19c57427-41aa-4279-8636-2ea1c54c5bc5 71 4 5 fraction

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

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 :

Poject collector: fill-in blanks
of project coponents [numerical] Project hello! π 0000000000033 mat 000033 mat :

r c M r c matrix :

33 M 00000000033 mat

M 33 M :

M 00000000033 mat

Project 11 el hello!

Project 12 el 3.1416

N 22 el M :

N 00000000000033 mat 22 mat

N 35 el M :

N 00000000000000033 mat 000000000000000033 mat 35 mat

# # # lim

Mapping an expression

<= Undocumented

M a 1 e x y * 22 mat :

ln M map maple a ln 01 x y * ln 22 mat

m a 1 e x y * 41 mat :

ln m map maple a ln 01 x y * ln 41 mat

m a 1 e x y * 41 mat :

f 2^m map maple a f 2^1 f 2^1 exp f 2^x y * f 2^41 mat

The matrix function

<= Undocumented

x y φ x y * :

<= The times table you can expand at will

Ω 55 φ matrix maple :

Ω 1234524681036912154812162051015202555 mat

Ω 3 row 369121515 mat

Ω 3 col 369121551 mat

V i 1 Ω rows range u i el i : v Ω 3 col : 21 line for u v augment 21 line :

Pts V 1 col V 2 col . 10 black plotG :

0 V Pts 3 1 sys

Two gorgeous utility companions

If you have two matrices [one design, one reference] ... square or rectangular, unwrap each one and plot each one to view as "absolute" . Also, substract one to the other and plot their deviation. Augment each "unwraped", view them element wise ... Much more matching can be done [not offered].

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

V p q Pack j 1 q range i 1 p range i p j * + V rows 1 + < M i j el V i p j * + el : M i j el 0 : if for for u V 1 p 11 submatrix : v M 1 p 1 p 1 - submatrix : u v augment 41 line :

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

V data Unwrap :

______________________ V below this line __________________

Q i 1 V rows range u i el i : v V : 21 line for u v augment 21 line :

mcol 10 :

M data Unwrap 55 Pack (:

V 012342 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 0.59 251 mat

sep mcol 5 - mcol 1022 mat :

M 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

xyQ Q 1 col Q 2 col . 10 black plotG :

The "mcol" separator goes in multiple of of the row(data)... mcol => 5,10,15,20,25

M tr 21100 / -

View the column wise matrix c/w col separator

0 Q xyQ 2 sep 5 1 sys

Analyse matrix via truth table

W 1.22 7.36 7.96 7.94 1.26 2.36 4.74 3.07 5.83 4.11 3.78 8.14 3.64 9.93 8.95 9.73 5.13 3.71 5.78 3.16 54 mat :

x c x 2 ≥ (x 4 ≤ (* :

Truth(M,a,b) ... truth table within [a, b] M a b Truth count 0 : i 1 M rows range j 1 M cols range M i j el a ≥ (M i j el b ≤ (* Z i j el M i j el : 11 line count count 1 + : if for for Z 31 line :

W 24 Truth 00000 2.36 0 3.07 00 3.78 0 3.64 0000 3.71 0 3.16 54 mat

M Min Max Scale Max Min - (M M min - M max M min - / (* Min + :

<= Scale a Matrix

Scaled M 1 - 1 Scale :

Scaled 0.8561 - 0.3921 - 0.0719 0.536 1 0.6241 - 0.7865 - 0.7517 - 0.7169 - 0.7169 - 0.3921 - 0.7633 - 0.9722 - 1 - 0.7053 - 0.1601 - 0.7633 - 0.9675 - 0.9954 - 0.7146 - 0.0719 0.768 - 0.7517 - 0.7285 - 0.7193 - 55 mat

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 :

Scaled 2 Mround 0.86 - 0.39 - 0.07 0.54 1 0.62 - 0.79 - 0.75 - 0.72 - 0.72 - 0.39 - 0.76 - 0.97 - 1 - 0.71 - 0.16 - 0.76 - 0.97 - 1 - 0.71 - 0.07 0.77 - 0.75 - 0.73 - 0.72 - 55 mat

Strip 0's in vector

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 :

v 10305616 mat transpose :

data 000012340000567800910102 mat :

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 :

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

U 0520458012561 mat 520458012518061 mat 458012518024532061 mat 8012518024532040561 mat 12518024532040550061 mat 15 mat :

U UnestH 05458012552080125180204512518024545801802453208012524532040512518032040550065 mat

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

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 :

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 :

XW 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

data lo hi CollectRange collects range select lo hi range : 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 :

Here, submatrix is a better candidate

XW 37 CollectRange 0.525 130.1479 0.7 61.3415 0.875 32.4856 1.05 47.5163 1.225 90.6788 52 mat

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 :

RemToDo: the other diagonal

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 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.

index [indices] of integer search matrix/vector. No match = "idx_not defined"

h 611526672316357553377120 mat transpose :

M 12345648933 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 No match if for index 31 line :

4 M 1 lookup 2321 mat

6 h 1 lookup 1561141 mat

13 h 1 lookup #

Create a 2D matrix of function values

We make it symmetric by equating both directions. You may range V otherwise for an asymmetric option

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 21 line 41 line :

L 1 - :

H 1 :

N 10 :

U L H N xd :

V U :

x y f 1 x 2^y 2^+ (- 0.25 / exp * :

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 :

N x y f U V Sym :

N 0.0003 0.0014 0.0043 0.0097 0.0156 0.0183 0.0156 0.0097 0.0043 0.0014 0.0003 0.0014 0.006 0.0183 0.0408 0.0659 0.0773 0.0659 0.0408 0.0183 0.006 0.0014 0.0043 0.0183 0.0561 0.1249 0.2019 0.2369 0.2019 0.1249 0.0561 0.0183 0.0043 0.0097 0.0408 0.1249 0.278 0.4493 0.5273 0.4493 0.278 0.1249 0.0408 0.0097 0.0156 0.0659 0.2019 0.4493 0.7261 0.8521 0.7261 0.4493 0.2019 0.0659 0.0156 0.0183 0.0773 0.2369 0.5273 0.8521 1 0.8521 0.5273 0.2369 0.0773 0.0183 0.0156 0.0659 0.2019 0.4493 0.7261 0.8521 0.7261 0.4493 0.2019 0.0659 0.0156 0.0097 0.0408 0.1249 0.278 0.4493 0.5273 0.4493 0.278 0.1249 0.0408 0.0097 0.0043 0.0183 0.0561 0.1249 0.2019 0.2369 0.2019 0.1249 0.0561 0.0183 0.0043 0.0014 0.006 0.0183 0.0408 0.0659 0.0773 0.0659 0.0408 0.0183 0.006 0.0014 0.0003 0.0014 0.0043 0.0097 0.0156 0.0183 0.0156 0.0097 0.0043 0.0014 0.0003 1111 mat

A 1221 - 22 mat :

B 1 - 12122 mat :

C 1 - 21422 mat :

I 2 identity :

s f C s I * A - (1 -^(* B * :

s f 517522 mat s 100122 mat * 1221 - 22 mat - /

s f maple Stack empty.

s 5 :

s f 1.6 0.7 2.6 1.7 22 mat

Accumulate in Vector/Matrix

V 109875 π 61 mat :

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 :

Accumulate  a range in selected "col" M s e col Σ Ranged accumulator vy M s e col col submatrix : Σ 1 el vy 1 el : i 2 vy rows range Σ i el vy i el Σ i 1 - el + : for Σ 51 line :

Optimise: numeric V 161 Σ 1019273439 42.1416 61 mat

M 143 Σ 720 / 3150 / 8225 / 21925 / 41 mat

vx 46542844125225260416 mat transpose :

vx 161 Σ transpose 465893133415861838244216 mat

Kind of create fill-in blank

bb 127349251133 mat :

i bb aa i 1 el i : for

It executes the code for every element in the matrix, 6 & 8 & 10 don't exist.

aa transpose 123450709011111 mat

aa i el i 1 aa rows sum 42

bb Unwrap transpose 132245791119 mat

M aa 44 Pack (:

M 1590200037110400044 mat

M M 1 M rows 13 submatrix :

M 159200371140043 mat

12 + 34 + 21 sys 3721 sys

"sys"

M 12345678933 mat :

M 22 vminor 137922 mat

"vminor"

M 22 minor 12 -

"minor"

collect x sqrt x ln x exp 31 sys :

If A is a square matrix, then the minor of the entry in the i-th row and j-th column (also called the (i,j) minor, or a first minor) is the determinant of the submatrix formed by deleting the i-th row and j-th column. This number is often denoted Mi,j.

collect 2 el x ln

# # # # # 51 sys

System 12 + 34 + 21 sys sys2mat :

# # 21 sys sys ≡

System 3721 mat

12 + 34 + 21 sys π e 21 sys e ln E sqrt π sqrt 31 sys 31 mat

MySystem 12 + 34 + 21 sys π e 21 sys e ln e sqrt π sqrt 31 sys 31 mat :

MySystem sys2mat.1 3721 sys 3.1416 2.7183 21 sys 1 1.6487 1.7725 31 mat 31 mat

MySystem 3 el 1 1.6487 1.7725 31 sys

<= ?

Why not nested ?

Nest MySystem sys2mat :

Nest 3721 mat 3.1416 2.7183 21 mat 1 1.6487 1.7725 31 mat 31 mat

Matrix editor

Link to download the matrix editor: http://smath.info/wiki/GetFile.aspx?File=Tools%2fSMathMatrixEditor%20V1.2.zip 1. Download 2. Click "Run" 3. Copy/Paste in SmathSudio directory

http://smath.info/wiki/GetFile.aspx?File=Tools%2fSMathMatrixEditor%20V1.2.zip

M 05204580125520458012518020458012518024545801251802453208012518024532040512518024532040550066 mat :

Order extraction

We demonstrate with a square matix 1. Extract specific col/row. 2. Order col/row

I 2 identity :

v 12365416 mat :

Extract an ordered col data col v Order select order in col ct 1 : i v a ct el data i col el : ct ct 1 + : 21 line for a 41 line :

M 2 v Order 520451801258061 mat

M transpose 3 v Order transpose 20458024518012516 mat

order cols in matrix data OrderCol v 12365416 mat : r 1 : i 1 v cols range c 1 : j v M r c el data i j el : c c 1 + : 21 line for r r 1 + : 31 line for M 41 line :

M OrderCol 05201258045520451801258020458024518012545801253202451808012518040532024512518024550040532066 mat

<= Delete col/row from these two

order rows in matrix data OrderRow v 12365416 mat : r 1 : i v c 1 : j 1 v rows range M r c el data i j el : c c 1 + : 21 line for r r 1 + : 31 line for M 41 line :

M OrderRow 05204580125520458012518020458012518024512580125180245320801251802453204054518024532040550066 mat

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 I 22 replace 0520458012552045801251802045101802454580012453208012518024532040512518024532040550066 mat

Enter 'r' [row], 'c' [col] where the upper left element of the smaller array anchors.

v 12345678910110 mat transpose :

Something is not completely correct in the program but it works .

idx i 1 v rows range idx i el i : for idx 21 line :

n 1 n ≤ (n 9 ≤ (& (≡ working range of 'n' ≡

n v rotate λ v length : p n abs λ mod : p 1 ≡ rot v : λ v length : i 1 n range rot i λ n - (+ el v i el : for rot 21 line i n 1 + λ range rot i n - el v i el : for rot 21 line 31 line if rot 51 line :

<= Pilot the rotation

n 5 :

v transpose n v rotate transpose 21 mat 12345678910110 mat 67891012345110 mat 21 mat

MyRot n v rotate :

XY idx MyRot augment :

rot idx MyRot . 10 black plotG :

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

Circular rotation of vector

5 XY rot 3 1 sys

θ π - 4 / :

θ 0.7854 -

Rot XY θ rot * :

You can rotate a matrix as well. The transit is via Smath built-in CreateMesh that converts the matrix in 3 cols vector suitable to Euler system ... exposed elsewhere.

xyRot Rot 1 col Rot 2 col . 10 black plotG :

Angular rotation of the !rotation! ... a bit circular !

5 Rot xyRot 3 1 sys

Finding the first element in a vector that exceeds a certain treshold

v u 02 π * 0.1 range : i 1 u rows range vy i el u i el sin : for vy 31 line :

v tresh first j 1 : v j el abs tresh ≤ j j 1 + : while j v j el 21 mat 31 line :

v 0.5 first 7 0.5646 21 mat

v 0.875 first 12 0.8912 21 mat

Locating the first occurrence of a specific element in a vector

w v 1500 range : i 1 v rows range vy i el 1000 v i el cos * Ceil : for vy 31 line :

check if occurrence is unique

400 w 1 lookup 39711 mat

v x search j 1 : v j el x ≠ j j 1 + : while j v j el 21 mat 31 line :

w 400 search 39740021 mat

w 700 search 15070021 mat

w 145 - search 8145 - 21 mat

w 400 - search #

<= No match

Delete 'ONE specified' col/row

M c DelColRow N 0 : ct 1 : n 1 M cols range n c ≠ N 1 ct el M n col : ct ct 1 + : 21 line if for Unest v N 1 el : j 2 N cols range v v N j el augment : for v 71 line :

M 05204580125520458012518020458012518024545801251802453208012518024532040512518024532040550066 mat :

M 3 DelColRow 05458012552080125180204512518024545801802453208012524532040512518032040550065 mat

M transpose 2 DelColRow transpose 0520458012520458012518024545801251802453208012518024532040512518024532040550056 mat

Mat Frame i 1 Mat rows 2 + range j 1 Mat cols 2 + range frame i j el : for for frame 21 line :

Frame M Frame :

FRAME Frame M 11 replace :

FRAME 05204580125 5204580125180 204580125180245 4580125180245320 80125180245320405 125180245320405500 88 mat

Insert valued matrix: row/col

M 05204580125520458012518020458012518024545801251802453208012518024532040512518024532040550066 mat :

COL 0000 00000062 mat :

ROW 00 00000 00026 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 :

M COL 3 InsertCol 0520004580125520450080125180204580 12518024545801250018024532080125180002453204051251802450032040550068 mat

M ROW 4 InsertRow 052045801255204580125180204580125180245458012518024532000 00000 0008012518024532040512518024532040550086 mat

something hello :

----------the working project is above this region ------

Insert an addressable system

aa 2 sqrt : bb 2 : cc 3 : dd something : ee 5 : ff 6 : gg 1 exp : hh 8 : ii 9 : jj 10 : kk 11 : ll 2 ln : 121 sys

Elements are collected [from the palette]. What a simplified way, compared to indexed addressing ! We just demonstrate inserting "colelement". As the project progresses, the conclusion follows below.

A M 1613 submatrix :

C M 1646 submatrix :

B aa bb cc dd ee ff gg hh ii jj kk ll 62 mat :

New A B C augment : 05202 sqrt 24580125520453 hello 801251802045805612518024545801251 exp 818024532080125180910245320405125180245112 ln 32040550068 mat

New 0520 1.4142 24580125520453 hello 80125180204580561251802454580125 2.7183 81802453208012518091024532040512518024511 0.6931 32040550068 mat

Jean Giraud: 2015/12/19

Nest Matrix ... Row/Col

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 0520458012561 mat 520458012518061 mat 20458012518024561 mat 458012518024532061 mat 8012518024532040561 mat 12518024532040550061 mat 16 mat

M NestRow 0520458012516 mat 520458012518016 mat 20458012518024516 mat 458012518024532016 mat 8012518024532040516 mat 12518024532040550016 mat 61 mat

un-Nest Matrix ... Row/col

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

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

U M NestCol UnestCol :

V M NestRow UnestRow :

U 05204580125520458012518020458012518024545801251802453208012518024532040512518024532040550066 mat

V 05204580125520458012518020458012518024545801251802453208012518024532040512518024532040550066 mat

Delete 'seletion' col/row

The module DelColRow(M,c) is not suited to delete a selection. Instead, reverse engineer :=> select the conservation. You can pilot the selection in-situ and collapse the module in utilities area above the work sheet

M 05204580125520458012518020458012518024545801251802453208012518024532040512518024532040550066 mat :

data v DeleteListCol collects index list select v : ct 1 : i select a 1 ct el data i col : ct ct 1 + : 21 line for a a collect is nested next:-> unest a V a 1 el : j 2 a cols range V V a j el augment : for V 101 line :

v 13513 mat :

Delete from in-situ

M v DeleteListCol 020805451252080180451252458018032012524540563 mat

M transpose v DeleteListCol transpose 052045801252045801251802458012518024532040536 mat