Hatch

76c9015b-9d3b-4ebd-9545-b82c967d1e8d 49 6 9 decimal

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Hatch

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 :

The normalised integrator

The "normalised integrator" is of range style [0 .. 1] Advanced integrators are based on that kind of reduction: 1. The multi dimensions "Simplex". 2. The "Quadratures" [Legendre, ChebyShev, Lobatto, Hermite, Jacobi ...] 3. The more common "Romberg" numerical integrator. Though this demo function looks naive and simple to integrate, it is NOT numerically. Mathcad advanced integrator has several answers. If the function has a symbolic int(f(x),x) expansion, you don't need the numerical integrator as demonstrated below, which we use as sanity check wrt the "normalised integrator". ____________________________________________ One question => 2 answers ! Why do we need a numerical integrator if not really needed ? 1. NOT needed ? Yes, but as long as you will have the Maple/Smath companions and as long as the function is a "mathematical function". 2. If you don't have a mathematical function like a spline interpolating function, you won't have a "mathematical derivative" therefore: NO exploitabe derivative. Now, there is a distinction of prime importance like this: a CAS like Mathcad, outputs "splines" as analytical to the 2nd order derivative ... that means interpretable for the "normalised integrator". The converse is not true in Smath, i.e: Smath does not output a scalar spline function, just a graph interpolation function on the graph canvas and/or in-situ interpolated value(s) or collected in discrete form via an indexed discrete module ... BUT here is the NUANCE => Smath DOES recognise the reduced form of the integrand G(w)dw

32 / b a - (* w G w 01 int *

1. The reduction: ... => define the transform domain

a 0 : b 2 : 21 line

x f x sqrt exp :

w s b a - (34 / 2 w * 1 - (* 14 / 2 w * 1 - (3^* - 12 / + (* a + :

x f

w ds 32 / b a - (* 12 w * 1 - (2^- (* :

--> 31 - 1 - 2 w * + (2^+ (* b a - (* 2 / - ≡

x s x ds 0 2 1 sys

2. Integration in the reduced domain

x f x a b int 32 / b a - (* w G w 01 int * ≡

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

where =>

x f x a b int 5.407526

x G 0 2 1 sys

The reduced system is lot more accurate 32 / b a - (* w G w 01 int * 5.40752818465657

Smath/Maple agrees with Mathematica x f x int maple 2 x sqrt exp * 1 - x sqrt + (*

x m 2 x sqrt exp * 1 - x sqrt + (* :

sanity check in  absolute accuracy 0 m abs 2 m + 5.40752818465627

a sense of scale/accuracy 1. "fm = fermi = 10e-15" 2. classical diameter electron ...... 5.635882 fm ...... In this example, the native Smath 5 true decimals is now 12

sanity check the triplet Mathcad 11 Romberg 5.40752818465632 ≡ Technical accuracy 5.40752818465627 ≡ Smath Romberg 5.40752818465657 ≡ 31 line

3. Highlight the demo

x f x m :

x g 0 :

<= activate the hatching "algo"

ω x f x m : x g 0 : blank the green hatching H x f x g 0230 hatch : 41 sys :

4. Integrate a splined data set

Discrete construct of ainterp

chr char ≡

sz size ≡

clr color ≡

vx vy chr sz clr plotG n vx length : plot vx 1 el vy 1 el chr sz clr augment : i 2 n range plot plot vx i el vy i el chr sz clr augment stack : for plot 41 line :

Low High step x fnct inc U Low High step range : i 1 U rows range f i el U i el fnct eval : for U f augment 31 line :

M Min Max Scale Max Min - (M M min - M max M min - / (* Min + :

data 40 0.4 - 80 0.461 - 120 0.483 - 160 1.53 - 200 1.065 240 0.135 280 0.008 - 320 0.039 - 360 0.068 400 0.171 - 102 mat :

The ainterp [Akima cubic spline] looks like the universal 'lspline'. For the spline plot, f(x) behaves like a scalar function but it is not wrt direct derivative/integration. The algo module inc(low,high,step,f(x)) => in collapsed.

x f x f Clear :

X data 1 col 440 Scale :

Y data 2 col :

x f X Y x ainterp :

Setup the plot

data X Y . 14 red plotG :

Spline 5 - 45 4.5 - x f inc :

<= spline in the X direction

Akima cubic spline

Spline data 2 1 sys

a 4 : b 40 : 21 line

Optimiz => symbolic w G w s f 12 w * 1 - (2^- (* :

Normalised cumulative area. 32 / b a - (* w G w 01 int * 11.817 -

x φ 12 x * 1 - (2^- :

time 45 sec * ≡

Disambiguation: In this spline demo, nothing is "true" except the data set itself [supposedly]. The spline f(x) is an "interpolation guesstimate", i.e: the best, maths can do for you. Consequently, the cumulative integration depends upon the "representativeness" of the unknown values along the running spline f(x).

x G 0 2 1 sys x s f 0.05 x φ * 2 1 sys

... faked φ(x)