Utilities...ListCollect(data,list)...plot(data,char,size,clr)

60bb7a7c-d54b-4197-ae68-1c506ffe6378 62 4 5 decimal

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

Solve/track an f(x,y) ... demo.

composite centered x y axes x y / - :

timing 7.5 min ≡

vertical x y f x y atan :

a 5 :

manual search for demo angle datum 8.08012 :

y x cardioide x 2^y 2^+ a x * - (2^a 2^x 2^y 2^+ (* - :

Strictly demo from compliant 'cardioide'.

X x datum cardioide x 410 solve : 4.6651

∠ 0 X 21 mat 0 datum 21 mat Slope : 1.732

x y track x X - y datum - / - :

complement
normal X datum atan datum X atan 21 sys ° 30.000 60.000 21 sys

cardioide track axes ∠ f * 4 1 sys

conventional trig direction X i datum * + arg ° 60.000

Ydatum 0.5 123456789 9.75 111 mat transpose :

Ω j 1 Ydatum rows range ω j el x Ydatum j el cardioide x 010 solve : for 0512 mat Ydatum ω augment 10012 mat stack 21 line :

X X : Y datum : 12 mat 4.6651 8.0801 12 mat

cardioide track axes ∠ f * 41 sys

Ω 05 0.5 5.4351 1 5.7681 2 6.2174 3 6.4458 4 6.4898 5 6.3601 6 6.0501 7 5.5329 8 4.7438 9 3.5064 9.75 1.8079 100132 mat

You will need 'support' wrt ... ListCollect(data,list) a too abundant imported XY

Ω 10 4 10 2 - 2 2 mat 2 1 sys

1.Plug your Thiele data set 2.Copy/Export Thiele companion.

Thiele below XY Ω : support Ω : 21 sys

Thiele below

Thiele auto-fit this project ...

Utilities...ListCollect(data,list)...plot(data,char,size,clr)

data list ListCollect 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 31 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 :

Thiele(X,Y,Δ) ..... plot(data,char,size,clr)

X Y Δ Thiele ε 1018 -^: n X rows : i 1 n range M i 1 el Y i el : for j 2 n range i j n range dM M i j 1 - el M j 1 - j 1 - el - : dM dM abs ε < ε M i j el X i el X j 1 - el - dM / : if : 21 line for for i 1 n range c i 1 el X i el : c i 2 el M i i el : 21 line for Δ 0 ≡ c M if 61 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 :

j XY rows : 13

Thiele on length of support points 'j' K X x Cfr N j 1 + el K j 1 + el : c j el N j el : a j el K j el : jj j 1 - (0 range c jj el a jj 1 + el : a jj el K jj el x X jj el - c jj el / + : 21 line for a 1 el 51 line :

symbolic expansion of Thiele continued fraction algo style K X x Cfr K 1 el x 4.66507034213009 1 el - K 2 el x 4.66507034213009 2 el - K 3 el x 4.66507034213009 3 el - K 4 el x 4.66507034213009 4 el - K 5 el x 4.66507034213009 5 el - K 6 el x 4.66507034213009 6 el - K 7 el x 4.66507034213009 7 el - K 8 el x 4.66507034213009 8 el - K 9 el x 4.66507034213009 9 el - K 10 el x 4.66507034213009 10 el - K 11 el x 4.66507034213009 11 el - K 12 el x 4.66507034213009 12 el - K 13 el / + / + / + / + / + / + / + / + / + / + / + / +

X support 1 col : Y support 2 col : 12 mat

Thiele K's coefficients @ support points [X,Y] X Y 0 Thiele

Expand Thiele Table @ support points  [X,Y] X Y 1 Thiele

Explode Tiele Table

Diagonal elements are Thiele 'K' coefficients ... column 1 'Y' values @ 'X' support points X Y 1 Thiele 11 mat 5000000000000 5.43509 1.149188 00000000000 5.768059 1.301983 3.272377 0000000000 6.217366 1.642891 3.038268 4.271512 - 000000000 6.445841 2.074918 2.700574 3.497708 - 1.292317 00000000 6.48979 2.684942 2.279012 3.020038 - 1.598116 3.270127 0000000 6.360098 3.676205 1.780756 2.681647 - 1.886953 3.363405 10.72062 000000 6.050123 5.713617 1.20497 2.418489 - 2.158635 3.462933 10.373135 2.87782 - 00000 5.532909 13.135452 0.542287 2.19773 - 2.411054 3.57546 9.825357 2.233978 - 1.553177 0000 4.743804 31.226079 - 0.231658 - 1.997697 - 2.638737 3.713551 9.020715 1.764804 - 1.79692 4.10267 000 3.506358 6.025541 - 1.184714 - 1.794893 - 2.826434 3.911044 7.801319 1.370191 - 1.98988 4.57977 2.095998 00 1.807897 3.054413 - 2.200494 - 1.598795 - 2.899672 4.199447 6.187322 1.047802 - 2.049161 5.544534 1.213707 0.85006 - 002 - 3.01665 - 1.431064 - 2.816457 4.592753 4.536431 0.808513 - 1.933015 7.898098 0.52695 0.637329 - 1.175195 1313 mat 11 mat

Thiele K's continued fraction K X Y 0 Thiele 2 col :

store/isolate ... 9 D generally optimum K K 9 round vectorize eval :

x λ 10 x ≤ (x 10 ≤ (& cases :

Thiele on length of support points 'j' K X x Cfr j K rows 1 - : N j 1 + el K j 1 + el : c j el N j el : a j el K j el : jj j 1 - (1 range c jj el a jj 1 + el : a jj el K jj el x X jj el - c jj el / + : 21 line for a 1 el 61 line :

recast  Cf(x) on the system  [K,X,x] x Cf K X x Cfr :

◄ NUMERIC ▲

x Cf 0 x ≤ ( x 10 ≤ ( & * support 2 1 sys

x D1 x Cf x diff eval :

t0 1 time :

Slow solve x D1 x 3.5 4 solve 3.75

x Cf x 010 int 54.5

1 time t0 - min 3.5

Slow plot D1(x) big Thiele ...

x D1 x λ * 3.75 0 + 40 red 1 5 mat 2 1 sys

1 time t0 - min 7

Thiele takes clean data from collection/import.

A system of 10 pairs is huge for the derivative.

Thiele is a scalar function, command line style.

If the imported data set is too huge, ListCollect..

D1(x), will reveal eventual glitche(s).

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Solving a circle ... bivalent demo

composite centered x y axes x y / - :

vertical x y f x y atan :

Y 1 :

d 4 :

60 ° Y 1.7320508 ≡

x y q x 2^y 2^+ d - :

alternate solve

X x Y q x 0 d 2 / solve : 1.7320508

The bivalent solver produces two alternate solutions,associated with the complementary angles ... atan(x,y),atan(y,x).

x y track x X - y Y - / - :

∠ 0 X 21 mat 0 Y 21 mat Slope : 0.5774

Trigonometric angle ...
depends solve accuracy X i Y * + arg ° 30.0000002

q track ∠ f * axes 4 1 sys

X Y atan ° 60.000000

q track ∠ f * axes 41 sys

Y X atan ° 30.000000

22.0243129 67.9756855 + (° * ° 90

jmG: 20230619