<p> plotG(data,char,size,color) ... UnestH(data) </p>

179324b9-6b32-4a5a-96ba-e7f9dffa5c65 70 3 12 decimal

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

1. The 'Δ incremental' discretizer.

3 appVersion 0.98.6179 ≡ Timing_up_to_15 ~ 35 s ≡ 21 mat

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 :

isolated  end use Δ 2 - 0.1 100 inc :

Δ transpose 2 - 1.9 - 1.8 - 1.7 - 1.6 - 1.5 - 1.4 - 17 mat

# f L Δ N Δinc X 0 : i 1 N range X i 1 + el L Δ i * + : X 1 el L : 21 line for i 1 X rows range vy i el X i el f : for X vy augment 41 line :

x F 11 x 2^+ / :

L 2 - : Δ 0.1 : N 101 : 13 mat F x F L Δ N Δinc : 21 sys

Discrete cumulative integration always starts valued [0],

not to be be confused with symbolic integral curve.

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data Σ Finite diff's 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 :

Σ F Σ :

Σ N row 8 2.55 12 mat

Σ F 2 1 sys

2. The 'step-in range' discretiZer.

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

x F 11 x 2^+ / :

Plot 5 - 5 4.9 - x F inc :

Plot range

Plot

Plot rows 101

# f Q i 1 U rows range vx i el U i el : vy i el U i el f : 21 line for vx vy augment 21 line :

3. The 'N-in-range' discretizer.

L H N xd U 0 : dx H L - N / : 12 mat i 1 N range U i el L dx - (dx i * + eval : for U 31 line :

L 5 - : H 5 : N 100 : 13 mat

U L H N xd :

x F 11 x 2^+ / :

x q 0.01 x 3^(* :

x F Q x q Q 2 1 sys

4. The 'N-mesh-reverse-circulation'.

Algo style "circulation" discretizer [piston discretizer]
1. collect branches of Maple symbolic solve.
2. highlight start/end of plot # f x0 x1 xmesh fmesh U 0 : count 1 : i 1 xmesh range x i el x0 i xmesh / x1 x0 - (* + : U count el x i el f eval : count count 1 + : 31 line for x U augment 41 line :

data x F 5 - 0.5 1000 fmesh : datareverse x F 511000 fmesh : 21 sys

data datareverse 2 1 sys

5. The infinite vector scheduler

<p> plotG(data,char,size,color) ... UnestH(data) </p>

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

M Low High Scale High Low - (M M Min - M Max M Min - 106 -^+ / (* Low + :

Range 'n' ∞ - n ≤ (n ∞ ≤ (&

User scale xs 1 : ys 1 : 12 mat

v 012345678910111 mat transpose :

n v rotate N v length : p n N mod : p p p 0 ≥ p N + p 0 < ignored cases : Ω v : p 0 ≡ i 1 p range Ω i N + p - el v i el : for i p 1 + (N range Ω i p - el v i el : for 21 line cases Ω 51 line :

n 15 - :

<= Pilot the rotation

v transpose n v rotate transpose 21 mat 012345678910111 mat 789100123456111 mat 21 mat

Vrot n v rotate :

XY xs v * ys Vrot * augment :

rot xs v * ys Vrot * . 10 black plotG :

C 55 + 200 black 15 mat :

Reversible rotation of vector. You can shrink/expand either direction via [xs, ys]. Illustrative plot wrt 'n'.

x n v rotate x 1 + el rot C 4 1 sys

Typical Matrix Scheduler M v transpose 19 - v rotate transpose 0 v rotate transpose 1 v rotate transpose 7 v rotate transpose 51 mat transpose : 012345678910111 mat 345678910012111 mat 012345678910111 mat 123456789100111 mat 789100123456111 mat 15 mat

The "rotated" XY 071829310405162738495106112 mat

U M UnestH : Schedule day Pet Jos Art Leslie 15 mat : Assign Schedule U stack : 31 line

A possible application:

Assume you are looking at aspecific star, you need several

parameters [Azimut, Horizon, Focus ...]. If the vector 'v' is

wrt parameters, easy to address and automate the system.

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Assign day Pet Jos Art Leslie 030171412825239363410474505856169672710783808949191051021006125 mat

'R' is subscripted to explode  100*Random(1) v transpose R 1001 Random 1 el * Floor : R v rotate transpose 31 mat 012345678910111 mat 41891001234567111 mat 31 mat

U U 0255 Scale :

Implicit 'zoom fractional' Up/Down Ω a b zoom a b ≡ (a b | (1 ≠ (& (=> ≡ => implicit conditions ≡ i 1 Ω rows a * (range j 1 Ω cols b * (range Nu i j el Ω i a / Ceil j b / Ceil el : a 1 ≥ Nu i j el Ω i a / Floor j b / Floor el : cases for for Nu 41 line :

iMage U 1010 zoom :

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 iMage

6. piecewise-suite range

L 1012 -^: H 10 : N 100 : 13 mat

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 :

range Plot L H N x ln range Algo :

plot b 1 el Plot b 2 el Plot b 3 el Plot 31 sys :

b 306010031 sys :

plot b 1 el H / 2 b 1 el H / 1 - 2 2 mat b 2 el H / 3 b 2 el H / 1 - 2 2 mat b 3 el H / 5 b 3 el H / 1 - 2 2 mat 4 1 sys

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

7. range-integrate-single-piece

L 1012 -^: H 10 : N 100 : 13 mat

lo 1 : hi 50 : 12 mat

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

Plot L H N x ln lo hi Algo :

X1 lo H / : X2 hi H / : 12 mat 0.1 512 mat

Plot X1 2 X1 1 - 2 2 mat X2 3 X2 1 - 2 2 mat 3 1 sys

data Σ Finite diff's 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 :

Σ Plot Σ :

Σ Σ rows row 4.9 3.2 12 mat

Σ X1 2 X1 1 - 2 2 mat X2 10 X2 1 - 2 2 mat 3 1 sys

8. n-intervals-in-range.

'n' intervals in range n range interval j 1 n 1 + (range h j el range - range j * + n / : for h 21 line :

J 123 interval transpose : 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3113 mat

9. otherwise N-intervals-in-range.

β 300010000000012 mat transpose :

plug your fnct ...  export discretized x f β 1 el 1200 x * 150 x * + (* + (* 10000 β 2 el x 2^* + / :

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 :

U 0 0.125 299 xd : x fnct Q i 1 U rows range vx i el U i el : vy i el U i el fnct : 21 line for vx vy augment 21 line : 21 line

x f Q 0 0.3 0.0004 0.325 0.0008 0.3498 0.0013 0.3741 0.0017 0.3976 0.0021 0.4202 0.0025 0.4416 0.0029 0.4617 82 mat

x f Q

10. otherwise-address.

x f x 3^x erf 1 x 1 + / 1 - 13 mat transpose :

L H N Φ i 1 N range x i el L i N / H L - (* + : Y i el x i el f j el eval : 21 line for x Y augment 21 line :

Lo 1 - : Hi 1 : 12 mat

j 3 :

N 0.5 - 1100 Φ :

===================================================

j 1 :

create matrix of some cols [0's,"] ... 
specific user end use  wrt [0's,"] cols . L H N c1 c2 c3 Φ i 1 N range x i el L i N / H L - (* + : M i c1 el x i el : M i c2 el x i el f j el eval : M i c3 el : 41 line for M 21 line :

data 1 - 140375 Φ : collect data 3 col data 7 col augment : 21 line

collect N 2 1 sys

11. else-inner-range... [Lo Hi inner L H]

x f x 3^x erf 1 x 1 + / 1 - 13 mat transpose :

Lo 0.625 - : Hi 1.5 : 12 mat

j 3 :

x fnct break a trace 0 x Lo ≤ 0 Hi x ≤ x f j el cases 21 line :

L H N Φ i 1 N range x i el L i N / H L - (* + : A i 1 el x i el : A i 2 el x i el fnct eval : 31 line for A 21 line :

data 3 - 3100 Φ : plot data 1 col data 2 col augment : 21 sys

Discrete outer exclusive outer u plot 1 col : v plot 2 col : k 0 : i 1 plot rows 1 - (range v i 1 + el v i el ≤ (v i 1 + el v i el ≥ (¤ P k k 1 + : el u i 1 + el v i 1 + el 12 mat : 11 line out search region if 11 line for 21 line v P 1 el : j 2 P rows 1 - (range v v P j el stack : for v 61 line :

plot outer

scalar inner exclusive x f 11 x 2^+ / : r 1 : d 2 : 13 mat datum d - r f d r f 22 mat : x c 1 x r - x ≤ (r x ≥ (& (¤ cases : 31 sys

x f x c * x f datum 3 1 sys

scalar outer exclusive x f 11 x 2^+ / : r 1 : d 2 : 13 mat datum d - r f d r f 22 mat : x c 1 x x r ≥ (¤ (x x r - ≤ (¤ (& cases : 31 sys

x f x c * x f datum 3 1 sys

12. box-mesh-style...square/rectangular/mirror

mirror scalar style ... export o end inc u k 1 : i o end o inc + range v k el i : k k 1 + : 21 line for v 31 line :

o 4 :

end 4 :

mesh 50 :

vx o - o o mesh / u :

vy end - end end mesh / u :

XY vx vy augment :

box o - end - o - end o end o end - o - end - 52 mat :

f vx vx ln vectorize Re augment : q vx vx exp vectorize augment : sqr vx 2 vx * vectorize (sqrt vectorize Re augment : 31 sys

XY box f q sqr 6 1 sys

13. Comprehensive 2D matrix algo

t0 1 time :

x y f L H N Sym U 0 : dx H L - N / : 12 mat i 1 N range U i el L dx i * dx - (+ : for r 1 : i 1 U rows range c 1 : j 1 U rows range M r c el U i el U j el f eval : c c 1 + : 21 line for r r 1 + : 31 line for 21 line M 41 line :

either direction

L 1 - :

H 1 :

N 32 :

εxy 2 :

x y f 255 x 2^εxy y * (2^+ (- 0.25 / exp * : N x y f L H N Sym : 21 sys

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 N

1 time t0 - 3 s *

14. 2D rotation of a data set

L H N xd i 1 N range x i el L i H L - N / * + : for x 21 line :

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

θ π - 2 / :

θ 1.5708 -

t 02 π * 50 xd :

a 0.5 :

G i 1 t rows range gx i el a t i el sin * : gy i el a t i el sin * t i el cos * : 21 line for gy gx augment 21 line :

NOT commutative Rot G θ rot * :

G Rot 2 1 sys G Rot 2 1 sys

15. 3D rotation of a solid

Sphere uu vv f uu sin vv cos * uu sin vv sin * uu cos 31 mat :

S uu vv f 02 π * 02 π * 2424 CreateMesh :

ω Ω1 1000 ω cos ω sin - 0 ω sin ω cos 33 mat :

Матрица преобразования для поворота вокруг оси y ω Ω2 ω cos 0 ω sin 010 ω sin - 0 ω cos 33 mat :

Матрица преобразования для поворота вокруг оси z ω Ω3 ω cos ω sin - 0 ω sin ω cos 000133 mat :

α 0 (π * 4 / :

β 1 (π * 4 / :

φ 1 (π * 4 / :

γ α Ω1 β Ω2 * φ Ω3 * eval :

S γ * 1 1 sys

γ 0.5 0.5 - 0.7071 0.7071 0.7071 0 0.5 - 0.5 0.7071 33 mat

S γ * NOT commutative > ≡

... Revisit the Discrete-Circulation-Extractor ... [ 1.8 min]

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Algo style "circulation" discretizer [piston discretizer] x f x0 x1 xmesh fmesh U 0 : count 1 : j 1 xmesh range x j el x0 j xmesh / x1 x0 - (* + : U count el x j el f eval : count count 1 + : 31 line for x U augment 41 line :

x y f x 4^2 x 2^* y * + 3 x * y 2^* - y 3^+ :

<= observe the construction y sol y x f 0 ≡ y solve allvalues maple :

x c 1 x 0.625 ≤ cases :

x f1 x sol 1 el Re : x f2 x sol 2 el Re : x f3 x sol 3 el Re : x f4 x sol 4 el Re : 41 sys

Observe . . .

The 4 branches are real but evaluate with

residual imaginary [ie-16], to be removed.

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x u 1 0.21 - x ≤ (x 0 ≤ (& cases :

escape divide by zero ε 1012 -^:

t0 1 time :

Discretize/collect b1 x f1 1 - 0.625 100 fmesh : b2 x f2 0.625 1 - 100 fmesh : b3 x f3 ε 0.21 - 100 fmesh : b4 x f4 0.21 - ε 100 fmesh : 41 sys

b1 rows b2 rows b3 rows b4 rows 41 mat 10010010010041 mat

cleaned of residuals 'Im"

x f1 x c * x f2 x c * x f3 x u * x f4 x u * 4 1 sys

infected of residuals 'Im'

x sol 1 el x c * x sol 2 el x c * x sol 3 el x u * x sol 4 el x u * 4 1 sys

clean discrete [0 level]

b1 b2 b3 b4 4 1 sys

zeroLevel b1 b2 b3 b4 stack :

Xzero zeroLevel 1 col : Yzero zeroLevel 2 col : YXzero Yzero Xzero augment : 31 sys

zeroLevel YXzero

1 time t0 - min 1.9