Domain !

x f 1 x - ln - x / :

TOL 1012 -^:

Zeta(2) => 1.644934066848226

x f 1 10 1 1 - 2 2 mat 2 1 sys

t0 1 time :

f TOL 1 TOL - (50000 Int #

1 time t0 - min 0

x f x TOL 1 TOL - (int 1.645187342

Intergrate wisely . . .

From a spline [no matter the type], nothing is true.

For Engineering purpose, minimal accuracy is enough.

]]> plotG(vx,vy,a,b,c)

vx vy a b c plotG a char ≡ b size ≡ c color ≡ 13 mat n vx length : plot vx 1 el vy 1 el a b c augment : i 2 n range plot plot vx i el vy i el a b c augment stack : for plot 51 line :

X 1 1.5 2 2.5 41 mat :

Y 1.3 3.1 5.2 0.8 41 mat :

Q(j,x)

D 1 el 0 : i 2 X rows range D i el 2 Y i el Y i 1 - el - X i el X i 1 - el - / (* D i 1 - el - : for 21 line

Composition of the quadratic segments j x Q D j el 2 / x X j 1 - el - (2^X j el X j 1 - el - (/ * D j 1 - el 2 / x X j el - (2^X j el X j 1 - el - (/ * - Y j el + D j el 2 / X j el X j 1 - el - (* - :

Scalar interpolation

The companions scalar interpolation

Local interpolation wrt segments.
The "if" does not recognise. x Quad j local assignment n X rows : x X 2 el ≤ n 1 : x X n el ≥ n n : i 2 n range x X i el ≤ break return 'j' < (i-1) if for if if i (41 line : j x Q 21 line :

Scalar algo style interpolation 
over all 'J' segment(s) x Xdata J search index n X rows : x Xdata 2 el ≤ n 1 : x Xdata n el ≥ n n : resume next if if i 2 n range x Xdata i el ≤ break close if for J i (: 51 line :

data X Y . 14 black plotG :

u Interp u X J u Q :

Quadratic segments

2 x Q 3 x Q 4 x Q data 4 1 sys

Don't mesh too fine, not to overkill plot t 1 2.5 1.05 range : i 1 t rows range vy i el t i el Interp eval : for t vy augment 31 line :

plot plot :

plot rows 31

Interpolate the Quad data vx plot 1 col : vy plot 2 col : x New vx vy x cinterp : 31 line

plot X Y augment 2 1 sys

t0 1 time :

u New u 1 2.5 int 5.066667

====================

1 time t0 - 0.28 s *

acc sec result 50 0.3 5.06666665333335 1000049 5.06666666666661 33 mat

h 0.01 :

u f x Interp :

x Φ x h + f x h - f + 2 / :