data

e844114f-b0c9-41e4-8253-945fabcc5bcd 398 4 5 decimal

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

X a n Append text or numerical R X : X rows 1 ≡ R X a augment : R X length n + el a : if 12 mat R 31 line :

X 1234567824 mat :

a JoBlo :

X a 7 Append 12345678000000 JoBlo 044 mat

X a 1 - Append 123456 JoBlo 824 mat

X a n Append text/numerical/symbolic R X : X rows 1 ≡ R X a augment : R X length n + el a : if 12 mat R 31 line :

Symbolic Append/Augment

X x cos 2 y * 1 z / 32 y^2 ln * 3 z 2^* - 11133 mat :

a x sqrt :

'a = 0' n [1, 4, 7, 11 ...]
auto-append multiples of cols X a 5 Append x cos 2 y * 1 z / 32 y^2 ln * 3 z 2^* - 1110000 x sqrt 053 mat

X a 8 Append x cos 2 y * 1 z / 32 y^2 ln * 3 z 2^* - 1110000000 x sqrt 063 mat

X a 11 Append transpose x cos 3100002 y * 2 y^2 ln * 1000 x sqrt 1 z / 3 z 2^* - 1000037 mat

Generalize the function Utilities M m n MyStuf i 1 m range j 1 n range X i j el M i j el : for for M X : 21 line :

Jo 34 MyStuf Jo 11 el Jo 12 el Jo 13 el Jo 14 el Jo 21 el Jo 22 el Jo 23 el Jo 24 el Jo 31 el Jo 32 el Jo 33 el Jo 34 el 34 mat

indexed user-vector-creator
... as a function utility u n id V j u n range V j el id j el : for V 21 line :

myVect 37 β V :

myVect 00 β 3 el β 4 el β 5 el β 6 el β 7 el 71 mat

myVect 1 el Hello ! : myVect 2 el Davide : 21 sys

myVect Hello ! Davide β 3 el β 4 el β 5 el β 6 el β 7 el 71 mat

data Strip nest stripped in 'R' u data 1 col : v data 2 col : k 0 : i 1 data rows 1 - range v i 1 + el v i el ≤ (v i 1 + el v i el ≥ (¤ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

data Strip3 u data 1 col : v data 2 col : k 0 : i 1 data rows range u i el v i el * 0 ≠ R k k 1 + : 1 el u i el : R k 2 el v i el : 21 line End of strip if 11 line for 21 line R 41 line :

data

data 0.175 317.303 0.1783 315.2141 0.2013 300.5918 0.2013 300.5918 0.2578 265.3591 0.3383 219.2655 0.3383 219.2655 0.4291 174.2471 0.5221 133.1614 0.5221 133.1614 0.6121 96.9537 0.7 67.9999 0.7 67.9999 0.7875 48.6602 0.875 39.8 0.875 39.8 0.9625 41.5013 1.05 52.2049 1.05 52.2049 1.1375 69.8787 1.225 92.2395 1.225 92.2395 1.3125 116.6746 1.4 139.5052 1.4 139.5052 1.4875 156.9021 1.575 165.5015 1.575 165.5015 1.6625 162.9198 1.75 150.2288 1.75 150.2288 1.8375 129.7698 1.925 105.5054 1.925 105.5054 2.0125 81.5624 2.1 61.1021 2.1 61.1021 2.1875 46.8501 2.275 40.756 2.275 40.756 2.3625 44.4316 2.45 58.9132 2.45 58.9132 2.5375 84.7074 2.625 120.7769 2.625 120.7769 2.7125 165.0492 2.8 212.8538 2.8 212.8538 2.8875 258.7324 2.975 296.6717 2.975 296.6717 3.0625 321.2105 3.15 329.6506 3.15 329.6506 3.2375 320.7014 3.325 295.9398 3.325 295.9398 3.4125 258.155 3.5 212.1164 3.5 212.1164 3.5875 163.176 3.675 117.0368 3.675 117.0368 3.7625 78.6706 3.85 49.7748 3.85 49.7748 3.9375 30.6099 4.025 18.9652 4.025 18.9652 4.1089 12.265 4.1708 8.9445 4.1708 8.9445 4.1964 7.7382 4.2 7.5659 752 mat :

S data Strip :

Before stripping data set, make sure values that look alike aren't from decimal places. This example is not equivoke from the origin, it comes from Hermite parametric and we know duplicates result from the return carriage,

S 0.175 317.303 0.1783 315.2141 0.2013 300.5918 0.2578 265.3591 0.3383 219.2655 0.4291 174.2471 0.5221 133.1614 0.6121 96.9537 0.7 67.9999 0.7875 48.6602 0.875 39.8 0.9625 41.5013 1.05 52.2049 1.1375 69.8787 1.225 92.2395 1.3125 116.6746 1.4 139.5052 1.4875 156.9021 1.575 165.5015 1.6625 162.9198 1.75 150.2288 1.8375 129.7698 1.925 105.5054 232 mat

XY 024681 .3 .45 .6 .6 2 .4 .5 - .62 - .65 3 .4 .48 - .6 - .61 4 .38 .45 .55 .59 55 mat :

M Unwrap V M 1 col : i 2 M cols range V V M i col stack : for V 31 line :

V r c Pack i 1 r range v i el V i el : for j 1 c range i 1 r range i r j * + V rows ≤ m i j el V i r j * + el : m i j el 0 : if for for v m augment 31 line :

V XY Unwrap :

V V :

P V 54 Pack :

V 012342 0.3 0.4 0.4 0.38 4 0.45 0.5 - 0.48 - 0.45 6 0.6 171 mat

P 024681 0.3 0.45 0.6 0.6 2 0.4 0.5 - 0.62 - 0.65 3 0.4 0.48 - 0.6 - 0.61 4 0.38 0.45 0.55 0.59 55 mat

Padd RHS New V 57 Pack :

New 024680001 0.3 0.45 0.6 0.6 0002 0.4 0.5 - 0.62 - 0.65 0003 0.4 0.48 - 0.6 - 0.61 0004 0.38 0.45 0.55 0.59 00058 mat

zeros 1 New cols range (0 * :

Pad zeros transpose New zeros transpose zeros transpose stack :

Pad 00000000024680001 0.3 0.45 0.6 0.6 0002 0.4 0.5 - 0.62 - 0.65 0003 0.4 0.48 - 0.6 - 0.61 0004 0.38 0.45 0.55 0.59 000000000000000000088 mat

data ListCollect collects index list select 12357916 mat : ct 1 : i select a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 51 line :

data n Decimate collect every 'n' pairs range 1 data rows n range : k 1 : i range u k el data i 1 el : v k el data i 2 el : k k 1 + : 31 line for u v augment 31 line 31 line :

S ListCollect transpose 0.175 0.1783 0.2013 0.3383 0.5221 0.7 317.303 315.2141 300.5918 219.2655 133.1614 67.9999 26 mat

S 10 Decimate transpose 0.175 0.7875 1.575 2.3625 3.15 3.9375 317.303 48.6602 165.5015 44.4316 329.6506 30.6099 26 mat

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 :

S 0.175 317.303 0.1783 315.2141 0.2013 300.5918 0.2578 265.3591 0.3383 219.2655 0.4291 174.2471 0.5221 133.1614 0.6121 96.9537 0.7 67.9999 0.7875 48.6602 0.875 39.8 0.9625 41.5013 1.05 52.2049 1.1375 69.8787 1.225 92.2395 1.3125 116.6746 1.4 139.5052 1.4875 156.9021 1.575 165.5015 1.6625 162.9198 1.75 150.2288 1.8375 129.7698 1.925 105.5054 2.0125 81.5624 2.1 61.1021 2.1875 46.8501 2.275 40.756 2.3625 44.4316 2.45 58.9132 2.5375 84.7074 2.625 120.7769 2.7125 165.0492 2.8 212.8538 2.8875 258.7324 2.975 296.6717 3.0625 321.2105 3.15 329.6506 3.2375 320.7014 3.325 295.9398 3.4125 258.155 3.5 212.1164 3.5875 163.176 3.675 117.0368 3.7625 78.6706 3.85 49.7748 3.9375 30.6099 4.025 18.9652 4.1089 12.265 4.1708 8.9445 4.1964 7.7382 502 mat :

XW 0.175 317.303 0.3325 217.0355 0.525 130.1479 0.7 61.3415 0.875 32.4856 1.05 47.5163 1.225 90.6788 1.4 143.2053 1.575 173.5314 1.75 155.6779 1.925 105.1301 2.1 56.8338 2.275 34.1471 2.45 51.1141 2.625 114.8755 2.8 214.0454 2.975 306.0658 3.15 341.7215 3.325 304.952 3.5 214.1093 3.675 111.3089 3.85 42.8756 4.025 15.8375 4.2 7.5659 242 mat :

X Y . 8 red plotG

XW 1 col XW 2 col . 14 black 15 mat

P XW 1 col XW 2 col . 14 black plotG :

idx 32 :

X S idx 1 el :

Y S idx 2 el :

XY X Y + 20 blue 15 mat :

Title 1360 Carbon 12 12 black 15 mat :

xy 5 - 050000505 - 52 mat :

<= centered crosshair

Ymarker

ykr 11001022 mat :

P S ykr XY Title 5 1 sys

X Y 12 mat 2.71 165.05 12 mat

M Lo Hi Scale Hi Lo - (M M Min - M Max M Min - 1012 -^+ / (* Lo + :

Round OFF a matrix

M n Mround i 1 M rows range j 1 M cols range m i j el M i j el n round : for for m 21 line :

Strip single 0's

Strip paired 0's

v Strip0 k 0 : i 1 v rows range v i el 0 ≠ R k k 1 + : el v i el : 11 line undefined if 11 line for R 31 line :

data Strip2 u data 1 col : v data 2 col : k 0 : i 1 data rows range u i el v i el * 0 ≠ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for R 31 line 31 line :

data 000012340000567800910102 mat :

v 10305616 mat transpose :

v Strip0 transpose 135614 mat

N data Strip2 :

N 3 el 5612 mat

<= Extract a particular row

N 1212 mat 3412 mat 5612 mat 7812 mat 91012 mat 51 mat

Strip2 (data) will strip 0's from dual vectors that have paired values and paired 0's. Very useful to clean "Contour Extract Advisor". Paired values need not be integer.

data Unest v data 1 el : j 2 data rows range v v data j el stack : for v 31 line :

N Unest 1234567891052 mat

data Strip2 nest striped in 'R' u data 1 col : v data 2 col : k 0 : i 1 data rows range u i el v i el * 0 ≠ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

data Strip2 1234567891052 mat

data Nest_1 u data 1 col : v data 2 col : k 0 : i 1 data rows range R k k 1 + : el u i el v i el 12 mat : for R 31 line 31 line :

data Nest_2 special unknown use u data 1 col : v data 2 col : k 0 : i 1 data rows 1 - range R k k 1 + : el u i el v i el u i 1 + el v i 1 + el 22 mat : for R 31 line 41 line :

On the principle of the rangeability of "rows(data)" you can multiple nest. if necessary: delete some rows in data.

XW 0.175 317.303 0.3325 217.0355 0.525 130.1479 0.7 61.3415 0.875 32.4856 1.05 47.5163 1.225 90.6788 1.4 143.2053 1.575 173.5314 1.75 155.6779 1.925 105.1301 2.1 56.8338 2.275 34.1471 2.45 51.1141 2.625 114.8755 2.8 214.0454 2.975 306.0658 3.15 341.7215 3.325 304.952 3.5 214.1093 3.675 111.3089 3.85 42.8756 4.025 15.8375 4.2 7.5659 242 mat :

XW Nest_1 0.175 317.303 12 mat 0.3325 217.0355 12 mat 0.525 130.1479 12 mat 0.7 61.3415 12 mat 0.875 32.4856 12 mat 1.05 47.5163 12 mat 1.225 90.6788 12 mat 1.4 143.2053 12 mat 1.575 173.5314 12 mat 1.75 155.6779 12 mat 1.925 105.1301 12 mat 2.1 56.8338 12 mat 2.275 34.1471 12 mat 2.45 51.1141 12 mat 2.625 114.8755 12 mat 2.8 214.0454 12 mat 2.975 306.0658 12 mat 3.15 341.7215 12 mat 3.325 304.952 12 mat 3.5 214.1093 12 mat 201 mat

XW Nest_2 0.175 317.303 0.3325 217.0355 22 mat 0.3325 217.0355 0.525 130.1479 22 mat 0.525 130.1479 0.7 61.3415 22 mat 0.7 61.3415 0.875 32.4856 22 mat 0.875 32.4856 1.05 47.5163 22 mat 1.05 47.5163 1.225 90.6788 22 mat 1.225 90.6788 1.4 143.2053 22 mat 1.4 143.2053 1.575 173.5314 22 mat 1.575 173.5314 1.75 155.6779 22 mat 1.75 155.6779 1.925 105.1301 22 mat 1.925 105.1301 2.1 56.8338 22 mat 111 mat

data Nest_3 special unknown use u data 1 col : v data 2 col : k 0 : i 1 data rows 2 - (range R k k 1 + : el u i el v i el u i 1 + el v i 1 + el u i 2 + el v i 2 + el 32 mat : for R 31 line 41 line :

XW Nest_1 5 el 0.875 32.4856 12 mat

XW Nest_2 5 el 0.875 32.4856 1.05 47.5163 22 mat

XW Nest_3 5 el 0.875 32.4856 1.05 47.5163 1.225 90.6788 32 mat

data r1 r2 CollectRange suite collect 1... r2 v data Nest_1 : v data Nest_1 r1 el : j 1 r2 range v v data Nest_1 j el stack : for v 2 v rows range 12 range el 51 line :

0.175 317.303 0.3325 217.0355 0.525 130.1479 0.7 61.3415 0.875 32.4856 1.05 47.5163 1.225 90.6788 1.4 143.2053 1.575 173.5314 1.75 155.6779 1.925 105.1301 2.1 56.8338 2.275 34.1471 2.45 51.1141 2.625 114.8755 2.8 214.0454 2.975 306.0658 3.15 341.7215 3.325 304.952 3.5 214.1093 3.675 111.3089 3.85 42.8756 4.025 15.8375 4.2 7.5659 242 mat

XW 17 CollectRange 0.175 317.303 0.3325 217.0355 0.525 130.1479 0.7 61.3415 0.875 32.4856 1.05 47.5163 1.225 90.6788 72 mat

data list ListCollect collects index list ct 1 : i list a ct el data i 1 el : b ct el data i 2 el : ct ct 1 + : 31 line for a b augment 41 line :

list 12357916 mat :

XW list ListCollect 0.175 317.303 0.3325 217.0355 0.525 130.1479 0.875 32.4856 1.225 90.6788 1.575 173.5314 62 mat

vec Stack V 1 el vec 1 el : i 2 vec rows range u i 1 - el vec i el : for V u stack 31 line :

v π 2 sqrt 2 ln 1.23456789 π sqrt 51 mat :

v Stack transpose 3.1416 12342 0.3 0.4 0.4 0.38 4 0.45 0.5 - 0.48 - 114 mat

v Stack 11 mat 3.141593 12342 0.3 0.4 0.4 0.38 4 0.45 0.5 - 0.48 - 0.45 6 0.6 0.62 - 0.6 - 0.55 8 0.6 0.65 0.61 0.59 1.414214 0.693147 1.234568 1.772454 291 mat 11 mat

data Strip strip duplicates Y add idx v data 2 col : k 0 : i 1 data rows 1 - (range v i 1 + el v i el ≠ R k k 1 + : el v i el : 11 line Undefined if 11 line for 21 line V R v v rows el stack : u 1 V rows range : i 1 V rows range u i el i : for u V augment 71 line :

data StripXY strip pairwise XY duplicates u data 1 col : v data 2 col : k 0 : i 1 data rows 1 - (range v i 1 + el v i el ≤ (v i 1 + el v i el ≥ (¤ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

data

data 0.175 317.303 0.1783 315.2141 0.2013 300.5918 0.2013 300.5918 0.2578 265.3591 0.3383 219.2655 0.3383 219.2655 0.4291 174.2471 0.5221 133.1614 0.5221 133.1614 0.6121 96.9537 0.7 67.9999 0.7 67.9999 0.7875 48.6602 0.875 39.8 0.875 39.8 0.9625 41.5013 1.05 52.2049 1.05 52.2049 1.1375 69.8787 1.225 92.2395 1.225 92.2395 1.3125 116.6746 1.4 139.5052 1.4 139.5052 1.4875 156.9021 1.575 165.5015 1.575 165.5015 1.6625 162.9198 1.75 150.2288 1.75 150.2288 1.8375 129.7698 1.925 105.5054 1.925 105.5054 2.0125 81.5624 2.1 61.1021 2.1 61.1021 2.1875 46.8501 2.275 40.756 2.275 40.756 2.3625 44.4316 2.45 58.9132 2.45 58.9132 2.5375 84.7074 2.625 120.7769 2.625 120.7769 2.7125 165.0492 2.8 212.8538 2.8 212.8538 2.8875 258.7324 2.975 296.6717 2.975 296.6717 3.0625 321.2105 3.15 329.6506 3.15 329.6506 3.2375 320.7014 3.325 295.9398 3.325 295.9398 3.4125 258.155 3.5 212.1164 3.5 212.1164 3.5875 163.176 3.675 117.0368 3.675 117.0368 3.7625 78.6706 3.85 49.7748 3.85 49.7748 3.9375 30.6099 4.025 18.9652 4.025 18.9652 4.1089 12.265 4.1708 8.9445 4.1708 8.9445 4.1964 7.7382 4.2 7.5659 752 mat :

data StripXY 0.175 317.303 0.1783 315.2141 0.2013 300.5918 0.2578 265.3591 0.3383 219.2655 0.4291 174.2471 0.5221 133.1614 0.6121 96.9537 0.7 67.9999 0.7875 48.6602 0.875 39.8 0.9625 41.5013 122 mat

create/collect/nest i 16 range : j 13 range : 12 mat X i 1 - : Y X 2^vectorize : U X 0.5^vectorize : D X Y U augment : M D i j el : N M 13 range 23 range el : 71 sys

Export X Y U M N 15 mat :

my 4 :

index 1  has property
Σ over all 'my' cols mySum Export my col 1 el sum : 78.382

Export 01234561 mat 0149162561 mat 01 1.414 1.732 2 2.236 61 mat 00011124 1.414 39 1.732 4162525 2.236 63 mat 00114 1.414 32 mat 15 mat

Subdivide a range in 'n' intervals ... demo n pivot interval j 1 n 1 + (range h j el pivot - pivot j * + n / : for h 21 line :

This demo illustrates a wisely designed export Pro. style ... ie: it populates high/mid/low a very NON-linear project similar to a user project function.

X 015914 mat : u1 32 X 2 el X 1 el - interval : u2 16 X 3 el X 2 el - interval : u3 8 X 4 el X 3 el - interval : data u1 X 1 el + u2 X 2 el + u3 X 3 el + stack : 51 sys

data Strip strip repeated in  col S X 1 el : u data 1 col : k 0 : i 1 u rows 1 - (range u i 1 + el u i el ≤ (u i 1 + el u i el ≥ (¤ S k k 1 + : el u i el : 11 line end of strip recover last u 21 line if 11 line for 21 line S X 4 el stack 51 line :

New data Strip : j New : collect j New sqrt vectorize augment : 31 sys

collect collect rows row 9312 mat

Publish your NON-linear project standard table style ... wisely populated wrt NON-linear ranges. Typical demo.

collect 1 1 + 20 red 1 5 mat 5 5 sqrt + 20 red 1 5 mat 3 1 sys

New 0 0.0312 0.0625 0.0938 0.125 0.1562 0.1875 0.2188 0.25 0.2812 0.3125 0.3438 0.375 131 mat

uv Strip collect 0 : uv uv 1 csort : uu uv 1 col : vv uv 2 col : k 0 : i 1 uu rows 1 - (range uu i 1 + el uu i el ≥ (vv i 1 + el vv i el ≠ (& collect k k 1 + : el uu i el vv i el 12 mat : 11 line end of strip if 11 line for 21 line Un-nest S Un collect 1 el : j 2 collect rows range Un Un collect j el stack : for Un 91 line :

Srip(data) on 2 if/else conditons ... NUMERIC Mydata Strip Mydata 0 : u Mydata 1 col : v Mydata 2 col : k 0 : i 2 Mydata rows (range u i el u i 1 - el ≥ (v i el v i 1 - el - 0 ≠ (& R k k 1 + : el u i el v i el 12 mat : u i el u i 1 - el ≥ (v i el v i 1 - el - 0 ≡ (& R k k 1 + : el u i el v i el 12 mat : end of strip collect/Unest 21 line if if 11 line for j 2 R rows range New j 1 - el R j el : for Unest New 1 el : j 2 New rows range Unest Unest New j el stack : for Unest 91 line :

Subdivide a range in 'n' intervals ... demo n pivot interval j 1 n 1 + (range h j el pivot - pivot j * + n / : for h 21 line :

NUMERIC hi 483 interval : lo 162 - interval reverse : 21 sys

rename/isolate Lo lo :

x f x 3^:

data_hi hi hi f vectorize augment : data_lo Lo Lo f vectorize augment : 21 sys

alternate plot: line/points

data_lo data_hi 2 1 sys

data data_lo data_hi stack :

XY Strip strip col pairwise repeated '0' end data data rows row : XY XY 1 csort : uu XY 1 col : vv XY 2 col : 12 mat k 0 : i 1 uu rows 1 - (range uu i 1 + el uu i el ≥ (vv i 1 + el vv i el ≠ (& collect k k 1 + : el uu i el vv i el 12 mat : 11 line end of strip if 11 line for 21 line Un-nest the stripped collect Un collect 1 el : j 2 collect rows range Un Un collect j el stack : for Un end stack 91 line :

Export data Strip :

V n Mround M V 1 el V 2 el stack : i 1 M rows range j 1 M cols range m i j el M i j el n round : for for m 31 line :

V n rounded i 1 V rows range j 1 V cols range m i j el V i j el n round : for for m 31 line :

C 1.235 2.124 3.554 4.555 22 mat 1.111 2.222 3.235 4.489 5.645 6.015 7.777 8.888 42 mat 21 mat :

C Mean vectorize 2.867 4.92275 21 mat

data C 2 Mround :

data 1.24 2.12 3.55 4.56 1.11 2.22 3.24 4.49 5.65 6.02 7.78 8.89 62 mat

C 1 el 2 rounded C 2 el 1 rounded 21 mat 1.24 2.12 3.55 4.56 22 mat 1.1 2.2 3.2 4.5 5.6 6 7.8 8.9 42 mat 21 mat

data 161 - 157 - 153 - 148 - 145 - 141 - 137 - 133 - 130 - 126 - 123 - 119 - 115 - 112 - 108 - 104 - 101 - 97 - 93 - 89 - 85 - 80 - 76 - 72 - 68 - 64 - 59 - 55 - 50 - 45 - 41 - 35 - 25 - 24 - 21 - 05 - 12 - 04 - 006 - 7 - 24 - 38 - 51 - 64 - 75 - 86 - 95 - 105 - 113 - 121 - 129 - 136 - 143 - 150 - 156 - 162 - 168 - 174 - 179 - 185 - 190 - 196 - 154 - 150 - 146 - 142 - 138 - 134 - 130 - 126 - 123 - 119 - 115 - 112 - 108 - 104 - 101 - 97 - 93 - 89 - 84 - 80 - 76 - 71 - 67 - 62 - 57 - 52 - 47 - 42 - 37 - 29 - 25 - 22 - 0017 - 05 - 18 - 11 - 15 - 12 - 8 - 0013 - 27 - 42 - 56 - 69 - 80 - 90 - 100 - 108 - 117 - 124 - 132 - 139 - 145 - 152 - 158 - 164 - 170 - 175 - 181 - 186 - 192 - 148 - 143 - 139 - 134 - 130 - 126 - 123 - 119 - 115 - 112 - 108 - 104 - 101 - 97 - 93 - 89 - 85 - 80 - 76 - 71 - 66 - 61 - 56 - 51 - 46 - 40 - 33 - 27 - 23 - 6 - 017 - 3 - 6 - 24 - 24 - 26 - 31 - 31 - 30 - 27 - 23 - 14 - 5 - 014 - 35 - 50 - 63 - 75 - 86 - 95 - 104 - 113 - 120 - 128 - 135 - 141 - 148 - 154 - 160 - 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159 - 154 - 149 - 144 - 139 - 134 - 129 - 123 - 118 - 112 - 106 - 99 - 92 - 84 - 75 - 66 - 56 - 44 - 31 - 13 - 2 - 02 - 11 - 13 - 17 - 08 - 31 - 30 - 28 - 42 - 61 - 74 - 84 - 92 - 101 - 110 - 119 - 127 - 135 - 143 - 150 - 157 - 163 - 170 - 176 - 182 - 188 - 193 - 198 - 204 - 209 - 214 - 219 - 223 - 228 - 233 - 237 - 242 - 246 - 251 - 255 - 6666 mat :

data data 1 - (* :

data length 4356

M Unwrap V M 1 col : i 2 M cols range V V M i col stack : for V 31 line :

data n Decimate collect every 'n' pairs range 1 data rows n range : k 1 : i range u k el data i 1 el : k k 1 + : 21 line for u 31 line 31 line :

V r c Pack i 1 r range v i el V i el : for j 1 c range i 1 r range i r j * + V rows ≤ m i j el V i r j * + el : m i j el 0 : if for for v m augment 31 line :

U data Unwrap eval : V U 5 Decimate eval : M V 3431 Pack : 31 sys

Decimation is not an exact terminattion procedure. You have to best manage your project in cause, mostly avoid extra '0'.

V rows sqrt 33

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V ct 1 : i 1 r rows range r i el 0 ≡ r i el : ct ct 1 + : if for r 31 line :

Address/replace Vector/Matrix

r 000 9.9 00 11.3 000 17.1 00113 mat transpose :

V transpose 9.9 11.3 17.1 113 mat

M ct 1 : i 1 m rows range j 1 m cols range m i j el 0 ≡ m i j el : ct ct 1 + : if for for m 31 line :

m 00 9.9 000 11.1 000 17.3 000 21.7 00 33.3 00 19.2 0099 - 055 mat :

M 9.9 11.1 17.3 21.7 33.3 19.2 99 - 55 mat

value table col lookup k 1 : j k table rows range table j col el value ≡ index k el j : k k 1 + : 21 line continue if for k 1 ≡ index NO match : index if 31 line :

M 12345648933 mat :

4 M 1 lookup 2321 mat

r M 3 col Max M 3 lookup : 311 mat

M r row 48913 mat

c 5 M 2 lookup : 211 mat

h 611526672316357553377120 mat transpose :

M c col 25831 mat

13 h 1 lookup NO match

6 h 1 lookup transpose 1561114 mat

M r c Matrix i 1 M rows range j 1 M cols range Ω i j el i r + j c + 12 mat : for for Ω 21 line :

Matrix inventor

Truncated times table M 1234524681036912154812162045 mat :

data UnestV v data 1 el : j 2 data rows range v v data j el stack : for v 31 line :

data UnestH v data 1 el : j 2 data cols range v v data j el augment : for v 31 line :

j 51 range : k 14 range : jj 15 range : kk 41 range : 14 mat

up/dn shifter n 3 - :

left/right λ 2.5 - :

N1 M λ n Matrix 1 row : N2 M λ n Matrix 1 col : N3 M λ n Matrix 4 row : N4 M λ n Matrix 5 col : 41 sys U N1 j el UnestV : V N2 k el UnestV : W N3 jj el UnestV : Z N4 kk el UnestV : 41 sys 12 mat

Rectangle U V W Z stack :

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 :

pts Rectangle o 5 black plot :

U V W Z pts 5 1 sys

U V W Z 14 mat 1.5 - 2 1.5 - 1 1.5 - 0 1.5 - 1 - 1.5 - 2 - 52 mat 1.5 - 2 - 0.5 - 2 - 0.5 2 - 1.5 2 - 42 mat 1.5 2 - 1.5 1 - 1.5 0 1.5 1 1.5 252 mat 1.5 2 0.5 2 0.5 - 2 1.5 - 242 mat 14 mat

Otherwise user rectangle M 08 Matrix 1912 mat 11012 mat 11112 mat 11212 mat 11312 mat 2912 mat 21012 mat 21112 mat 21212 mat 21312 mat 3912 mat 31012 mat 31112 mat 31212 mat 31312 mat 4912 mat 41012 mat 41112 mat 41212 mat 41312 mat 45 mat

That's your rectangle above M 01 Matrix 1212 mat 1312 mat 1412 mat 1512 mat 1612 mat 2212 mat 2312 mat 2412 mat 2512 mat 2612 mat 3212 mat 3312 mat 3412 mat 3512 mat 3612 mat 4212 mat 4312 mat 4412 mat 4512 mat 4612 mat 45 mat

Advanced convenience . . .

]]> Utilities

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 :

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 :

data spec pts i 1 data rows range p i el a b c spec : for v p 1 el : Unest 'p' j 2 p rows range v v p j el stack : for data v augment 51 line :

Vect(sizechar,n)... Vect(colorchar,n) size n Vect n 1 ≡ size ct 1 : j 1 n range a ct 1 el size : ct ct 1 + : 21 line for v a 1 el : k 2 a rows range v v a k el stack : for v 51 line if :

Γ(x) [15 D] Alvaro Same otherwise

Γ 0.99999999999999709182 57.156235665862923517 59.597960355475491248 - 14.136097974741747174 0.49191381609762019978 - 0.33994649984811888699 104 -^* 0.46523628927048575665 104 -^* 0.98374475304879564677 104 -^* - 0.15808870322491248884 103 -^* 0.21026444172410488319 103 -^* - 0.21743961811521264320 103 -^* 0.16431810653676389022 103 -^* - 0.84418223983852743293 104 -^* 0.26190838401581408670 104 -^* - 0.36899182659531622704 105 -^* 151 mat :

ζ Γ z ζ 1 - : zh z 0.5 + : zg zh 607128 / + : zp zg 0.5 zh *^: 14 mat 2 π * sqrt Γ 1 el Γ κ 1 + el z κ + / κ 114 sum + (* zp 2^* zg - exp * eval 21 line :

Valid for Re(ζ)>0

User 'command line'

4 appVersion 0.98.6179.21440 ≡

Observe

constrain to validate PDF hypergeometric x Γ x Γ 0 x ≤ (* :

scalar 3 param
Hypergeometric N n p x hypergeom N 1 + Γ n 1 + Γ * N n + p - 1 + Γ * p 1 + Γ * x 1 + Γ x - n + 1 + Γ * N n + 1 + Γ * x N + p - 1 + Γ * x - p + 1 + Γ * / eval :

3 parameters function N 80 : n 10 : p 5 : 13 mat

Continuous PDF Hypergeometric x hypergeom N n p x hypergeom :

x f N n p x hypergeom :

ESSENTIALLY ... this document is a 'command line'. 1. display single user point value. 2. user vector for export. Both styles wrt to 'n' rounded. It associates more graphic displays. Rounded displays can't be preserved. NodesLabel ... added convenience.

User command line ▼

set drop down menu 15 D ... NO trailing '0'
it recasts the system for command line x n result x f n round eval :

user 1.5 :

command line result user 7 result 0.1939456

vector x 051 range : i 1 x rows range y i el x i el 6 result : for x y augment 31 line :

vector 0 0.546995 1 0.359865 2 0.084124 3 0.008628 4 0.000382 56106 -^* 62 mat

0.125 - f 0 f 0.125 f 31 mat 0.51781289492909 0.546994684871668 0.560540393057634 31 mat

x f x hypergeom :

Golden-Ratio search [min/max]

TOL 1012 -^:

Golden ratio

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 71 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ 21 sys

X f 0 0.25 y1 y2 ≥ TOL Extrema 1 el :

X 0.1772

X f 0.562

Reconciliate/local for publishing

'data' points specs vx vector 1 col : vy vector 2 col : 12 mat data vx vy o 7 black plotG : 21 sys

nodes vx : NodesLabel nodes var2str vectorize : X vx : Y vy : 12 mat m X rows : 41 line

NodesLabel convenience

size

plot symbol converted2Numbers

symbol color

Labels specs nodes_label X Y NodesLabel 10 m Vect red m Vect augment :

Discrete mode

user 1.5 :

Hypergeom x 05 0.025 range : i 1 x rows range y i el x i el hypergeom eval : for x y augment 31 line :

XY user user 3 result 12 mat : 1.5 0.194 12 mat

track 10 0.5 2 0.5 user user 3 result 32 mat :

x y axes x y / - :

plot Hypergeom data nodes_label user user 3 result + 30 red 15 mat track 71 sys :

plot

XY 1.5 0.194 12 mat

Tools/Options
...  D ≥ 'n' vector 0 0.546995 1 0.359865 2 0.084124 3 0.008628 4 0.000382 56106 -^* 62 mat

Hypergeom track XY axes 4 1 sys

XY 1.5 0.194 12 mat

r c matrix i 1 r range j 1 c range M i j el 0 : for for M 21 line :

blank matrix mat 77 matrix (: 000000000000000000000000000000000000000000000000077 mat

X x cos 2 y * 1 z / 32 y^2 ln * 3 z 2^* - 11133 mat :

Smath index style A B r c replace U A : i 1 B rows range j 1 B cols range U r i + 1 - (c j + 1 - (el B i j el : for for U 31 line :

mat X 32 replace 000000000000000 x cos 2 y * 1 z / 000032 y^2 ln * 3 z 2^* - 00001110000000000000000077 mat

data Explode data UnestRow v data 1 el : j 2 data rows range v v data j el stack : for v 31 line : j 1 data cols range myU 1 j el data j col UnestRow : for v myU 1 el : j 2 myU cols range v v myU j el augment : for v 41 line 21 line :

M 0 : M 57 el 0 : 21 line

M 0000000000000000000000000000000000057 mat

data 32.1014 11 mat 1.9591 11 mat 0.2438 11 mat 25.7124 11 mat 1.9591 11 mat 0.1254 11 mat 0.0155 11 mat 1.5539 11 mat 0.2438 11 mat 0.0155 11 mat 0.0077 11 mat 0.1927 11 mat 25.7124 11 mat 1.5539 11 mat 0.1927 11 mat 22.4646 11 mat 44 mat :

data Explode 32.101 1.959 0.244 25.712 1.959 0.125 0.016 1.554 0.244 0.016 0.008 0.193 25.712 1.554 0.193 22.465 44 mat

Insert col/row on key condition

M 11212 4.5 42321144245 7.2 6454 mat :

M B c InsertCol a M 1 M rows 1 c submatrix : c M 1 M rows c 1 + M cols submatrix : a B c augment 31 line :

M B r InsertRow a M 1 r 1 M cols submatrix : c M r 1 + M rows 1 M cols submatrix : a B c stack 31 line :

col 51 mat :

row 14 mat :

M key FindIndex r M rows : c M cols : 12 mat i 1 : j 1 : 12 mat j c ≤ (M i j el key ≠ (* i r < i i 1 + : i 1 : j j 1 + : 21 line if while j c ≤ i j 21 mat 0 if 41 line :

key 3 :

row
col idx M key FindIndex (: 3121 mat

N M row idx 1 el InsertRow : U M col idx 2 el InsertCol : 21 sys

N 11212 4.5 423211 44245 7.2 6464 mat

U 1 1212 4.5 423 2114 4245 7.2 6455 mat

Border/Frame a matrix

associae with replace(Frame,M,#,#) f 2 :

Mat Frame i 1 Mat rows f + (range j 1 Mat cols f + (range frame i j el : for for frame 21 line :

A B r c replace U A : i 1 B rows range j 1 B cols range U r i + c j + el B i j el : for for U 31 line :

M 05204580125520458012518020458012518024545801251802453208012518024532040512518024532040550066 mat :

Frame M Frame :

FRAME Frame M 11 replace :

FRAME 05204580125 5204580125180 204580125180245 4580125180245320 80125180245320405 125180245320405500 88 mat

Delete 'ONE specified' col/row Square/Rectangular matrix

M 05204580125520458012518020458012518024545801251802453208012518024532040512518024532040550066 mat :

M c DelCol N 0 : ct 1 : n 1 M cols range n c ≠ N 1 ct el M n col : ct ct 1 + : 21 line Unest 'N' if for v N 1 el : j 2 N cols range v v N j el augment : for v 61 line :

M 3 DelCol 05458012552080125180204512518024545801802453208012524532040512518032040550065 mat

M c DelRow N 0 : ct 1 : n 1 M rows range n c ≠ N ct 1 el M n row : ct ct 1 + : 21 line Unest 'N' if for v N 1 el : j 2 N rows range v v N j el stack : for v 61 line :

M 3 DelRow 05204580125520458012518045801251802453208012518024532040512518024532040550056 mat

N 05204580125520458012518045801251802453208012518024532040512518024532040550056 mat :

N 3 DelCol 0545801255208012518045801802453208012524532040512518032040550055 mat

Minor Square Matrix

M 12345678933 mat :

minor from f(x) drop down list
delete row 2 ... col 3 M 22 vminor 137922 mat

det("matrix") 137922 mat det 12 -

palette ... determinant of minor M 22 minor 12 -

Populate selected discontinuous vector ... 'sub' has zoom effect in the X ► direction.

Sub-spces System matrix

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 :

data sub subspace j 1 data rows 1 - (range Vsub j el i 1 sub range vsub i el data j el i sub / data 1 j + el data j el - (* + : for vsub 21 line eval : 11 line for v Vsub 1 el : j 2 Vsub rows range v v Vsub j el stack : for data 1 el v stack 31 line 21 line :

matrix sub populate M 0 : j sub P matrix j col sub subspace : j 1 matrix cols range M 1 j el j sub P : for M 41 line :

data n Decimate collect every 'n' pairs range 1 data rows n range : k 1 : i range u k el data i 1 el : v k el data i 2 el : k k 1 + : 31 line for u v augment 31 line 31 line :

select column col 2 :

◄ fragment sub 4 :

U 04613721831104195916101781873964103 mat :

Mysub U sub populate eval : vy Mysub col col 1 el : (idx 1 explode ≡ i 1 vy rows range : SUB i vy augment : plot SUB . 12 red plot : x f i vy x linterp : Origin SUB sub 1 + Decimate : O Origin o 9 black plot : 81 sys

x f plot O 3 1 sys

Int x f x 1 vy rows int :

Int 180

Int vy rows 1 - / 5.0000

Subdivide 'U' ... Export

Mysub 11 mat 1 el 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 4.25 4.5 4.75 5 5.25 5.5 5.75 6 6.25 6.5 6.75 7 7.25 7.5 7.75 8 8.25 8.5 8.75 9371 mat 4 3.75 3.5 3.25 3 2.5 2 1.5 1111111113579 9.25 9.5 9.75 10 9.5 9 8.5 8 7.75 7.5 7.25 7 6.75 6.5 6.25 6371 mat 6 6.25 6.5 6.75 7 7.25 7.5 7.75 8 8.5 9 9.5 10 9.75 9.5 9.25 9753111111111 1.5 2 2.5 3 3.25 3.5 3.75 4371 mat 13 mat

Otherwise unary system ...

Unary system 
convenience U SUB 1 col : λ U Max U Min - : iX U 11 mat λ / U 11 mat λ / Min - : unary iX 1 el SUB 2 col augment : Unary unary . 10 black plot : 51 sys

Unary unary 2 1 sys

vx unary 1 col (: vy unary 2 col : 12 mat x Φ vx vy x linterp : 21 sys

x Φ x 01 int 5.0000

Discrete circle/ellipse

L Δ N inc X 0 : i 1 N range X i 1 + el L Δ i * + : X 1 el L : 21 line for X 31 line :

n 99 : Δ 02 π * n / n inc : 12 mat x u i Clear X 1 : Y 0 : 12 mat u Cir i 2 * π * u * exp : x Δ : svg plot svg x Cir Im vectorize X + (* svg x Cir Re vectorize Y + (* augment : 61 sys

mesh fine enough, odd

local command  X or Y svg 0.5 X * :

0.5 svg plot * 1 svg plot * 2 1 sys

x u i Clear u Cir i 2 * π * u * exp : x Δ : svg xc yc C svg x Cir Im vectorize xc + (* 0.5 x Cir Re vectorize yc + (* augment : 41 sys

plot arc at will from x:= set oblate/prolate ellipse

[xc, yc] wrt unary circle svg 1 : xc 1 - : yc 1 : 13 mat

x y axes x y / - :

svg xc yc C 0.5 svg xc yc C * 2 1 sys svg xc yc C axes 2 1 sys

r x circle r 2^x 2^- sqrt ± :

local command r 1.5 :

r x circle

Ravel Times Table Matrix

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 :

M 0 :

V 0 :

Times Table n T i 1 n range j 1 n range M i j el i j * : for for M 21 line :

M ravel cnt 1 : i 1 M rows range j 1 M cols range Vr cnt el M i j el : cnt cnt 1 + : 21 line for for Vr 31 line :

n 6 :

V n T ravel :

n T 123456246810123691215184812162024510152025306121824303666 mat

j 1 n T ravel rows range : Ravel j n T ravel augment : R Ravel o 5 black plot : 31 sys

113 / + (1 -^0.75

Ravel x R 1 n / x * 1 1 3 / + ( 1 - ^ + 4 1 sys

V 11 mat 1234562468101236912151848121620245101520253061218243036361 mat 11 mat

Ravel 1122334455667284961081110112 mat

β 1.25 :

Times Table n β T i 1 n range j 1 n range M i j el i β * (j * : for for M 21 line :

j 1 n β T ravel rows range : Ravel j n β T ravel augment : R Ravel o 5 black plot : 31 sys

Ravel β x * R β 1 n / x * 1 1 3 / + ( 1 - ^ + ( * 4 1 sys

Diagonal/Mirror Matrix

M 0 :

M 0.19 0.59 0.35 0.82 0.71 0.61 0.27 0.59 0.12 0.03 2.66 4.21 0.45 0.23 5.48 5.20 44 mat :

Vectorise Diagonal M Dia Vector elements i 1 M rows range Dia i el M i i el : for Dia 31 line :

Smath  built-in M Diag 0.19 0000 0.61 0000 2.66 0000 5.2 44 mat

M Dia 0.19 0.61 2.66 5.2 41 mat

Smath 6179 M mirror M 1 M rows range M cols 1 range el :

M mirror Dia 0.82 0.27 0.03 0.45 41 mat

M mirror 0.82 0.35 0.59 0.19 0.59 0.27 0.61 0.71 4.21 2.66 0.03 0.12 5.2 5.48 0.23 0.45 44 mat

M NestCol v 0 : a 0 : 12 mat v M 1 col : ct 1 : j 1 M cols range a 1 ct el M j col : ct ct 1 + : 21 line for a 51 line :

M NestRow v 0 : a 0 : 12 mat v M 1 row : ct 1 : j 1 M rows range a ct 1 el M j row : ct ct 1 + : 21 line for a 51 line :

M NestCol 0.19 0.71 0.12 0.45 41 mat 0.59 0.61 0.03 0.23 41 mat 0.35 0.27 2.66 5.48 41 mat 0.82 0.59 4.21 5.2 41 mat 14 mat

M NestRow 0.19 0.59 0.35 0.82 14 mat 0.71 0.61 0.27 0.59 14 mat 0.12 0.03 2.66 4.21 14 mat 0.45 0.23 5.48 5.2 14 mat 41 mat

Matrix/Vector Convolution

K 1.123456 2.2 3.3 4.4 00000 1.123456 2.2 3.3 4.4 00000 1.123456 2.2 3.3 4.4 00000 1.123456 2.2 3.3 4.4 048 mat :

x 0.1 0.2 0.3 0.4 41 mat :

t0 1 time :

Convolve matrix over vector i 1 x transpose K * (transpose rows range : Convol i x transpose K * (transpose augment : XY Convol : 31 line

i 1 Convol rows range pt i el . : for

data XY 1 col XY 2 col pt augment :

data XY 2 1 sys

Numeric, Decimal Convol 1 0.1123 2 0.4447 3 1.107 4 2.2094 5 2.75 6 2.64 7 1.76 8082 mat

vx XY 1 col : vy XY 2 col (: 12 mat x cnv vx vy x ainterp : x λ 10 x ≤ (x 8 ≤ (& cases : 31 sys

data x cnv x λ * 2 1 sys

1 time t0 - 0.26 s *

plot(data,char,size,clr)

Universal version  [Smath 6179]. It supports "data" sets from 
source ... data sets subject to be decimated from auxilliary
module Decimate(data,n)... reasonably fast on primitive PC. 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 :

M mirror M transpose reverse transpose :

Vectorise Diagonal M Dia Diagonal elements i 1 M rows range Dia i el M i i el : for Dia 31 line :

n T n i 1 n range : n i n i transpose * 21 line :

n 32 :

vx n T Dia : vy n T mirror Dia : idx 1 n range : m vx vy augment : Arc idx vy augment : 51 line

M Lo Hi Scale Hi Lo - (M M Min - M Max M Min - 1012 -^+ / (* Lo + :

image n T 0255 Scale :

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 image

data m . 12 red plot :

arc Arc . 12 red plot :

m data 2 1 sys idx vy augment arc 1 idx vx augment ( * 3 1 sys

Pad/Center a Matrix

floccinaucinihilipilification The longest English idiotic word !

x y φ x y * :

Ω 812 φ matrix maple :

Ω 12345678910111224681012141618202224369121518212427303336481216202428323640444851015202530354045505560612182430364248546066727142128354249566370778481624324048566472808896812 mat

Explain ::

This padding is the basic padding to zoom a grayscale image. RGB requires each R,G,B similar padding. This demo is for a matrix. If the matrix is a grayscale, in 'A' we plug its Fourier 3D "cfft(M)" and recover the zoomed image "scale[Re(icfft(S))]. Note the symmetry for even row/col in Ω .

M nc nr Pad A M : insert 'nc' cols in 'A' i 1 A rows range j 1 nc range vert i j el 0 : for for a A 1 A rows 1 A cols 2 / Floor submatrix : b A 1 A rows A cols 2 / Floor 1 + A cols submatrix : S a vert b augment transpose : insert 'nr' rows in the transpose 'S' i 1 S rows range j 1 nr range horiz i j el 0 : for for a S 1 S rows 1 S cols 2 / Floor submatrix : b S 1 S rows S cols 2 / Floor 1 + S cols submatrix : S a horiz b augment transpose : 111 line :

Numeric

Here, we padd
3 rows/cols 
centered in Ω P Ω 33 Pad :

P 123456000789101112246810120001416182022243691215180002124273033364812162024000283236404448000000000000000000000000000000000000000000000510152025300003540455055606121824303600042485460667271421283542000495663707784816243240480005664728088961115 mat

Specific pivot

Symmetric/asymmetric "pivot" pivot n m S U 0 : V 0 : 12 mat i 0 m range U i el i : for j 1 n range V j 1 + el j - : for V reverse U stack pivot + 41 line :

pivot 6 : n 10 : m 5 : 13 mat

local assign jdx pivot n m S 11 line :

j 1 jdx rows (range :

jdx j el sqrt vectorize 2 i * 1.7321 i * 1.4142 i * i 01 1.4142 1.7321 2 2.2361 2.4495 2.6458 2.8284 3 3.1623 3.3166 161 mat

jdx 4 - 3 - 2 - 1 - 01234567891011161 mat

The Delta Matrix

JPG Δelta matrix upper triangle n s Δ Δ 0 : i 1 n range j 1 n range Δ i j el i j + s 1 + ≤ : for for Δ 31 line :

's' sets upper/lower around 'n' n 32 : s 32 : 12 mat

Δ1 255 n s Δ (* : Δ 235 n s Δ (* : flip Δ reverse : mirror Δ 1 Δ rows range Δ cols 1 range el : mydiagonal 255 n identity * : 51 sys

Vectorise Diagonal M diag Vector elements i 1 M rows range Dia i el M i i el : for Dia 31 line :

iVBORw0KGgoAAAANSUhEUgAAACAAAAAgCAIAAAD8GO2jAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAClSURBVEhLtc1bFoIwDEDB7n+rbsKqgweFQh/JfHGa5FIeyXJ/UCqfCV71vB+oVx5CSX94i6P75TmI6J5JBMU/hsvkjszXaJ2yskCoxdYslQsWp0hcszvO/S3rgxz3cDHCZSdH3Zz1c9fHzRCnHRyMcn3H9gSBS1bnaLTZmybTYGmF0hkbi8QOjNfp/TILIbljEEV14zWQ8JunWNpJ9Sq3XuXWS3kCn1iuXMONMHwAAAAASUVORK5CYII= Δ iVBORw0KGgoAAAANSUhEUgAAACAAAAAgCAIAAAD8GO2jAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAACoSURBVEhLtc1REoMgDAVA739VLlGmbju2KgYI+wfJe9lKKdtS9cDaG/uByjud+jdfuXR/+E2k+MAgi9ZfZilUnhjP03fFxiRlNyzN0HTP3jA1TVbH6Hhie4CCAIFe0jEyXUTDxOLkekgGCXUSjpDoJ//I+hAVbXZHaWmwOEHRHVtzdF2yMk3dmXkGjX8Mkyg9Msmj98t3KtU7f9m0Vz4WWH6gWn6glPICImauXD1/eloAAAAASUVORK5CYII= flip iVBORw0KGgoAAAANSUhEUgAAACAAAAAgCAIAAAD8GO2jAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAACnSURBVEhLtc1REsIgDAXA3v+qXELU1am2tEDCfjEk72Uri22V5xrPA0tvOFD5yKb9zV8q1V++8+jdM0mi9I9hBo1H5mHqTlmJ0dViK0DRBYuztFyzO0XFLevj5HtIDBLuJDRCsp9cN7Ehon1kRkl3EJig4I7tOTouWZ2mps1ehKYGS0HKztiI03dgnELlL7MsWncMEin+8JtL94uvdOrXHaiWH6hKKQ+BGK5cOkr3DgAAAABJRU5ErkJggg== mirror iVBORw0KGgoAAAANSUhEUgAAACAAAAAgCAIAAAD8GO2jAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAAnSURBVEhL7c0xAQAwDAIw/JtmFnh6LTGQtM01x86xc+wcOwcAX0sebNkv0edlZW4AAAAASUVORK5CYII= Δ1 Diag

bck 235 : m bck 2 / Floor 32 identity * bck + : 21 sys

iVBORw0KGgoAAAANSUhEUgAAACAAAAAgCAIAAAD8GO2jAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAAtSURBVEhL7c0xAQAADIOw+jfNLPDsIwYyYN86vA6vw+vwOrwOr8Pr8Dq8Dg846A1focwoWa8AAAAASUVORK5CYII= mydiagonal

mirror from native SS Δ mir Δ transpose reverse transpose :

iVBORw0KGgoAAAANSUhEUgAAACAAAAAgCAIAAAD8GO2jAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAACZSURBVEhLtc1bDoMwEMVQljr7ZxN1BaItDck87ljy99nMbO9s41bjDVCfcQLUZHwA6jB+AJIbd4C0xgAgoTEGSGU8AiQxZgDVjQVARWMNUMVwAZQ2vADljABACSMGUNQIAxQyMgD5jSRATiMPkMcoAbQ0qgDNDQFAE0MD0JMhA2hoKAH6N8QA3Qw9QN9GC0CX0QXQYTQCZGYvIZDQd/7d3K8AAAAASUVORK5CYII= m iVBORw0KGgoAAAANSUhEUgAAACAAAAAgCAYAAABzenr0AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAC1SURBVFhHxdfJEcNQCARRpGgdgkJQuMrAdtsqlRctf4GhDxQ33pXhZnafLa+R49N7T2lkZCJeAMpCbADKQHwBSI34A5ASsQsgFeIQQArEKYCiEZcAikQUASgKUQygCEQVgLwR1QDyRDQByAvRDCAPRBeAehHdAOpBuACoFeEGoBaEK4BqEe4AqkGEAKgUEQagEkQogK4Q4QA6Q0gAdISQAWgPIQXQL0IOoE/EsDzf83WXN5nZA43CIpVFnTj2AAAAAElFTkSuQmCC

NFG !

1.025 m * 0 m * 0 m * 1 3 mat

m 0 :

n 0 :

s 0 :

◄ clear after use

replicate(rep,v) applications

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Core component of the stem style rep v replicate k 0 : i 1 v length range rep i el 0 ≡ continue j 1 rep i el range T k j + el v i el : for k k rep i el + : 21 line if for T 31 line :

v 1.2 2.2 3.3 31 mat :

n_rep 12331 mat :

n_rep v replicate 1.2 2.2 2.2 3.3 3.3 3.3 61 mat

P π 2 sqrt 2 ln 13 mat transpose : v 21213 mat transpose : W v P replicate : 31 sys

W transpose 3.142 3.142 1.414 0.693 0.693 15 mat

v 20213 mat transpose : W v P replicate : 21 sys

Random stem style

W transpose 3.142 3.142 0.693 0.693 14 mat

Stem Construction Utility

Core component of the stem style P# v# replicate k 1 : i 1 v# length range P# i el 1 ≡ continue j 1 P# i el range T k j + el v# i el : for k k P# i el + : 21 line if for T 31 line :

Stem construction vy# stem S 0 : N vy# length : i 1 N range q i el 3 : q N el 2 : 21 line for 11 line i 1 N range v i el i : for i 1 N range S 3 i * el vy# i el : for u q v replicate : V u S augment : V V 3 V rows 12 submatrix : 81 line :

Generate the random N rnd Uniform ± 1 idx 1 N range : R 2 N Random * 1 - : 31 line :

Collect the project N 100 : U N rnd : Random N rnd stem : 31 line

OBSERVE ... 1. canvas style [blue] el(U,x) 2. stem style [red] 3. each plot is independent

U x el 0 2 1 sys 0 Random 2 1 sys

replicate/delete command line rep v replicate k 0 : i 1 v rows range rep i el 0 ≡ continue j 1 rep i el range T k j + el v i 1 el v i 2 el augment : for k k rep i el + : 21 line if for v# T 1 el : j 2 T rows range v# v# T j el stack : for v# 31 line 31 line :

v 1234567891052 mat :

n_rep 1201051 mat :

rep n_rep v replicate :

rep 1234347842 mat

Two objectives ... 1. multiple stack. 2. mutiple delete.

Matrix border quadrants '0' ... border composite '0'

M 0 :

N 0 :

M 05204580125152045801251802204580125180245345801251802453204801251802453204055125180245320405500667 mat :

i M rows 1 + : j M cols 1 + : N i j el 0 : 31 line

A B r c replace U A : i 1 B rows range j 1 B cols range U r i + c j + el B i j el : for for U 31 line :

style_1 N M 01 replace : style_2 N M 11 replace : style_3 N M 10 replace : style_4 N M 00 replace : style_5 style_2 style_4 11 replace : style_6 N M 22 replace : and so on 71 line

style_1 0052045801251052045801251802020458012518024530458012518024532040801251802453204055012518024532040550060000000078 mat

style_2 0000000000520458012510520458012518020204580125180245304580125180245320408012518024532040550125180245320405500678 mat

style_3 0000000005204580125105204580125180202045801251802453045801251802453204080125180245320405501251802453204055006078 mat

style_4 0520458012510520458012518020204580125180245304580125180245320408012518024532040550125180245320405500600000000078 mat

style_5 00000000000520458012510052045801251802002045801251802453004580125180245320400801251802453204055001251802453204055006000000000089 mat

style_6 00000000000000000000052045801251005204580125180200204580125180245300458012518024532040080125180245320405500125180245320405500689 mat

Setup/fill critical region

b1 3.4445 - : b2 1.5 : β 4 - : 13 mat x spline1 β x * 2 β * b1 + (x 2^* - β b1 + (x 3^* + : x spline2 β x * 2 β * b2 + (x 2^* - β b2 + (x 3^* + : 31 sys

x λ 10 x ≤ (x 1 ≤ (& cases :

x arc1 1 x spline1 x diff (2^+ sqrt : x arc2 1 x spline2 x diff (2^+ sqrt : 21 sys

x spline1 x λ * x spline2 x λ * 2 1 sys

dorso x arc1 x λ * x 01 int : 1.82603

torso x arc2 x λ * x 01 int : 1.82604

1. Prepare the project

2. Points specification

N 1000 : (Density of random ≡

data spec pts i 1 data rows range p i el a b c spec : for Unest 'p' v p 1 el : j 2 p rows range v v p j el stack : for data v augment 51 line :

xmax 1 : xmin 0 : 21 line

ymax 0.5 : ymin 1 - : 21 line

space_x xmax xmin - (: space_y ymax ymin - (: 21 line

x y1 x spline1 : x y2 x spline2 : 21 line

3. Fill critical region

A function that tests an arbitrary point 
for its membership in the critical region. x y Test y x y2 - 0 ≥ (y x y1 - 0 ≥ (- :

#A0A0A0

t0 1 time :

A routine that finds an array of N random 
pairs in the critical region. Pts the routine is a compressor it maintains density 'N' within the variable region i 1 N range x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 1 x y Test - x 1 Random space_x * xmin + 1 el : y 1 Random space_y * ymin + 1 el : 21 line while P i 1 el x : P i 2 el y : 51 line for P 51 line :

a b c spec . 6 #A0A0A0 13 mat :

Pts Pts :

pts Pts a b c spec pts :

1 time t0 - 36 s *

x y1 0 ≡ x 0.5 0.6 solve 0.53731

Elastic Dolphin [Jean 4 ► Sophie Germain]

x y1 x λ * x y2 x λ * pts 0.875 0.125 . 15 black 1 5 mat 4 1 sys

dorso torso 21 mat 1.826 1.826 21 mat

Golden-Ratio search [min/max]

TOL 1012 -^:

Golden ratio

Code adapted from Valery Ochov Mathcad 11 f a b test TOL Extrema a b G 32 / b * 12 / a * - 12 / 5 sqrt * a b - (* + : y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ x1 a b G eval : y1 x1 f eval : x2 b a G eval : y2 x2 f eval : 41 line iter 0 : iter 1000 < a b - abs TOL ≤ break test b x2 : x2 x1 : y2 y1 : x1 a b G eval : y1 x1 f eval : 51 line a x1 : x1 x2 : y1 y2 : x2 b a G eval : y2 x2 f eval : 51 line if iter iter 1 + : 21 line if while a b + 2 / iter 21 mat 71 line :

1. All modules/functions in "Symbolic" optimiz. 2. Once in target: shrink the range 3. respect the search y1≥y2≡"max in range" y1≤y2≡"min in range"

x f x y1 :

y1 y2 ≥ (max in range ≡ y1 y2 ≤ (min in range ≡ 21 sys

X1 f 0.75 0.825 y1 y2 ≥ TOL Extrema 1 el f 0.125 0.375 y1 y2 ≤ (TOL Extrema 1 el 21 mat :

x f x y2 :

X2 f 0.25 0.5 y1 y2 ≤ (TOL Extrema 1 el :

X1 X2 21 mat 0.8014 0.2235 21 mat 0.4000 21 mat

Bolean algo style

f x hi lo Clear 1

else " is the NO plot code = Mathcad 11 f bol else If f bol else cases :

f 0.5 x ln * :

algo style x f f lo x ≤ (x hi ≤ (& If :

local command line lo 0.125 : hi 8 : 12 mat

x f

f 0.5 x sqrt * :

algo style x f f lo x ≤ (x hi ≤ (& If :

f 5 - x ≤ ( x 0 ≤ ( & If Im f 0 x ≤ ( x 5 ≤ ( & If 2 1 sys

Intersection of vectors ... ref ► Smath Forum

Intersection System

These functions have been written to implement in Smath some features that are natively provided by either Scilab or Matlab in order to make available in Smath a function originally written in Matlab to find intersections between lines. The function in subject is P = InterX(L1, L2), written by NS. See below link: https://de.mathworks.com/matlabcentral/fileexchange/22441-curve-intersections

Function mdiff(V) Input: a vector (either a row or a column) Output: a vector of the same nature of input (a row or a column) containing the differences V(i+1)-V(i). Number of output elements are lenght(V)-1.

Function mdiff(V)
Input: a vector (either a row or a column)
Output: a vector of the same nature of input (a row or a column)
                  containing the differences V(i+1)-V(i).
                   Number of output elements are lenght(V). V mdiff R V rows V rows 1 > (- V cols V cols 1 > (- matrix : ii 1 R length range R ii el V ii 1 + el V ii el - : for R 31 line :

Examples:

z2 4273151 mat :

z2 mdiff 2 - 54 - 2 - 41 mat

z1 13589416 mat :

z1 mdiff 22315 - 15 mat

Function index(x; cond) Input: x: a vector (row or column). cond: expression evaluating to an array of T/F values. Output: a vector of same nature of input array whose components are those corresponding to a TRUE value for the given condition.

Function index(x; cond)
Input: x: a vector (row or column). cond: expression evaluating to an array of T/F values.
Output: a vector of same nature of input array whose components are those corresponding
                to a TRUE value for the given condition. V cond index R V rows V cols matrix : j 0 : V.end V length : 12 mat i 1 V.end range cond vectorize i el j j 1 + : R j el V i el : 21 line continue if for R rows 1 > (R 1 j 11 submatrix R 111 j submatrix if 41 line :

Examples:

z1 z1 4 > index 58913 mat

z2 z1 mdiff 0 > index 427341 mat

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 :

Function map(x; y; fun(2)) Input: x, y: vectors or matrices with compatible sizes. fun: a function of 2 argumens Output: vector or matrix whose elements are fun(el(x);el(y)) Apply a given function to corresponding elements of compatible arrays. For info on the Matlab concept of "arrays of compatible sizes" please refer to: https://it.mathworks.com/help/matlab/matlab_prog/compatible-array-sizes-for-basic-operations.html

a b plus a b + :

a b times a b * :

a b minus a b - :

a b divide a b / :

x y 2 fun map xh x cols : xv x rows : yh y cols : yv y rows : 22 mat okx xh yh ≡ (xh 1 ≡ (yh 1 ≡ (| (| (: oky xv yv ≡ (xv 1 ≡ (yv 1 ≡ (| (| (: okx oky & xx x : yy y : xh yh < ii 2 yh range xx xx x augment : for xh yh > ii 2 xh range yy yy y augment : for 1 if if xv yv < ii 2 yv range xx xx x stack : for xv yv > ii 2 xv range yy yy y stack : for 1 if if hv xv yv Max : hx xh yh Max : 12 mat ii 1 hv range jj 1 hx range zz ii jj el xx ii jj el yy ii jj el fun : for for zz 71 line Incompatible sizes error if 41 line :

z1 z2 A B times map 4122032361626101618872135566328391524271213589456 mat

Examples:

z1 z1 A B plus map 26101618816 mat

z2 z2 A B plus map 84146251 mat

Function D(x; y) Helper function for InterX. See below.

x y D x 1 x rows 1 x cols 1 - submatrix y A B minus map x 1 x rows 2 x cols submatrix y A B minus map A B times map :

Function InterX(L1; L2) Input: L1; L2 : 2-rows matrices of coords (Row1 = x-coord; Row2 = y-coord) Output: 2-rows matricx of coords of intersecting points

L1 L2 InterX x1 L1 1 row transpose : x2 L2 1 row : y1 L1 2 row transpose : y2 L2 2 row : dx1 x1 mdiff eval : dy1 y1 mdiff eval : dx2 x2 mdiff eval : dy2 y2 mdiff eval : ii 1 dx1 length range S1 ii el dx1 ii el y1 ii el * dy1 ii el x1 ii el * - : for ii 1 dx2 length range S2 ii el dx2 ii el y2 ii el * dy2 ii el x2 ii el * - : for S2 S2 transpose eval : C1 dx1 y2 A B times map dy1 x2 A B times map - S1 D 0 mwle eval : C2 y1 dx2 A B times map x1 dy2 A B times map - (transpose S2 transpose D 0 mwle transpose eval : ff C1 C2 mwand mwfind eval : ii ff 1 el eval : jj ff 2 el eval : ii length 0 ≡ 21 Zeros ii ii transpose eval : dx2 dx2 transpose eval : dy2 dy2 transpose eval : S2 S2 transpose eval : L dy2 jj el dx1 ii el A B times map dy1 ii el dx2 jj el A B times map - eval : ii ii L 0 ≠ index eval : jj jj L 0 ≠ index eval : L L L 0 ≠ index eval : PP1 dx2 jj el S1 ii el A B times map dx1 ii el S2 jj el A B times map - eval : PP2 dy2 jj el S1 ii el A B times map dy1 ii el S2 jj el A B times map - eval : PP PP1 PP2 augment L L augment A B divide map eval : PP 121 line if 171 line :

D1, D2

D1 4 - 1.593 3.027 - 1.473 2.338 - 1.34 1.878 - 1.204 1.581 - 1.076 1.393 - 0.958 1.277 - 0.846 1.207 - 0.74 1.168 - 0.637 1.147 - 0.536 1.137 - 0.435 1.133 - 0.336 1.131 - 0.236 1.131 - 0.137 1.131 - 0.037 1.131 - 0.062 - 1.131 - 0.162 - 1.131 - 0.262 - 1.129 - 0.361 - 1.124 - 0.46 - 1.112 - 0.559 - 1.088 - 0.657 - 1.044 - 0.752 - 0.973 - 0.843 - 0.864 - 0.928 - 0.712 - 1.002 - 0.512 - 1.061 - 0.271 - 1.1 - 0.006 - 1.114 - 0.259 1.101 - 0.501 1.064 - 0.703 1.005 - 0.858 0.932 - 0.969 0.847 - 1.042 0.756 - 1.086 0.661 - 1.111 0.564 - 1.124 0.465 - 1.129 0.366 - 1.131 0.266 - 1.131 0.167 - 1.131 0.067 - 1.131 0.032 1.131 0.132 1.131 0.231 1.133 0.331 1.136 0.431 1.146 0.531 1.166 0.632 1.205 0.735 1.273 0.841 1.386 0.952 1.57 1.071 1.861 1.198 2.312 1.333 2.989 1.467 3.947 1.588 5.241 1.715 7.171 1.951 10.869 2.327 19.032 2.924 612 mat :

D2 0.1 2.175 - 0.507 2.103 - 0.848 1.929 - 1.089 1.66 - 1.228 1.317 - 1.288 0.932 - 1.306 0.529 - 1.308 0.122 - 1.307 0.284 1.302 0.69 1.271 1.088 1.184 1.461 1.006 1.778 0.722 2.01 0.35 2.143 0.069 - 2.177 0.478 - 2.111 0.826 - 1.945 1.075 - 1.683 1.221 - 1.345 1.286 - 0.961 1.305 - 0.559 1.308 - 0.153 1.307 - 0.254 - 1.303 - 0.66 - 1.275 - 1.059 - 1.193 - 1.435 - 1.023 - 1.757 - 0.747 - 1.996 - 0.38 - 2.137 - 0.037 2.177 - 312 mat :

Strip paired 0's

data Strip2 data 0 : u data 1 col : v data 2 col : k 0 : i 1 data rows range u i el v i el * 0 ≠ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

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 :

t0 1 time :

D1 rows D2 rows 21 mat 613121 mat

basket D1 transpose D2 transpose InterX :

1 time t0 - min 2

basket 1.289 0.8567 1.2908 - 0.8593 22 mat

BK basket o 7 black plot :

The data set must be clean of multiple intersecting junctions. Vectors need NOT be of same length.

D1 D2 BK 3 1 sys

Let's test with an off hand example ▼

t0 1 time :

L1 123456890112203 0.5 28 mat :

L2 34566579701332021429 mat :

ω L1 L2 InterX : 4.5 2 5.5 1 5.6667 0.6667 7.25 1.875 8.75 1.125 7.8333 2.75 8.5 1.75 72 mat

plot1 ω o 6 black plot :

1 time t0 - 0.9 s *

L1 transpose L2 transpose plot1 3 1 sys

C1 1 - 11111 - 1 - 1 - 1 - 152 mat :

θ rot θ cos θ sin - θ sin θ cos 22 mat :

θ π - 4 / : 0.7854 -

clear Imaginary i Clear 1

NOT commutative C2 C1 θ rot * :

Ω C1 transpose C2 transpose InterX : 0.4142 - 11 - 0.4142 0.4142 11 0.4142 1 0.4142 - 0.4142 1 - 0.4142 - 1 - 1 - 0.4142 - 82 mat

plot Ω o 6 black plot :

Rotated a paired geometric vector

C1 C2 plot 3 1 sys

retrieves 45° rotation Ω 12 el i Ω 22 el * + arg Ω 22 el i Ω 12 el * + arg 21 mat ° 22.5 67.5 21 mat

Export Folium ▼ ... enable 16 min

Folium Intersection System

Function mdiff(V)
Input: a vector (either a row or a column)
Output: a vector of the same nature of input (a row or a column)
                  containing the differences V(i+1)-V(i).
                   Number of output elements are lenght(V). V mdiff R V rows V rows 1 > (- V cols V cols 1 > (- matrix : ii 1 R length range R ii el V ii 1 + el V ii el - : for R 31 line :

Examples:

z2 4273151 mat :

z2 mdiff 2 - 54 - 2 - 41 mat

z1 13589416 mat :

z1 mdiff 22315 - 15 mat

Function index(x; cond)
Input: x: a vector (row or column). cond: expression evaluating to an array of T/F values.
Output: a vector of same nature of input array whose components are those corresponding
                to a TRUE value for the given condition. V cond index R V rows V cols matrix : j 0 : V.end V length : 12 mat i 1 V.end range cond vectorize i el j j 1 + : R j el V i el : 21 line continue if for R rows 1 > (R 1 j 11 submatrix R 111 j submatrix if 41 line :

Examples:

z1 z1 4 > index 58913 mat

z2 z1 mdiff 0 > index 427341 mat

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 :

a b plus a b + :

a b times a b * :

a b minus a b - :

a b divide a b / :

x y 2 fun map xh x cols : xv x rows : yh y cols : yv y rows : 22 mat okx xh yh ≡ (xh 1 ≡ (yh 1 ≡ (| (| (: oky xv yv ≡ (xv 1 ≡ (yv 1 ≡ (| (| (: okx oky & xx x : yy y : xh yh < ii 2 yh range xx xx x augment : for xh yh > ii 2 xh range yy yy y augment : for 1 if if xv yv < ii 2 yv range xx xx x stack : for xv yv > ii 2 xv range yy yy y stack : for 1 if if hv xv yv Max : hx xh yh Max : 12 mat ii 1 hv range jj 1 hx range zz ii jj el xx ii jj el yy ii jj el fun : for for zz 71 line Incompatible sizes error if 41 line :

z1 z2 A B times map 4122032361626101618872135566328391524271213589456 mat

Examples:

z1 z1 A B plus map 26101618816 mat

z2 z2 A B plus map 84146251 mat

Function D(x; y) Helper function for InterX. See below.

x y D x 1 x rows 1 x cols 1 - submatrix y A B minus map x 1 x rows 2 x cols submatrix y A B minus map A B times map :

Function InterX(L1; L2) Input: L1; L2 : 2-rows matrices of coords (Row1 = x-coord; Row2 = y-coord) Output: 2-rows matricx of coords of intersecting points

L1 L2 InterX x1 L1 1 row transpose : x2 L2 1 row : y1 L1 2 row transpose : y2 L2 2 row : dx1 x1 mdiff eval : dy1 y1 mdiff eval : dx2 x2 mdiff eval : dy2 y2 mdiff eval : ii 1 dx1 length range S1 ii el dx1 ii el y1 ii el * dy1 ii el x1 ii el * - : for ii 1 dx2 length range S2 ii el dx2 ii el y2 ii el * dy2 ii el x2 ii el * - : for S2 S2 transpose eval : C1 dx1 y2 A B times map dy1 x2 A B times map - S1 D 0 mwle eval : C2 y1 dx2 A B times map x1 dy2 A B times map - (transpose S2 transpose D 0 mwle transpose eval : ff C1 C2 mwand mwfind eval : ii ff 1 el eval : jj ff 2 el eval : ii length 0 ≡ 21 Zeros ii ii transpose eval : dx2 dx2 transpose eval : dy2 dy2 transpose eval : S2 S2 transpose eval : L dy2 jj el dx1 ii el A B times map dy1 ii el dx2 jj el A B times map - eval : ii ii L 0 ≠ index eval : jj jj L 0 ≠ index eval : L L L 0 ≠ index eval : PP1 dx2 jj el S1 ii el A B times map dx1 ii el S2 jj el A B times map - eval : PP2 dy2 jj el S1 ii el A B times map dy1 ii el S2 jj el A B times map - eval : PP PP1 PP2 augment L L augment A B divide map eval : PP 121 line if 171 line :

Fo1, Fo2

Fo1 00 0.038 - 0.022 0.075 - 0.046 0.112 - 0.072 0.15 - 0.099 0.188 - 0.128 0.225 - 0.16 0.262 - 0.193 0.3 - 0.229 0.338 - 0.269 0.375 - 0.312 0.412 - 0.359 0.45 - 0.413 0.488 - 0.476 0.525 - 0.555 0.525 - 0.846 0.488 - 0.872 0.45 - 0.88 0.412 - 0.876 0.375 - 0.864 0.338 - 0.844 0.3 - 0.817 0.262 - 0.783 0.225 - 0.74 0.188 - 0.69 0.15 - 0.629 0.112 - 0.554 0.075 - 0.46 0.038 - 0.33 00 0.067 0.036 0.133 0.068 0.2 0.096 0.267 0.121 0.333 0.141 0.4 0.158 0.467 0.17 0.533 0.179 0.6 0.184 0.667 0.184 0.733 0.178 0.8 0.166 0.867 0.145 0.933 0.109 1010 0.933 0.109 - 0.867 0.145 - 0.8 0.166 - 0.733 0.178 - 0.667 0.184 - 0.6 0.184 - 0.533 0.179 - 0.467 0.17 - 0.4 0.158 - 0.333 0.141 - 0.267 0.121 - 0.2 0.096 - 0.133 0.068 - 0.067 0.036 - 00 0.038 - 0.022 - 0.075 - 0.046 - 0.112 - 0.072 - 0.15 - 0.099 - 0.188 - 0.128 - 0.225 - 0.16 - 0.262 - 0.193 - 0.3 - 0.229 - 0.338 - 0.269 - 0.375 - 0.312 - 0.412 - 0.359 - 0.45 - 0.413 - 0.488 - 0.476 - 0.525 - 0.555 - 0.525 - 0.846 - 0.488 - 0.872 - 0.45 - 0.88 - 0.412 - 0.876 - 0.375 - 0.864 - 0.338 - 0.844 - 0.3 - 0.817 - 0.262 - 0.783 - 0.225 - 0.74 - 0.188 - 0.69 - 0.15 - 0.629 - 0.112 - 0.554 - 0.075 - 0.46 - 0.038 - 0.33 - 00902 mat :

Fo2 00 0.044 - 0.001 0.088 - 0.003 0.133 - 0.006 0.179 - 0.011 0.227 - 0.017 0.275 - 0.026 0.324 - 0.036 0.374 - 0.049 0.427 - 0.064 0.481 - 0.082 0.537 - 0.105 0.596 - 0.133 0.66 - 0.168 0.732 - 0.219 0.878 - 0.47 0.858 - 0.512 0.83 - 0.537 0.795 - 0.553 0.757 - 0.561 0.714 - 0.562 0.668 - 0.558 0.619 - 0.546 0.565 - 0.529 0.507 - 0.504 0.444 - 0.469 0.374 - 0.424 0.295 - 0.361 0.198 - 0.267 00 0.04 0.065 0.081 0.126 0.125 0.183 0.171 0.238 0.218 0.289 0.268 0.337 0.319 0.381 0.372 0.422 0.428 0.459 0.485 0.493 0.546 0.521 0.61 0.544 0.678 0.559 0.754 0.561 0.866 0.5 0.866 0.5 0.863 0.372 0.823 0.308 0.776 0.256 0.724 0.212 0.669 0.174 0.612 0.141 0.552 0.111 0.489 0.086 0.425 0.063 0.359 0.045 0.291 0.029 0.221 0.017 0.15 0.008 0.076 0.002 00 0.021 - 0.038 - 0.042 - 0.078 - 0.062 - 0.118 - 0.08 - 0.161 - 0.098 - 0.205 - 0.115 - 0.251 - 0.131 - 0.298 - 0.145 - 0.349 - 0.158 - 0.401 - 0.169 - 0.457 - 0.178 - 0.517 - 0.183 - 0.583 - 0.184 - 0.656 - 0.177 - 0.744 - 0.032 - 0.995 - 0.014 0.999 - 0.05 0.987 - 0.081 0.965 - 0.107 0.936 - 0.13 0.9 - 0.149 0.858 - 0.164 0.809 - 0.175 0.754 - 0.182 0.691 - 0.184 0.619 - 0.18 0.536 - 0.165 0.436 - 0.133 0.305 - 00902 mat :

Strip paired 0's

data Strip2 nest striped in 'R' u data 1 col : v data 2 col : k 0 : i 1 data rows range u i el v i el * 0 ≠ R k k 1 + : el u i el v i el 12 mat : 11 line Undefined if 11 line for 21 line Un-nest R v R 1 el : j 2 R rows range v v R j el stack : for v 81 line :

Fo1 rows Fo2 rows 21 mat 909021 mat

greedy ... 16 min folium Fo1 transpose Fo2 transpose InterX :

folium folium : folium folium Strip2 : folium folium 0012 mat stack : Folium folium o 6 black plot : 41 sys

folium 0.499 - 0.4996 0.6816 0.1827 0.1821 - 0.6805 - 0042 mat

OBSERVE ... If you pre-Strip2 before InterX you get the 5 intersections however not desired result.

Fo1 Fo2 Folium 3 1 sys

Fo1 Strip2 Fo2 Strip2 21 sys

. . . Scilab example [30 min] . . .

]]>

t 02 π * (2 π 100 / * range :

r1 4 t * sin vectorize 2 + :

r2 8 t * sin vectorize 2 + :

x1 r1 t cos vectorize A B times map :

y1 r1 t sin vectorize A B times map :

x2 r2 t cos vectorize A B times map :

y2 r2 t sin vectorize A B times map :

arg1 x1 transpose y1 transpose stack : arg2 x2 transpose y2 transpose stack : 21 sys

Scilab arg1 arg2 InterX :

plot2 Scilab o 12 Green plot :

arg1 transpose arg2 transpose plot2 3 1 sys

arg1 transpose arg2 transpose plot2 31 sys

Animation ... rotation/circulation

Eol data

Eol 00 0.0062 - 0.0036 0.0125 - 0.0073 0.0188 - 0.011 0.025 - 0.0148 0.0312 - 0.0186 0.0375 - 0.0224 0.0438 - 0.0263 0.05 - 0.0302 0.0562 - 0.0341 0.0625 - 0.0381 0.0688 - 0.0422 0.075 - 0.0463 0.0812 - 0.0504 0.0875 - 0.0546 0.0938 - 0.0589 0.1 - 0.0631 0.1062 - 0.0675 0.1125 - 0.0718 0.1188 - 0.0762 0.125 - 0.0807 0.1312 - 0.0852 0.1375 - 0.0898 0.1438 - 0.0944 0.15 - 0.0991 0.1562 - 0.1038 0.1625 - 0.1086 0.1688 - 0.1134 0.175 - 0.1183 0.1812 - 0.1233 0.1875 - 0.1283 0.1938 - 0.1333 0.2 - 0.1384 0.2062 - 0.1436 0.2125 - 0.1489 0.2188 - 0.1542 0.225 - 0.1595 0.2312 - 0.165 0.2375 - 0.1704 0.2438 - 0.176 0.25 - 0.1816 0.2562 - 0.1873 0.2625 - 0.1931 0.2688 - 0.1989 0.275 - 0.2049 0.2812 - 0.2109 0.2875 - 0.2169 0.2938 - 0.2231 0.3 - 0.2293 0.3062 - 0.2357 0.3125 - 0.2421 0.3188 - 0.2486 0.325 - 0.2552 0.3312 - 0.2618 0.3375 - 0.2686 0.3438 - 0.2755 0.35 - 0.2825 0.3562 - 0.2896 0.3625 - 0.2968 0.3688 - 0.3042 0.375 - 0.3116 0.3812 - 0.3192 0.3875 - 0.3269 0.3938 - 0.3348 0.4 - 0.3428 0.4062 - 0.351 0.4125 - 0.3593 0.4188 - 0.3678 0.425 - 0.3764 0.4312 - 0.3853 0.4375 - 0.3943 0.4438 - 0.4036 0.45 - 0.4131 0.4562 - 0.4228 0.4625 - 0.4328 0.4688 - 0.4431 0.475 - 0.4537 0.4812 - 0.4647 0.4875 - 0.476 0.4938 - 0.4878 0.5 - 0.5 0.5062 - 0.5128 0.5125 - 0.5262 0.5188 - 0.5404 0.525 - 0.5555 0.5312 - 0.5717 0.5375 - 0.5894 0.5438 - 0.6091 0.55 - 0.6321 0.5562 - 0.6611 0.5562 - 0.7829 0.55 - 0.8034 0.5438 - 0.8178 0.5375 - 0.829 0.5312 - 0.8381 0.525 - 0.8457 0.5188 - 0.8521 0.5125 - 0.8575 0.5062 - 0.8621 0.5 - 0.866 0.4938 - 0.8693 0.4875 - 0.8721 0.4812 - 0.8744 0.475 - 0.8763 0.4688 - 0.8778 0.4625 - 0.8788 0.4562 - 0.8796 0.45 - 0.88 0.4438 - 0.8801 0.4375 - 0.8799 0.4312 - 0.8794 0.425 - 0.8787 0.4188 - 0.8777 0.4125 - 0.8764 0.4062 - 0.8749 0.4 - 0.8732 0.3938 - 0.8713 0.3875 - 0.8691 0.3812 - 0.8667 0.375 - 0.8641 0.3688 - 0.8613 0.3625 - 0.8582 0.3562 - 0.855 0.35 - 0.8516 0.3438 - 0.8479 0.3375 - 0.8441 0.3312 - 0.8401 0.325 - 0.8359 0.3188 - 0.8314 0.3125 - 0.8268 0.3062 - 0.822 0.3 - 0.817 0.2938 - 0.8117 0.2875 - 0.8063 0.2812 - 0.8007 0.275 - 0.7948 0.2688 - 0.7888 0.2625 - 0.7826 0.2562 - 0.7761 0.25 - 0.7694 0.2438 - 0.7625 0.2375 - 0.7554 0.2312 - 0.7481 0.225 - 0.7405 0.2188 - 0.7327 0.2125 - 0.7246 0.2062 - 0.7163 0.2 - 0.7077 0.1938 - 0.6989 0.1875 - 0.6897 0.1812 - 0.6803 0.175 - 0.6706 0.1688 - 0.6606 0.1625 - 0.6503 0.1562 - 0.6396 0.15 - 0.6286 0.1438 - 0.6173 0.1375 - 0.6055 0.1312 - 0.5933 0.125 - 0.5807 0.1188 - 0.5677 0.1125 - 0.5541 0.1062 - 0.54 0.1 - 0.5254 0.0938 - 0.5101 0.0875 - 0.4942 0.0812 - 0.4775 0.075 - 0.46 0.0688 - 0.4416 0.0625 - 0.4222 0.0562 - 0.4016 0.05 - 0.3796 0.0438 - 0.356 0.0375 - 0.3304 0.0312 - 0.3024 0.025 - 0.2712 0.0188 - 0.2354 0.0125 - 0.1927 0.0062 - 0.1366 00 0.0111 0.0064 0.0222 0.0126 0.0333 0.0187 0.0444 0.0247 0.0556 0.0305 0.0667 0.0363 0.0778 0.0419 0.0889 0.0474 0.1 0.0528 0.1111 0.0581 0.1222 0.0633 0.1333 0.0683 0.1444 0.0733 0.1556 0.0781 0.1667 0.0828 0.1778 0.0874 0.1889 0.092 0.2 0.0964 0.2111 0.1007 0.2222 0.1048 0.2333 0.1089 0.2444 0.1129 0.2556 0.1168 0.2667 0.1206 0.2778 0.1242 0.2889 0.1278 0.3 0.1312 0.3111 0.1346 0.3222 0.1378 0.3333 0.141 0.3444 0.144 0.3556 0.147 0.3667 0.1498 0.3778 0.1525 0.3889 0.1551 0.4 0.1577 0.4111 0.1601 0.4222 0.1624 0.4333 0.1646 0.4444 0.1667 0.4556 0.1686 0.4667 0.1705 0.4778 0.1723 0.4889 0.1739 0.5 0.1754 0.5111 0.1769 0.5222 0.1782 0.5333 0.1793 0.5444 0.1804 0.5556 0.1814 0.5667 0.1822 0.5778 0.1829 0.5889 0.1835 0.6 0.1839 0.6111 0.1843 0.6222 0.1844 0.6333 0.1845 0.6444 0.1844 0.6556 0.1842 0.6667 0.1838 0.6778 0.1833 0.6889 0.1826 0.7 0.1818 0.7111 0.1808 0.7222 0.1797 0.7333 0.1783 0.7444 0.1768 0.7556 0.1751 0.7667 0.1732 0.7778 0.1711 0.7889 0.1687 0.8 0.1662 0.8111 0.1634 0.8222 0.1603 0.8333 0.157 0.8444 0.1534 0.8556 0.1494 0.8667 0.1451 0.8778 0.1404 0.8889 0.1353 0.9 0.1297 0.9111 0.1235 0.9222 0.1167 0.9333 0.1091 0.9444 0.1006 0.9556 0.0908 0.9667 0.0794 0.9778 0.0654 0.9889 0.0467 1010 0.9889 0.0467 - 0.9778 0.0654 - 0.9667 0.0794 - 0.9556 0.0908 - 0.9444 0.1006 - 0.9333 0.1091 - 0.9222 0.1167 - 0.9111 0.1235 - 0.9 0.1297 - 0.8889 0.1353 - 0.8778 0.1404 - 0.8667 0.1451 - 0.8556 0.1494 - 0.8444 0.1534 - 0.8333 0.157 - 0.8222 0.1603 - 0.8111 0.1634 - 0.8 0.1662 - 0.7889 0.1687 - 0.7778 0.1711 - 0.7667 0.1732 - 0.7556 0.1751 - 0.7444 0.1768 - 0.7333 0.1783 - 0.7222 0.1797 - 0.7111 0.1808 - 0.7 0.1818 - 0.6889 0.1826 - 0.6778 0.1833 - 0.6667 0.1838 - 0.6556 0.1842 - 0.6444 0.1844 - 0.6333 0.1845 - 0.6222 0.1844 - 0.6111 0.1843 - 0.6 0.1839 - 0.5889 0.1835 - 0.5778 0.1829 - 0.5667 0.1822 - 0.5556 0.1814 - 0.5444 0.1804 - 0.5333 0.1793 - 0.5222 0.1782 - 0.5111 0.1769 - 0.5 0.1754 - 0.4889 0.1739 - 0.4778 0.1723 - 0.4667 0.1705 - 0.4556 0.1686 - 0.4444 0.1667 - 0.4333 0.1646 - 0.4222 0.1624 - 0.4111 0.1601 - 0.4 0.1577 - 0.3889 0.1551 - 0.3778 0.1525 - 0.3667 0.1498 - 0.3556 0.147 - 0.3444 0.144 - 0.3333 0.141 - 0.3222 0.1378 - 0.3111 0.1346 - 0.3 0.1312 - 0.2889 0.1278 - 0.2778 0.1242 - 0.2667 0.1206 - 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Eol rows 540

x c 1 x 2^- sqrt ± :

t0 1 time :

θ rot θ cos θ sin - θ sin θ cos 22 mat :

θ π 5 / - :

greedy ... 3 min τ 0540 range :

t EolX Eol 1 col t el : t EolY Eol 2 col t el : 12 mat

step1 060 range :

Smath 2D vectros rotation system t Eol Eol θ π 30 / t * + rot * eval :

Smath "Wind Mill" project

t Eol 0 0 . 20 red 1 5 mat x c 3 1 sys

Smath "Wind Mill" project

t 1 ≡ path t EolX t EolY 1 2 mat : path path t EolX t EolY 1 2 mat stack : if 0 0 . 14 red 1 5 mat x c 3 1 sys

1 time t0 - min 3.1

Hemi-circle "roots" at(2) . . . sector circle solve

]]>

full size hemi-circle N# is even lo 1 - : hi 1 : N# 100 : 13 mat

π*ζ is exact incremental

lo 0.5 : hi 1 : N# 100 : 13 mat

svg 1.25 :

clear Imaginary i Clear 1

r hi - : x y f x 2^y 2^+ r + : ζ lo hi lo hi lo - N# / + (range : 31 sys

Hemi j 1 ζ length range rho j el ρ ζ j el y f y roots at : for rho 21 line :

Auxilliary maths

∠

C½ svg ζ Hemi augment * :

Row 1 : αργ C½ Row row : ∠ αργ 11 el i αργ 12 el * + arg : 31 sys

C½ C½ - αργ 1 2 el αργ 1 1 el / x * 3 1 sys

1 - lo ≤ (lo 1 ≤ (& (lo hi < (& 0 hi ≤ (hi 1 ≤ (& 21 sys

αργ 0.625 1.0825 12 mat

∠ ° 60

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 circle r 2^x 2^- sqrt ± :

r 1 :

Sub-circle . . . division

]]>

lo 0 : hi 1 : N# 200 : 13 mat

svg 1 :

clear Imaginary i Clear 1

r hi - : x y f x 2^y 2^+ r + : ζ lo hi lo hi lo - N# 1 + / + (range : 31 sys

Hemi j 1 ζ length range rho j el ρ ζ j el y f y roots at : for rho 21 line :

Auxilliary maths

∠

C½ svg ζ Hemi augment * :

Row 1 : αργ C½ Row row : ∠ αργ 11 el i αργ 12 el * + arg : 31 sys

C½ C½ - 2 1 sys

αργ 0112 mat

∠ ° 90

x 2 π * ζ * : X 0 : Y 0 : 12 mat u Cir i 2 * π * u * exp : C svg x Cir Im vectorize X + (* svg x Cir Re vectorize Y + (* augment : 51 sys

'sub' division
depends upon N# sub C 933 range 12 range el : Sub sub o 7 black plot : 21 sys

x circle Sub 2 1 sys

C rows 202

x Clear 1

x circle r 2^x 2^- sqrt ± :

n 12 :

r 1 :

roots x n^1 - 0 ≡ x solve maple :

Unity roots Re roots Im augment r * :

rootsUnity Unity . 14 black plot :

'roots' aren't sequentially ordered thus, can't be immediately selected in sequential order ... can be sorted.

x circle rootsUnity 2 1 sys Zac

lo 1 - : hi 1 : N# 51 : 13 mat

svg 1 :

clear Imaginary i Clear 1

r hi - : x y f x 2^y 2^+ r + : ζ lo hi lo hi lo - N# 1 + / + (range : 31 sys

datum 1 - 010000101 - 52 mat :

Hemi j 1 ζ length range rho j el ρ ζ j el y f y roots at : for rho 21 line :

C½ svg ζ Hemi augment * :

Zac C½ C½ - stack :

clr Zac clr π Zac * red 2 clr 14 mat 12 mat : clr datum clr 6 datum * black 1 clr 14 mat 12 mat : templates #A0A0A0 Zac datum 21 sys : 31 sys

Native Modeller both scaled respectively by double click.

Zac

datum 1 - 010000101 - 52 mat

Modeller . . . demo

]]> Airfoil cleaned ... Streamlines native

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Streamlines rows 800

sub1 Streamlines 1162 range 12 range el : sub2 Streamlines 163242 range 12 range el : sub3 Streamlines 243322 range 12 range el : sub4 Streamlines 323401 range 12 range el : sub5 Streamlines 402481 range 12 range el : sub6 Streamlines 482561 range 12 range el : sub7 Streamlines 562641 range 12 range el : sub8 Streamlines 642720 range 12 range el : sub9 Streamlines 721800 range 12 range el : 91 sys

plot sub1 sub2 sub3 sub4 sub5 sub6 sub7 sub8 sub9 91 sys :

plot sub1

datum 1 - 010000101 - 52 mat :

clr Airfoil clr 5 Airfoil * red 1 clr 14 mat 12 mat : clr sub1 clr 6 sub1 * red 1 clr 14 mat 12 mat : clr sub3 clr 6 sub3 * cyan 1 clr 14 mat 12 mat : clr sub4 clr 6 sub4 * cyan 1 clr 14 mat 12 mat : clr sub5 clr 6 sub5 * cyan 1 clr 14 mat 12 mat : clr sub9 clr 6 sub9 * cyan 1 clr 14 mat 12 mat : clr datum clr 6 datum * black 1 clr 14 mat 12 mat : 71 sys

#A0A0A0 Web colors ≡

open style
fill black by  default ?
Web colors fill Airfoil . #A0A0A0 31 sys ≡

templates #A0A0A0 sub1 sub3 sub4 sub5 sub9 datum 61 sys :

0. Default templates icons 5 1. more templates drag down 2. check ► Show templates √ 3. check ► templates √ 4. 1 clr ► thickness 5. 6*sub3 ► size templates 6. scale in pixels .. by default Modeller 200.

Joukowsky

. . . NO more bananas in this compendium . . .

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