plotG

8c842277-4a31-461b-8ccd-1d1ab42707c7 79 4 5 decimal

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

4 appVersion 0.98.6179.21440 ≡

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

5 T 1234524681036912154812162051015202555 mat

M 1234524681036912154812162051015202555 mat :

V 12345246810369121548121620510152025251 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 1 - (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 55 Pack 1234524681036912154812162051015202555 mat

Pad 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

P Ω 33 Pad :

P 123456000789101112246810120001416182022243691215180002124273033364812162024000283236404448000000000000000000000000000000000000000000000510152025300003540455055606121824303600042485460667271421283542000495663707784816243240480005664728088961115 mat

Mesh U, V  within the bounds L, H L H N xd N intervals U 0 : dx H L - N / : i 1 N 1 + (range U i el L dx i * dx - (+ : for U 51 line :

Generate the symmetric square matrix x y f U V Sym r 1 : i 1 U rows range c 1 : j 1 V rows range M r c el U i el V j el f eval : c c 1 + : 21 line for r r 1 + : 31 line for M 31 line :

L 1 - :

H 1 :

N 49 :

xx yy f 255 xx 2^yy 2^+ (- 0.5 / exp * :

U L H N xd :

V U :

G xx yy f U V Sym :

pie G 11 Pad :

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 pie

Mirror Matrix

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 Diagonal 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 M 1 col : ct 1 : j 1 M cols range a 1 ct el M j col : ct ct 1 + : 21 line for a 41 line :

M NestRow 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 41 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 p i el . : for

data XY 1 col XY 2 col p augment :

data XY 2 1 sys

Optimiz: 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

1 time t0 - 0.09 s *

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

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 Mat dia Diagonal elements i 1 Mat rows range Dia i el Mat i i el : for Dia 31 line :

Smath built-in f(x) menu 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

M dia sum 8.66

M tr 8.66

The trace of a square matrix tr(M) is the Σ of the diagonal elements, i.e: Euclidean norm

Smath built-in   f(x) menu 123424683691248121644 mat norme 30

4 M Dia 1491641 mat

4 M tr 30

4 M dia sum 30

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

M diag2vec v M rows M cols 12 mat min : out v 1 matrix : n 1 v range out n el M n n el : for out 41 line :

a h k l b m n p c 33 mat diag2vec a b c 31 mat

a h k l b m n p c q r s 43 mat diag2vec a b c 31 mat

a h k q l b m r n p c s 34 mat diag2vec a b c 31 mat

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

Pivot via local assignment

]]>

vy 9.206 9.3 9.278 9.175 9.275 9.289 9.287 9.261 9.303 9.276 9.273 9.288 9.256 9.252 9.298 9.267 161 mat :

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 10 : n 5 : m 5 : 13 mat

local assign jdx pivot n m S 11 line :

vy jdx el 9.275 9.289 9.287 9.261 9.303 9.276 9.273 9.288 9.256 9.252 9.298 111 mat

jdx 56789101112131415111 mat

pivot 15 : n 5 : m 5 : c1 1 : c2 45 : 15 mat

local assign jdx pivot n m S 11 line :

Mat i 150 range j 150 range Mat i j el i 1 - sin j 1 - cos + eval : for for Mat 21 line :

stuff Mat jdx c1 c2 range el :

iVBORw0KGgoAAAANSUhEUgAAAC0AAAALCAIAAABkjZPmAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAACVSURBVDhPndFBDoMwDERRfP9Dtz/OVCq2iYz/pprXAItcZnbdQwJ2hDJ2hG7y8TS8vpC2N5PVVtI+foC0XwqJHt6/ytoX0vYmUt5WwI5Qxo7QkvDHnqTtiY7HtKev+v381RfSroQmEu6P4hVWQqUEzEJZdI60KyHR8Zj2+wdXsuphDW8Labc/8CSkHW6LslDn2PxVZl8AWTS/AI30iAAAAABJRU5ErkJggg==

crocodile

stuff

Else mirror of square matrix as a function

]]> plotG

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 plotG k 1 data rows range r3 k el char : r4 k el size : r5 k el clr : 31 line 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 - 106 -^+ / (* Lo + :

image n T 0255 Scale :

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 image

data m . 12 red plotG :

arc Arc . 12 red plotG :

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

The Delta Matrix

# size upper size triangle Δ ≡

JPG Δelta matrix upper triangle n s Δ i 1 n range j 1 n range Δ i j el i j + s 1 + ≤ : for for Δ 21 line :

n 32 :

iVBORw0KGgoAAAANSUhEUgAAACAAAAAgCAIAAAD8GO2jAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAClSURBVEhLtc1bFoIwDEDB7n+rbsKqgweFQh/JfHGa5FIeyXJ/UCqfCV71vB+oVx5CSX94i6P75TmI6J5JBMU/hsvkjszXaJ2yskCoxdYslQsWp0hcszvO/S3rgxz3cDHCZSdH3Zz1c9fHzRCnHRyMcn3H9gSBS1bnaLTZmybTYGmF0hkbi8QOjNfp/TILIbljEEV14zWQ8JunWNpJ9Sq3XuXWS3kCn1iuXMONMHwAAAAASUVORK5CYII= 235 n n Δ ( *

========== rank & trace ==========

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

The maximum number of linearly independent vectors in a matrix is equal to the number of non-zero rows in its row echelon matrix. Therefore, to find the rank of a matrix, we simply transform the matrix to its row echelon form and count the number of non-zero rows.

19 M rank 1

M 1212 - 3 - 135033 mat :

M rank 2

Q 7 - 232110721 - 1 - 3 - 34 mat :

N 11021 - 1 - 02 - 24 mat :

N rank 1

Q rank 2

rank of this matrix = 8 A 00101424 - 01 - 011000001 - 002203003 - 0111000 3.99 - 000007001 - 0042018010 - 6 - 004201808 - 6 - 01 - 000012 - 0000000710 - 002 - 99 mat :

H Au Cl K Fe C N Pt O 91 mat

A tr 15 -

optimiz symbolic A rank 8

wrong and confusing A mat2sys sys2mat rank 1

r A rank maple : 8

This goes in top work sheet as "Utilities"
This module is to be developped wrt to cols
Here we have preselected cols [1,3]. 
At this point, you can select any rows  in
cols [1,3] or as otherwise preselected. 
Next: add/delete/insert/shuffle rows at will.
Mathcad 11 could NOT do that. data row ListCollect ct 1 : i row a ct el data i 1 el : b ct el data i 3 el : ct ct 1 + : 31 line for a b augment 31 line :

M 142567234512375895678945 mat :

row 23413 mat :

M row ListCollect 242795732 mat

The col/row extractors utilities

add/remove/insert/shuffle
in col(,,,) argument ... at will in-situ data v Col ct 1 : i v a 1 ct el data i col : ct ct 1 + : 21 line for a V a 1 el : j 2 a cols range V V a j el augment : for V 61 line :

add/remove/insert/shuffle
in row(,,,) argument ... at will in-situ data v Row ct 1 : i v a 1 ct el data i row : ct ct 1 + : 21 line for a V a 1 el : j 2 a cols range V V a j el stack : for V 61 line :

M 05204580125520458012518020458012518024545801251802453208012518024532040512518024532040550066 mat :

Select from in-situ

col 13513 mat :

row 23413 mat :

M row Row 5204580125180204580125180245458012518024532036 mat

M col Col 020805451252080180451252458018032012524540563 mat

NEW M col Col row Row :

NEW 54512520801804512524533 mat

replicate(rep,v) application

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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

u v replicate k 1 : i 1 v length range u i el 0 ≡ continue j 0 u i el 1 - (range T k j + el v i el : for k k u i el + : 21 line if for T 31 line :

P π 2 sqrt 2 ln 13 mat transpose :

v 21213 mat transpose :

W v P replicate :

W transpose 3.1416 3.1416 1.4142 0.6931 0.6931 15 mat

v 20213 mat transpose :

W v P replicate :

W transpose 3.1416 3.1416 0.6931 0.6931 14 mat

To renew: click anywhere in any active region, press F9

Random stem style

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 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 : 71 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

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

Matrix Quadrants

]]> M Venturi

M 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mat :

Mat 85191762281917811131101286315187131771920182014101048 mat :

M2 Mat reverse : 71920182014101086315187131719178111311012851917622848 mat

M2 M2 1 M2 rows range M2 cols 1 range el :

M2 10101420182019717137181536812101131181719822617195848 mat

Rotate image 4 quadrants

Explode mirror Matrix [5346] M mirror ct 1 : j M cols ct range mir 1 ct el M j col : ct ct 1 + : 21 line for mir Unest N v mir 1 el : j 2 mir cols range v v mir j el augment : for v 71 line :

Mirror M mirror :

M Min M Max 21 mat 8616721 mat

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Inverse 91 M - :

M rotate_90 M mirror transpose :

M rotate90 M transpose :

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Inverse 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M 168 -

Davide Carpi: 10 April, 20017 M mirror M 1 M rows range M cols 1 range el :

N M mirror :

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166 M + 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N 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ljcKYfV32j/JpBTcQWM394nJfToRPh0qaTWo5aGCbRztL8e5M4JWgCmYt2yYzQ5GzbsCpr4UEkknacavC0ECDDwgp5yNwZVODpCrBx76q+4VdYwrMgGnpXqKELlpCdvp06f/BQcuQOJhT7MMAAAAAElFTkSuQmCC N M augment

Insert submatrix

M Clear

N 1234 678933 mat :

MM 99 el 0 :

MM 33 el N :

i 15 range j 15 range VV i j el MM i j el : for for

i 15 range j 15 range UU i j el MM i 1 + j 1 + el : for for

i 15 range j 15 range W i 2 + j 2 + el MM i 1 + j 1 + el : for for

VV 0000000000001234 678933 mat 00000000000055 mat

W 3.141593 000000 3.141593 000000 0.693147 000000 0.693147 001234 678933 mat 00000000000000000000000077 mat

UU 0000001234 678933 mat 00000000000000000055 mat

WL 0.85 0.74 0.74 0.75 0.73 0.72 0.72 0.72 0.73 0.72 0.71 0.69 0.7 0.71 0.72 0.7 0.7 0.7 0.67 0.71 0.7 0.7 0.68 0.68 0.69 0.69 0.69 0.67 0.67 0.67 0.68 0.7 0.68 0.67 0.68 0.67 0.67 0.67 0.66 0.68 0.68 0.67 0.67 0.66 0.65 0.86 0.9 0.9 0.89 0.93 0.9 0.9 1.11 1.13 0.89 0.9 0.95 0.98 0.9 0.9 0.87 0.87 0.89 0.64 0.63 0.63 0.65 0.64 0.63 0.62 0.63 0.62 0.85 0.92 0.9 0.9 0.91 0.91 0.9 0.93 0.94 99 mat :

value T col lookup k 1 : j k T rows range T j col el value ≡ index k el j : k k 1 + : 21 line 0 if for index IsDefined index No value(s) match error if 31 line :

t0 1 time :

0.75 WL 5 lookup #

0.68 WL 5 lookup 3521 mat

1 time t0 - 0.03 s *

Round OFF Matrix/Vector [Smath prior 6179] M n Round 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 :

Generate the symmetric square matrix x y f U V Sym r 1 : i 1 U rows range c 1 : j 1 V rows range M r c el U i el V j el f eval : c c 1 + : 21 line for r r 1 + : 31 line for M 31 line :

Mesh U, V  within the bounds L, H L H N xd N intervals U 0 : dx H L - N / : i 1 N 1 + (range U i el L dx i * dx - (+ : for U 51 line :

L 1 - :

H 1 :

N 24 :

xx yy f 255 xx 2^yy 2^+ (- 0.25 / exp * :

U L H N xd :

V U :

G xx yy f U V Sym :

N G 3 Round :

N 0.086 0.162 0.29 0.492 0.789 1.197 1.718 2.332 0.162 0.307 0.55 0.933 1.495 2.268 3.255 4.418 0.29 0.55 0.986 1.671 2.68 4.065 5.833 7.917 0.492 0.933 1.671 2.833 4.543 6.891 9.887 13.421 0.789 1.495 2.68 4.543 7.284 11.049 15.855 21.521 1.197 2.268 4.065 6.891 11.049 16.761 24.05 32.646 1.718 3.255 5.833 9.887 15.855 24.05 34.51 46.844 78 mat

iVBORw0KGgoAAAANSUhEUgAAAEAAAAAhCAIAAADmozp2AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAkgSURBVFhHrdg3jxXZFgXgnmGAhx2QQEKPBAiQEBIBYQsJERIiJLxvvPfee++9dwMxP4GEiL9Axh8gI2O+2+v2prjvaSa5OyhVnzpnn7XWXvtU3e7piN/64/f+GDRo0B+/hpE8yrT2moE4d+7c8ePHDxw4sGvXri1btqxbt27lypWLFy9etGjRkiVL3BsxvnPnzv379x85cuTUqVMXLly4evXqrVu3Hjx48PTp09evX//VHy9fvnz8+PG9e/du3Lhx+fLls2fPnjhx4tChQ3v37t2+ffumTZv6+vpWrFghbXvvRKAH9+DBg4cMGTJ06ND/DIR7I8aLSQcHm128ePH06dNHjx4FEdDNmzeHhp3QQMau9oYAjj179hw8ePDYsWNnzpyx8Nq1a7dv38bk2bNnYfL27dtXr14hZhBJExCW3xIyWb5t27b23iLoAx3W4cOHjxw5cvTo0X8OhHsjxj0NjQ4Ojx49unPnjm06aFB9/fr1q1evXrZsGRpi6dKlq1atwg3DHTt2FBOrrE1NyK8IL168ePPmDSauKcv9+/c9NSdk2nsXehoPGzZs1KhRY8eOHTdu3IQJE/47EO6NGPfUHDM7ONjm+fPnoXH9+nVQSMtUwEUtwq9du1ZBEEhBUAqT1GT37t2k5a6TJ0+yzaVLl1IWoJ88edIkk8o8fPiwtXGhJ+2IESOIPX78+IkTJ06ePHnq1KnTpk2bPn26q3sjxj01x0zzmxykfvfunaud4uCbN29euXLl/Pnz7A6WgoCIycaNGzEBHQE0BI9x15o1a9RKxVKWkOF+ZFIZCe/evUsjNlMQe/2iPUxjxoyh9KRJkyCeOXNmb2/v7Nmz58yZ4+reiHFPzTHT/GYdgFYB8qDBwSVSCqIXdTkmrBUHk5yFIIY7ZWmSYblUBmGnwr59+zSxtVWc8GkR0JE8zRV0hWzKlCkzZsyYNWvW3Llz58+fL+/y5ctd3Rsx7qk5ZppvlbUyyKOJYc15EnlSkGJCPFvGvtxF2pQlZAi/YcMGLY6MyqARMrY2gmTx0Vepz+HDh3vKPJzNG9SFj97z5s2TC/UU0dW9EeOemmOm+VaVkZrCwApxmKhJjhQ33MXNSBYZ3rCQrpqYxpSOzRQnfDhNQdAIGYFeUWoRIKGzRXfyN4fQGEo2hVsDpXVc3Rsx7qk5ZppvlbUyyEMPLcsksSz3q0lasFxb56OcKmOQ8fBROpPxIUGcJhXVyOzspx1KSpTm4S6s4ose1edj56MTRo9yOZ/gDavUHz58+Pjx46dPn1zdGzHuqTlmmm+VtTLIk8qyRMxqpmMOmpDRBjkcwZUHdATwQSZ83GOoaNjijJISOYU0D8sRpVg51lIoO/bEP854nnbO9Pb28jrGlIP48+fPX758+fr1q6t7I8Y9NcdM862yNi7SjhSiU9nUNjazJXswSfGBCTL4ZGtSUudiFUr+xMrZYAJPNokRRblaBLxltaOTniv4W11sKRfV4f727dv3799d3Rsx7qk5ZppvlbUyyGNQWRWXOxW6ycexSDBmUB+UmA0lJSKqbi5WLKRQ8EEZYqqhZ8INE3yKWLj9QsB578QEQmrzOIf20P/48cPVvRHjnppjpvlNAnnLJnKA6DYzNRzXMp6e05dOEg26detWJ49ChZhasTvvpVyskoo16ZGc8GGY0ilINytAeEDBdWiAbg4OApkmK+Mi3MxUsdBL0TBUNwyVrkgyLUOGZ2pYVLvZA6xiSxvbHghQABKEhy/0wMVQQI9DRViFcCIjHpkZqiJrJQltObt5CvGAoBCdBMHIJphEoE1LoR8EaUU4F23BYCHf5C/Ux74C6GiR6OZ7gHHV9J/D8doMNqjQ2c1gkgTDNCMyJYjVzTdxP4xWtCH8PxDtnfvD9s1y/W/FEqmbSOlEqicUsJvfQrVrNssGTYd02KPDGLFENUlMnwZInyT6W+NndPNrtANWoYGjuXF61ImU6OjX4LYqbSqPhNJKjnyO4BxQeb108/eA7YMPrJyboiAGX5DRG0nIAiuYAkgBVZKd+IrNuI4VdYsG42Qvvno5+L5ovYkTzTpwBWfrTicMpZ30CfdGjHtqTlP7JFm4cGHgwkrFAkq/oCQbXzFYIGqAJr68uXzz+AEEWfPFXK8tXxNOQt9L9W7O1q0oDjxNWmeL89EZT+yEeyPGPTWnA72AOBWna+BGUX0ZrPrbQVRASdjx+dCEGHy+GpyBor6LRL5h82nU3jsRDjoyNGgMq7dswr2RQDenA72IDQgcxA6fSJsP7IKbb2yKBis0BbT5DecDzldQfcBZlS9TnxX5MuUocrT3roApNIpJM4JbZFp7zUBwLZkDuj4861dBEEdaEGH1wyBwEWh+RdcPA16SR8XkJAdRSEMgtVVhdebPHpuJmqHibEpLj/mBKxhaLy5YsIDF9aIu5Gy2loLe5hdukmgvu9KJKwIaLOBAhLV+yhRiE/IPH0sCVxKpJNQkktsiWHOsaS14tFkOBvGTgDX4cS1+hd4MuKG3wGIpCrpGRFgLWm7jJm5C8gYPUBfo4A5oRfBjMhqrD8IR2IEDcdTV8RqJTMQquAKYyOep89RM89sEtFfkh8wzpWmiz0qUpLYsqqtYoDN3fFL/8ODg4CY2wyCDUpRmDA4G2o5Ak0zBKUIX+YM4R5mrezCAIShTELdU435twKVtAtDAhJB5clHa4iZ6WRCzmS3VyvqoDroDhBOghBXi/E+F3lqwcEfsbBSly6WS26LeGOpcFrXdv9a5RQAaeSMGlJLKIl2hVzLpCICkXGxKRYlKdW5hkvqvlpH4xBwzzbeqKRCIXmq2CG73Zc6AVhwlUijLkSdBfk+XM9NLbtoEiMoY7BH54YZeakxoL6+9IUhSMnCwjH7yMYxE79+/L+iOEQrZlWAygwJQ9VWkCe6Ibbxpy6hToMnc8U/F1NbWimCjFgE9pEwgAgquvDbgH8LYUk2iPfRsx/FSQxnPUF1I6k+qF/TKSXIFTEcFN3PSqHDHIXRh6I6qgsuNcBdolGgHQ9She4sAZPBBqUdJEvldbWNETT21R9CDqJTSKV+EVwSpdYLtGV3ekgN0DuRs0F3TS3GjTojeTdwRRc4cADI7taqk6aKcllDB1tfX9zfQVKP5icJYzwAAAABJRU5ErkJggg==

superpose 2 images

data n Decimate every 'n' points range 1 data rows n range : k 1 : i range u k el data i 1 el : k k 1 + : 21 line for u 51 line :

data n Decimate 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 51 line :

Observe:

You may escape the first data point(s) by setting the start range [2].

This happens for data from source that erroneously star [0,0].

If this pair is maintained, it may render the fitting session difficult

or simply not possible. Over populated data set are detrimental to

fit a model function ... thus decimation.

]]>

Nesting arrays

v 310 + 14 - 510 * 31 mat :

w 2 v * 72 i * 18 - 31 mat + :

M 01230253133 mat :

a 1 el v :

a 2 el M :

a 3 el w :

a 133 - 5031 mat 01230253133 mat 3348231 mat 31 mat

a transpose 133 - 5013 mat 03510322133 mat 3348213 mat 13 mat

M M 1 -^* 10001000133 mat

M 1 -^M * 10001000133 mat

v1 aa bb 21 mat :

v2 cc dd 21 mat :

Dot product v1 v2 * aa cc * bb dd * +

w 10 0.417 0.583 22 mat :

sum 01213 mat transpose :

wht 11113 mat transpose λ * :

λ 235 :

The rules are respected ... the Dot vector
comes first as scalar then  multiplied by vector
This rule is not commutative sum w 1 col w 2 col * wht * + 57.131085 58.131085 59.131085 31 mat

NOT commutative sum wht w 1 col w 2 col * (* + #

# # # alg

w 1 col 1 0.417 21 mat

w 2 col 0 0.583 21 mat

implicit of "dot product" => w 1 col w 2 col * 0.243

10 * 0.417 0.583 * + 0.243

w 1 col w 2 col * wht * 57.131 57.131 57.131 31 mat

v 310 + 14 - 510 * 31 mat :

w 2 v * 721 - sqrt * 18 - 31 mat + :

v 133 - 5031 mat

w 336 - 2 i * + 8231 mat

Complex magnitude w w * sqrt 88.572108 0.135483 i * -

Absolute magnitude w w * sqrt abs 88.572212

V

V 5.482 - 3.823 5.299 - 3.645 5.117 - 3.469 4.934 - 3.292 4.751 - 3.117 4.568 - 2.943 4.386 - 2.769 4.203 - 2.597 4.02 - 2.426 3.837 - 2.257 3.655 - 2.09 3.472 - 1.925 3.289 - 1.762 3.106 - 1.602 2.924 - 1.446 2.741 - 1.294 2.558 - 1.146 2.376 - 1.004 2.193 - 0.868 2.01 - 0.739 1.827 - 0.618 1.645 - 0.505 1.462 - 0.403 1.279 - 0.31 1.096 - 0.229 0.914 - 0.16 0.731 - 0.103 0.548 - 0.058 0.365 - 0.026 0.183 - 0.006 302 mat 11012 -^* 2.28 106 -^* 0.091 0.688 0.183 0.971 0.274 1.186 0.365 1.365 0.457 1.52 0.548 1.658 0.64 1.782 0.731 1.895 0.822 1.999 0.914 2.094 1.005 2.181 1.096 2.261 1.188 2.335 1.279 2.403 1.371 2.465 1.462 2.521 1.553 2.572 1.645 2.616 1.736 2.655 1.827 2.688 1.919 2.714 2.01 2.734 2.101 2.746 2.193 2.749 2.284 2.743 2.376 2.725 2.467 2.69 2.558 2.634 2.65 2.537 302 mat 2.741 2.057 2.65 1.722 2.558 1.528 2.467 1.373 2.376 1.241 2.284 1.124 2.193 1.018 2.101 0.922 2.01 0.833 1.919 0.751 1.827 0.675 1.736 0.604 1.645 0.539 1.553 0.478 1.462 0.421 1.37 0.368 1.279 0.32 1.188 0.275 1.096 0.234 1.005 0.196 0.914 0.162 0.822 0.131 0.731 0.103 0.64 0.079 0.548 0.058 0.457 0.04 0.365 0.026 0.274 0.014 0.183 0.006 0.091 0.002 302 mat 11012 -^* 2.28 106 -^* - 0.091 0.69 - 0.183 0.978 - 0.274 1.201 - 0.365 1.391 - 0.457 1.56 - 0.548 1.716 - 0.64 1.861 - 0.731 1.998 - 0.822 2.129 - 0.914 2.255 - 1.005 2.377 - 1.096 2.495 - 1.188 2.61 - 1.279 2.723 - 1.371 2.834 - 1.462 2.942 - 1.553 3.05 - 1.645 3.155 - 1.736 3.26 - 1.827 3.363 - 1.919 3.465 - 2.01 3.567 - 2.101 3.667 - 2.193 3.767 - 2.284 3.867 - 2.376 3.965 - 2.467 4.064 - 2.558 4.161 - 2.65 4.259 - 302 mat 41 mat :

'V 'is a global vector of 4 nests 'stack(,,,) will globaly stack from indexing

stack exploded nests N V 1 el V 2 el V 3 el V 4 el stack :

V 1 el V 2 el V 3 el V 4 el 4 1 sys N V 1 el V 2 el V 3 el V 4 el stack

V rows V 1 el rows N rows 31 mat 43012031 mat

Select 'j' row in 'V', stack 'n' times V n j stack U V j row : stack n rows c 1 : i 1 n range data c el U : c c 1 + : 21 line for data 41 line Unest the stack v data 1 el : j 2 data rows range v v data j el stack : for v 41 line 31 line :

V 123456789025 mat :

V 32 stack 67890678906789035 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 U 1 j el data j col UnestRow : for v U 1 el : j 2 U cols range v v U j el augment : for v 41 line 21 line :

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 :

M data Explode :

M 32.1014 1.9591 0.2438 25.7124 1.9591 0.1254 0.0155 1.5539 0.2438 0.0155 0.0077 0.1927 25.7124 1.5539 0.1927 22.4646 44 mat

Add col/row to be adpted from replace

M 0 :

N 0 :

M 05204580125152045801251802204580125180245345801251802453204801251802453204055125180245320405500667 mat :

1021 mat style_2 style_3 12 mat ≡

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

0121 mat style_0 style_3 12 mat ≡

1121 mat 4 corners pairwise ≡

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_0 N M 01 replace : style_1 N M 11 replace : style_2 N M 10 replace : style_3 N M 00 replace : 41 line

style_0 0052045801251052045801251802020458012518024530458012518024532040801251802453204055012518024532040550060000000078 mat

style_1 0000000000520458012510520458012518020204580125180245304580125180245320408012518024532040550125180245320405500678 mat

style_2 0000000005204580125105204580125180202045801251802453045801251802453204080125180245320405501251802453204055006078 mat

style_3 0520458012510520458012518020204580125180245304580125180245320408012518024532040550125180245320405500600000000078 mat

M Venturi

M 12612311611010397898693991061121191261321391451521581661651591521461391331261191131071019589889095101107112118123129135140145151156162165165160154149143138132127121116110105999490126123117111105999287899510110711412012613213914515115816416716115614914313613012411811210610195908889949910511011512012513113614114615115716216516416115615114514113513012511911510912612411911210710196898691971031081151201261311381431501551611661651591531471411361301241181121071029792888891961011061111151201251301361401451491541591631641631601541501451401351301261251201141091039892878793981041091151201251311361421471531581631661641581531471421361311261201151091051009591888992961011051101141191231281321371411461501551591631641631611561521261251211161111061009690868893981041081141191231291341391441491541591641661641591541491441391341291231181141091041009692898990949810210711111511912312713213614014414815315716016412612612211711210810399948986899398103107112117121126130135139144149154158163166165161156152147142138133129124119115110106102989591898990949710110510911311712112412813213714014412612612311911411010610198938986889296101105109114118122126130134139143147151156160164166164161156152148144140135131127123118115110107103999592908990919598101105109112115119122126126124121117112109105101979489868791949810210611011411812112512913213714014414815215616016316516416315915515114814414013613212812512111711410910610399969491909191929598101126126125122119115112108105101989591878688929598102105109112115119122126129132136139143146150153157160162165164163161157154151147143140137133129126123119116113109107104100979693911261261261231211171141111081051029996939087868891949610010310610911211511812112412713013313613914214514915115415716116216416416316216015615315014714414113813513212912612312011711411212612612612512212011711411210910710410199969391888787899194969910210410710911211511712012312512813013313613814114414614915115415715916116216416316216116015815415215014714414113913712612612612612412212011811511311110910710410299989593918887878890919396981001021041071091111141161181201221251271291311331361381401421441471491511531551571591601611631621611611601261261261261261241231211191171151131111101081061041021009997959391908888888889909293959799101103105106108110112114116117119121123125126128130131133136138139141142144146149150126126126126126126125124122121120118117115114112111110108107105104102101100989795949291909089888889909191929394969799100102103104106107108110112113115116117118120121123125126126126126126126126126125125124123122121120119118117115115114113112111110109108107106104103102102101999897969595939292929291919090909091919293939494959596969798126126126126126126126126126126126126126125125124124123123123122122121120120119119118117117116115115114114113112112111111110110109108108107106106105104103103102102101101100999998989797971261261261261261261261261261261261261261261261261261261261261261261261261261261251251251251251251251251251251251241241241241241231231231231221221221221221211211211201201201201201191191191191181261261261261261261261261261261261261261261261261261261251251251251241241241231231231221221221211211211201201201191191191191181181171171161161151151141141131131131121121121111111111101101101091261261261261261261261261261261261251241231231221211211201191181171161151151141131131121111101091081081071061051041031031021011009999989797969595949493939292929291919191911261261261261261261261251241231221211191181171151141131121111101081071061051031021011009997969594929191919090898989909191929394949596979910110210310410510610710810911112612612612612612512412212111911811611411211110910810610410310110098969593919089888888899091929395969810010110310410610710911111311411611711912012212412612712913013113313513712612612612612512312112011711511311110910710510310199979593918988878789909294969810010210410610811011211411611812012212412612813013213413713814014214414614915015215415615715916012612612612612312111911611311110910710410199979492898787888991949699101103106108111113115118120123125128130132135138140142144147150152154156159160162163163162161160160157154152150126126126124122119115113110107104101999693908786879093959810110410710911211511812112412612913213513814014314614915215415716016216316416316216115815515215014714314113813613212912712512612612612312011611311010710310097949087868891949710110410711011411712012412613013313613914314614915215615916216416516316216015615315014714314013713413012812412111811511210810610310012612612412111811311010710399969288868791959810210610911311712012412713113413814214614915315616016316516416316015615314914514113813413012712412011611310910610399979492909091911261261241201151111081039996918786909497102106109114118122126130134138142146150154158162165165164160156152148143140136131128124119116112108104101989592909090919598101105108112126126123118113109105100969187869195991041081131171211261301351391431481531571611651651631591541501451411371321281231191151101061029995928990899296100103107111114118122125129133126126121117112107102979287869195100105110115120125130134139144149154159163166165160155150145141136131126121116112107103999591888990949810210611011411812212613013413814214615015412612512111511010599948986919510110611111712212713213814314815315816416616415915314814313813212712211711210710298938988899397102106110114119123128132136141145149154158162164163162126125120113108102979186899410010611211712312813414014515115716316716415915314714213613012511911410810399948988899498103108113117121127132136141145150155159163164163160155151146142126124118112106100948887939910511111712312913514114715415916616616015414814213613012411811210710196918889929710310811311812212813313814314815315816316516416015515114514013513012512012612311711110498918691971041101171221291351421491551621671641571511441381311251181121061019590889095100106111116121127133138143148154159164165163157152147142136131126120115110104993364 mat :

Round OFF Matrix/Vector [Smath prior 6179] Mat n Round i 1 Mat rows range j 1 Mat cols range m i j el Mat i j el n round : for for m 21 line :

Generate the symmetric square matrix x y f U V Sym r 1 : i 1 U rows range c 1 : j 1 V rows range Ω r c el U i el V j el f eval : c c 1 + : 21 line for r r 1 + : 31 line for Ω 31 line :

Mesh U, V  within the bounds L, H L H N xd N intervals U 0 : dx H L - N / : i 1 N 1 + (range U i el L dx i * dx - (+ : for U 51 line :

L 1 - :

H 1 :

N 12 :

xx yy f 255 xx 2^yy 2^+ (- 0.25 / exp * :

U L H N xd :

V U :

G xx yy f U V Sym :

N G 3 Round :

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 :

Rep M N 100 replace :

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 Rep