<content> hatch, scale </content>

09450ea6-b091-46b4-9a58-38a038d7493d 71 4 5 false false 0 0 decimal

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

Technical limits demo ...

4 appVersion 1.0.8348.30405

try/on error from palette
encountered 3 times Smath
1. Thiele style 1
2. Laplace Admitance
3. rotate image typical RGB
4. more below ...

a x f 1 a x - / :

a 4 :

a x f a 100 a 100 - 2 2 mat 2 1 sys

a x f ∞ ± a 21 sys try ∞ ∞ - 431 sys

ε 1014 -^:

Simpson NFG !!! a x f x a ε + e int 21011^*

a x f 1 a x - / :

a 1 :

algo style ... command line wrt limits

# f a b ∫ x Ω x f x int maple : x Ω a b at 21 line :

x f a x f :_a 1 ε + :_b e : 15 mat

sanity WolframAlpha x f a b ∫ 32.7775161565296 -

11014^e 1 - (* / ln 32.7775161565296 -

<content> hatch, scale </content>

You need that one for scaling grayscale image M lo hi Scale hi lo - (M M Min - M Max M Min - / (* lo + :

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 g 0 :

Hatch f g a e 11 hatch :

Collect plot 1 a x - / Hatch 0 e 2.5 e 2.5 - 22 mat 41 sys :

Exact Integration

plot

data char size clr plot k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 13 mat for data r3 r4 r5 augment 21 line :

x y axes x y / - :

L H N xd U 0 : dx H L - N / : i 1 N 1 + (range U i el L dx - (dx i * + : for U 41 line :

L 1 - : H 1 : N 99 : 13 mat U L H N xd : 21 mat

N <= 99/100

result 1 U / vectorize Divide by zero try :

XY U result augment : data XY o 2 black plot : 21 sys

data axes 2 1 sys

If you set N 100
"Divide\0020\by\0020\zero"
may crash Smath
Don't export either

U rows 100

result rows 100

v 1 0.300737 - 0.991494 0.295478 - 0.982984 0.290227 - 0.974471 0.28498 - 0.965955 0.279738 - 0.957436 0.274502 - 0.948913 0.269271 - 0.940387 0.264044 - 0.931862 0.258818 - 0.923335 0.253593 - 0.914807 0.248371 - 0.906277 0.243153 - 0.897745 0.237937 - 0.889211 0.232725 - 0.880677 0.227512 - 0.872145 0.222296 - 0.863613 0.217081 - 0.855082 0.211862 - 0.846556 0.206637 - 0.838032 0.201408 - 0.829511 0.196173 - 0.820998 0.190927 - 0.812491 0.185672 - 232 mat :

v rows 23

n 0.5 :

i 1 v rows range :

Optimiz : Numeric collect v 1 col n i * el NOT integer index try :

1 x /

collect NOT integer index

j 1 - 1 0.99 - range :

data j 1 j / vectorize augment :

data 1 - 1 - 0.99 - 1.01 - 0.98 - 1.02 - 0.97 - 1.031 - 0.96 - 1.042 - 0.95 - 1.053 - 0.94 - 1.064 - 0.93 - 1.075 - 0.92 - 1.087 - 0.91 - 1.099 - 0.9 - 1.111 - 0.89 - 1.124 - 0.88 - 1.136 - 0.87 - 1.149 - 142 mat

pts data . 2 red plot :

data pts 2 1 sys

result Max result Min 21 sys 9999 - 21 sys

Solving 'divide by zero' ...

x Vf v f 12 x * g * v 2^/ - exp - sqrt atanh v 2^* g Vf * / :

g 9.8066 :

Vf 10 : v 0.625 : 12 mat

exessive unknown in advance x 01 0.001 range :

Division Zero 1 Vf v f #

We have no means to detect
the rangeability of x ..........
The code ▼ will find the range ..

0.723 Vf v f 0.0745

V k 1 : j 1 x rows range x j el Vf v f IsDefined 1 ≡ Y k el x k el Vf v f : k k 1 + : 21 line continue if break try for 21 line Y 21 line :

x x 1 V rows range el :

data x V augment :

V rows 724

XY data data rows row : 0.723 0.0745 12 mat

x rows 724

V V rows row V 724 row x x rows row 31 sys 0.0745 0.0745 0.723 31 sys

data rows 724

x Clear 1

0 0.723 21 mat 0 0.0745 21 mat Slope 0.103

x y axes x y / - :

u 0.04 :

sensitive to 'init' X x Vf v f u - x 0.01 roots : 0.3724

data 0.103 x * 2 1 sys

user 0 u X u X 032 mat : 0 0.04 0.3724 0.04 0.3724 032 mat

U user 1 col user 2 col augment :

data XY 1 1 el XY 1 2 el + 40 red 1 5 mat U 0.103 x * 4 1 sys

U 0 0.04 0.3724 0.04 0.3724 032 mat

data XY 11 el XY 12 el + 40 red 15 mat U 0.103 x * 41 sys

data data rows row 0.723 0.0745 12 mat

x Vf v f 12 x * g * v 2^/ - exp - sqrt atanh v 2^* g Vf * / :

Vf 10 : v 7 : 12 mat

g 9.8066 :

plot x Vf v f 1.9999 - x ≤ (x 0 ≤ (& x abs Vf v f if 21 sys :

plot

j 1 - 1 0.9 - range :

data j 1 j / vectorize augment :

data 1 - 1 - 0.9 - 1.111 - 0.8 - 1.25 - 0.7 - 1.429 - 0.6 - 1.667 - 0.5 - 2 - 0.4 - 2.5 - 0.3 - 3.333 - 0.2 - 5 - 0.1 - 10 - 00 0.1 10 0.2 5 0.3 3.333 0.4 2.5 0.5 2 0.6 1.667 0.7 1.429 0.8 1.25 0.9 1.111 11212 mat :

data axes 2 1 sys

N Clear 1

Range a Nested export ...

<content> Ω vector </content>

Ω 1 0.300737 - 0.991494 0.295478 - 0.982984 0.290227 - 0.974471 0.28498 - 0.965955 0.279738 - 0.957436 0.274502 - 0.948913 0.269271 - 0.940387 0.264044 - 0.931862 0.258818 - 0.923335 0.253593 - 0.914807 0.248371 - 0.906277 0.243153 - 0.897745 0.237937 - 0.889211 0.232725 - 0.880677 0.227512 - 0.872145 0.222296 - 0.863613 0.217081 - 0.855082 0.211862 - 0.846556 0.206637 - 0.838032 0.201408 - 0.829511 0.196173 - 0.820998 0.190927 - 0.812491 0.185672 - 232 mat :

Numeric ... NEST extractor companion     Copy/Paste  ...  zap  0's ... Strip0( ■ ) i collect N i el Ω 1 col i el Ω 2 col i el augment : break try N 21 line :

Ω rows 23

i 319 range :

Unest Ω i 12 range el :

Ω 31912 submatrix

i collect 00 0.982984 0.290227 - 12 mat 0.974471 0.28498 - 12 mat 0.965955 0.279738 - 12 mat 0.957436 0.274502 - 12 mat 0.948913 0.269271 - 12 mat 0.940387 0.264044 - 12 mat 0.931862 0.258818 - 12 mat 0.923335 0.253593 - 12 mat 0.914807 0.248371 - 12 mat 0.906277 0.243153 - 12 mat 0.897745 0.237937 - 12 mat 0.889211 0.232725 - 12 mat 0.880677 0.227512 - 12 mat 0.872145 0.222296 - 12 mat 0.863613 0.217081 - 12 mat 0.855082 0.211862 - 12 mat 0.846556 0.206637 - 12 mat 191 mat

Nested collection 0.982984 0.290227 - 12 mat 0.974471 0.28498 - 12 mat 0.965955 0.279738 - 12 mat 0.957436 0.274502 - 12 mat 0.948913 0.269271 - 12 mat 0.940387 0.264044 - 12 mat 0.931862 0.258818 - 12 mat 0.923335 0.253593 - 12 mat 0.914807 0.248371 - 12 mat 0.906277 0.243153 - 12 mat 0.897745 0.237937 - 12 mat 0.889211 0.232725 - 12 mat 0.880677 0.227512 - 12 mat 0.872145 0.222296 - 12 mat 0.863613 0.217081 - 12 mat 0.855082 0.211862 - 12 mat 0.846556 0.206637 - 12 mat 171 mat

Un-Nested collection Unest 0.982984 0.290227 - 0.974471 0.28498 - 0.965955 0.279738 - 0.957436 0.274502 - 0.948913 0.269271 - 0.940387 0.264044 - 0.931862 0.258818 - 0.923335 0.253593 - 0.914807 0.248371 - 0.906277 0.243153 - 0.897745 0.237937 - 0.889211 0.232725 - 0.880677 0.227512 - 0.872145 0.222296 - 0.863613 0.217081 - 0.855082 0.211862 - 0.846556 0.206637 - 172 mat

strip  0's  in vector v Strip0 k 0 : i 1 v rows range v i el 0 ≠ R k k 1 + : el v i el : End of strip if 11 line for R 31 line :

preserved NESTS ζ i collect Strip0 :

ζ 0.983 0.2902 - 12 mat 0.9745 0.285 - 12 mat 0.966 0.2797 - 12 mat 0.9574 0.2745 - 12 mat 0.9489 0.2693 - 12 mat 0.9404 0.264 - 12 mat 0.9319 0.2588 - 12 mat 0.9233 0.2536 - 12 mat 0.9148 0.2484 - 12 mat 0.9063 0.2432 - 12 mat 0.8977 0.2379 - 12 mat 0.8892 0.2327 - 12 mat 0.8807 0.2275 - 12 mat 0.8721 0.2223 - 12 mat 0.8636 0.2171 - 12 mat 0.8551 0.2119 - 12 mat 0.8466 0.2066 - 12 mat 171 mat

collect(i) ... 0's can be replaced
by else user stuff .

j 12 :

scan explorer wrt 'j'

Unest ζ j el 2 1 sys

j ζ j el 12 mat 12 0.8892 0.2327 - 12 mat 12 mat

Hyperbola

1 x /

Hyperbola white [254,254,254]

1 x / ∞ - x ≤ ( x 0 ≤ ( & * 1 x abs / 2 1 sys

Hyperbola

1 x / 1 x - / 2 1 sys

Hyperbola

1 x /

jmG: 20230914