plotG ...Hatch

29c3a25e-ed3d-48f0-89c7-d9d7c8b3a407 66 6 9 decimal

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

plotG ...Hatch

vx vy char size color plotG n vx length : plot vx 1 el vy 1 el char size color augment : i 2 n range plot plot vx i el vy i el char size color augment stack : for plot 41 line :

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 :

XY 0 0.00794 0.1 0.012 0.2 0.022 0.3 0.048 0.4 0.167 0.5 1 0.6 0.167 0.7 0.048 0.8 0.022 0.9 0.012 1 0.00794 112 mat :

X XY 1 col : Y XY 2 col : data X Y . 14 black plotG : 31 line

t0 1 time :

x φ 0.002 0.002 x 0.5 - (2^+ / :

D 1 el 0 : i 26 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

X Y x ainterp x φ 2 1 sys

Naturally, Smath does not takes an idex as argument. Transit via a fake j x Q i j : D i el 2 / x X i 1 - el - (2^X i el X i 1 - el - (/ * D i 1 - el 2 / x X i el - (2^X i el X i 1 - el - (/ * - Y i el + D i el 2 / X i el X i 1 - el - (* - 21 line :

2 x Q 3 x Q 4 x Q 5 x Q 6 x Q data 6 1 sys Scalar interpolation

The companions scalar interpolation

Local interpolation wrt segments.
The "if" does not recognise. u Quad j local assignment n 6 : u X 2 el ≤ n 1 : u X n el ≥ n n : i 2 n range u X i el ≤ break return 'j' < (i-1) if for if if i (41 line : j u Q 21 line :

Scalar algo style interpolation 
over all 'J' segment(s) u Xdata J search index n 6 : u Xdata 2 el ≤ n 1 : u Xdata n el ≥ n n : resume next if if i 2 n range u Xdata i el ≤ break close if for J i (: 51 line :

u Interp u X J u Q :

plot t 0 0.5 0.01 range : i 1 t rows range vy i el t i el Interp eval : for t vy augment 31 line :

Identical interpolation 0.375 Interp 0.375 Quad 21 mat 0.12246 0.12246 21 mat

plot plot :

u 0.458488 :

An excellent Quadratic spline iscalar interpolation.

plot data 0.4585 0.5 + 14 red 1 5 mat x φ 4 1 sys

u Interp 0.5 - 0.000001

Quad t 0 0.5 0.005 range : i 1 t rows range vy i el t i el Interp eval : for t vy augment t 0.5 + vy reverse augment 21 mat 31 line :

Akima t 0 0.5 0.005 range : i 1 t rows range vy i el X Y t i el ainterp eval : for t vy augment t 0.5 + vy reverse augment 21 mat 31 line :

Essentially, the quadratic segments Q(j,x) are about same as the ones in "Holistic". This last method [Holistic]is based on solving for coefficients. Alike Q(j,x) Holistic does not allow the complete scan. Reading the test model in the Akima method, their test function is a non-sense, just a fabricated fake. Data from this test fonction φ(x) is far superior, it is based on real possible data. On the other hand, the given XY data does not let presume of bell shape, thus the "cubic" is false presumption. As given, the quadratic approach is justified. We know from educated background that the "shape" needs more points, better distributed in the turning region.

Mirror quadratic spline of symmetric data

Quad 1 el Quad 2 el data 3 1 sys

Akima: NOT better... being cubic, it
invents bell shape [unknown from data]. 0.375 φ 0.375 Interp X Y 0.375 ainterp 31 mat 0.113475 0.12246 0.127678 31 mat

Holistic => a bit closer 10 0.375 f 0.121699 ≡

Integrate piecewise i 15 range Σ i el i 1 + x Q x X i el X i 1 + el int : for

Export individual segment via symbolic expansion 3 x Q 11100271 - 10 x * + (2^* + (* - 8121 - 5 x * + (2^* + 297 + 50000 / -

Σ Σ sum 21 mat 0.000929 0.001669 0.003265 0.009435 0.047765 51 mat 0.063063 21 mat

Derive individual segment ... plot/Export 4 x Q (x diff 20061 - 5 x * + (* - 2973 - 10 x * + (* + 2500 / -

x crap 0 x ≤ (x 1 ≤ (& X Y x cinterp out data range if :

x Akima 0 x ≤ (x 1 ≤ (& X Y x ainterp out data range if :

x crap x Akima 0 0.00794 . 14 black 0.1 0.012 . 14 black 0.2 0.022 . 14 black 0.3 0.048 . 14 black 0.4 0.167 . 14 black 0.5 1 . 14 black 0.6 0.167 . 14 black 0.7 0.048 . 14 black 0.8 0.022 . 14 black 0.9 0.012 . 14 black 1 0.00794 . 14 black 11 5 mat 3 1 sys

The cumulative Integral transit procedure

u F x Interp :

z Int u F u X 1 el x int :

Does NOT support compactum

x Int z Int :

x f x Int : x g 0 : H xx f xx g 0 xx 40 hatch : 31 sys

2nd derivatives are  numbers 2 x Q (x 2 diff 3 x Q (x 2 diff 4 x Q (x 2 diff 5 x Q (x 2 diff 6 x Q (x 2 diff 51 sys 0.812 0.376 2.824 15.776 127.024 51 sys

xx 0.475 :

xx Int 0.32

Plot valid low range x Σ 0 x ≤ (x 0.5 ≤ (& x Int zombie if :

Cumulative integral over 'xx'.

x f x Σ : x g 0 : H xx xx Int + 20 red 1 5 mat 6 1 sys

0.5 Int 0.5

1 time t0 - 14.6 sec *

Cumul_0 t 0 0.5 0.005 range : i 1 t rows range vy i el t i el Int eval : for t vy augment 31 line :

Cumul_1 t 0 0.5 0.005 range : i 1 t rows range vy i el t i el Int eval : for t 0.5 + 1 vy reverse - augment 31 line :

Cumul_0 Cumul_1 0.5 0.5 . 14 black 1 5 mat 3 1 sys

Cumul_1 101 row 1112 mat

data Σ Finite differences Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

Quad 1 el Σ 101 row 0.5 0.063093 12 mat

Useful Maple

x f x φ x int maple : 5 sqrt 532 /^1 - 2 x * + (* atan * 50 /

x IntCurve 0 x ≤ (x 1 ≤ (& x f N/A if :

x IntCurve

Naturally, Smath does not takes an idex as argument. Transit via a fake j x Q i j : D i el 2 / x X i 1 - el - (2^X i el X i 1 - el - (/ * D i 1 - el 2 / x X i el - (2^X i el X i 1 - el - (/ * - Y i el + D i el 2 / X i el X i 1 - el - (* - 21 line :

x f 5 x Q :

L H N # f range Algo U 0 : dx H L - N / : i 1 N 1 + range U i el L dx - (dx i * + : for j 1 range range u j el U j el : y j el U j el f eval : 21 line for u y augment 51 line :

This algo style will NOT accept some user function. For instance will reject "Interp(x)" from user interpolation function which is based on conditional statements.

range Plot 0.25 0.45 100 x f range Algo :

40 Plot 60 Plot 100 Plot 31 sys

40 Plot 60 Plot 100 Plot X 4 el Y 4 el . 14 black 1 5 mat X 5 el Y 5 el . 14 black 1 5 mat 5 1 sys

x f x Interp :

range Plot 0.25 0.45 100 x f range Algo :

Does NOT support user "if function" that constructs "Interp(x).

40 Plot 60 Plot 100 Plot X 4 el Y 4 el . 14 black 1 5 mat X 5 el Y 5 el . 14 black 1 5 mat 5 1 sys