hatch

524242d9-0c05-445b-ba78-d7a57f598a5a 42 4 5 decimal

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

Under "while", the module computes the last value(s). It can be used to interpolate on-line
and interpolate for a given vector. The project must be recast for other  Smath "RK".
Does NOT plot on canvas [not scalar] ... interpolate Rkadapt is defendable. init t1 t2 n f RK4 vx init : t t1 : h t2 t1 - n / : 13 mat t t2 < k1 t vx f h * eval : k2 t h 2 / + vx k1 2 / + f h * eval : 12 mat k3 t h 2 / + vx k2 2 / + f h * eval : k4 t h + vx k3 + f h * eval : 12 mat vx vx 16 / k1 2 k2 * + 2 k3 * + k4 + (* + eval : t t h + : 12 mat 31 line while vx 31 line :

x t f x t 0 + (+ :

init 0 : start 0 : end 2 : N 100 : 14 mat

last value @ 'end' RK4 init start end N f RK4 : 4.389056

sol init start end N x t f Rkadapt : u interp init start u N f RK4 start u ≤ (u end ≤ (& out range cases : 21 sys

u 3 sqrt : 1.7321

1.23456789 interp u interp 21 mat 1.2023 2.9202 21 mat

sol 1 col sol 2 col augment u u interp + 15 red 1 5 mat 2 1 sys

User project X 0 2.5 0.25 range : U 12 1.25 range : 21 sys

X X interp vectorize eval augment 11 mat 00 0.25 0.034 0.5 0.1487 0.75 0.367 1 0.7183 1.25 1.2403 1.5 1.9817 1.75 3.0046 2 4.3891 2.25 out range 2.5 out range 112 mat 11 mat

x f sol 1 col sol 2 col x linterp :

x c 10 x ≤ (x end ≤ (& cases :

x g 0 :

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 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 :

x f x c * u u interp + 15 red 1 5 mat x f x g 0 u 25 hatch 4 1 sys

x f x c * u u interp + 15 red 15 mat x f x g 0 u 25 hatch 41 sys

x f x 0 u int 1.4203

===== Working a user system =====

x t f x t sqrt 0 + (+ :

init 0 : start 0 : end 2 : N 200 : 14 mat

last value @ 'end' ... sanity MCD 11 RK4 init start end N f RK4 : 3.9998

u interp init start u N f RK4 start u ≤ (u end ≤ (& out range cases :

u 2.5 sqrt : 1.5811

1.23456789 interp u interp 21 mat 1.5241 2.5249 21 mat

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 :

t0 1 time :

X 02 0.25 range : Y X interp vectorize eval : XY X Y augment : x f X Y x ainterp 0 x ≤ (x end ≤ (& out range cases : 41 sys

Notes :

1. Takes no time to interpolate @ user point(s), plot canvas.

2. This solver is identical to Mathsoft Odesolve, which by

same token does not rescue to spline interpolation. Rather it

is based on ¬ if IsScalar. The Odesolve block is a wraper of

fixed steps rk, adaptive, stiff from menu option.

Observe ========================

RK4(init,t1,t2,n,f) ... is sensibly more accurate than Smath

built-in rkfixed. On the other hand, Rkadapt is generally

not equally spaced for export. The proposal will adequately

interpolate from on-line command. Would be rather foolish

linterp over 100 points from Rkadapt ... as well linterp from

less accurate rkfixed.

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2.3456789 f out range

X Y . 12 black plotG x f u u interp + 15 blue 1 5 mat 3 1 sys

1 time t0 - 10.5 s *

x t q x t sqrt 0 + (+ : t ω D t ω 1 el q : rkf 002 N t ω D rkfixed : rka 002 N t ω D Rkadapt : r rkf rows : 51 sys

RK4 rkf r row rka r row 31 mat 42 3.98 12 mat 2412 mat 31 mat

Observe: "if/otherwise" does not interpolate 
on the canvas. "if/else" => slow but plots. u interp start u ≤ (u end ≤ (& init start u N f RK4 eval out range if :

x interp

t0 1 time :

mesh at will to export X 02 0.1 range : Y X interp vectorize eval : XY X Y augment : 31 sys

XY 00 0.1 0.01 0.2 0.04 0.3 0.09 0.4 0.16 0.5 0.25 0.6 0.36 0.7 0.49 82 mat

1 time t0 - 23.7 s *