plotG(vx,vy,char,size,color)

f7e25d8c-1b6b-48b3-8526-0dd597a9b718 27 4 5 decimal

&[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 :

DATA 11227121941316514141151212055301140350 0.9 60 0.25 100 0.0004 200 1.3 106 -^* 3002107 -^* 400 4.4 108 -^* 500 1.1 108 -^* 152 mat :

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 List collects index list select 13567891018 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 :

select the support points XY DATA List :

XY 1122713165151212055301140350 0.9 60 0.25 82 mat

length Thiele j XY rows : 8

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 X 1 el - K 2 el x X 2 el - K 3 el x X 3 el - K 4 el x X 4 el - K 5 el x X 5 el - K 6 el x X 6 el - K 7 el x X 7 el - K 8 el / + / + / + / + / + / + / +

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

data set vx DATA 1 col : vy DATA 2 col : 21 sys

support points X XY 1 col : Y XY 2 col : 21 sys

examine Thiele K's X Y 0 Thiele 1122713 0.0323 - 15 365.1111 - 20 0.307 30 176.4421 - 40 0.0127 - 50 296.6245 60 1.8836 82 mat

Thiele K's to validate Cf(x) K X Y 0 Thiele 2 col :

Cont x 1160 11.5 range : i 1 x rows range v i el x i el Cf eval : for x v augment 31 line :

support X Y + 15 green plotG : data vx vy . 10 black plotG : 21 line

data Cont support 3 1 sys

K 227 0.0323 - 365.1111 - 0.307 176.4421 - 0.0127 - 296.6245 1.8836 81 mat

x Cf confrac x convert maple expand, collect/simplify the coefficients ≡

can't float, can't expand "bigprod" construct J_Frac from x Cf confrac x convert maple ≡

This next section . . .

is stand alone except for the Thiele(vx,vy,0) that has been

extracted independently. We use it to demonstrate how

to proceed with maple(convert(Cf(x),confrac,x)) ... which

will convert the continued fraction [K,X,x] into the J_Frac

that converts/combines the K's & X's into β coefficients.

. . . . Cf(K,X,x) . . . => . . . J(β,x)

]]> plotG(vx,vy,char,size,color)

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 :

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 11012 -^* 1.0475 0.25 0.4963 0.5 0.2755 0.75 0.1703 1 0.1136 1.5 0.0589 2 0.035 3 0.0157 4 0.0086 6 0.0035 102 mat :

vx data 1 col : vy data 2 col : data vx vy . 10 black plotG : 31 line

vx vy 0 Thiele 11012 -^* 1.0475 0.25 0.4536 - 0.5 1.2879 - 0.75 5.8979 1 0.2575 1.5 108.4294 - 2 0.0102 - 3 431.1187 - 4 0.0041 - 6 1343.1271 - 102 mat

Thiele 11012 -^* 1.0475 0.25 0.4536 - 0.5 1.2879 - 0.75 5.8979 1 0.2575 1.5 108.4294 - 2 0.0102 - 3 431.1187 - 4 0.0041 - 6 1343.1271 - 102 mat :

The full native decimals 'K' are NOT compatible with maple(convert(Cf(x),confrac,x)) Conserve less decimals

compatible Thiele system K Thiele 2 col : vx Thiele 1 col : 21 sys

x c 1 vx 1 el x ≤ (x 6 ≤ (& cases :

vx vy 0 Thiele

Thiele for maple(convert(Cf(x),confrac,x)) x Cf K 1 el x vx 1 el - K 2 el x vx 2 el - K 3 el x vx 3 el - K 4 el x vx 4 el - K 5 el x vx 5 el - K 6 el x vx 6 el - K 7 el / + / + / + / + / + / + :

data x Cf x c * 2 1 sys

x Cf confrac x convert maple 6910000 / 3083178490897975000000000000000 x 34956429271172973125981961663569817959400000000 / + 4157756796633422298484854394201887133398247239152095833707774138850191201934400000000000000 x 3884242410686621984633524407594160457479232708321748865746456809793813911512746984947076049325736868814442076513026305598933640000000 / - 130919076715954289329414323632153877191848493823541207335129720269004251407791456145265451421591701363455446843638681019259549059352875374021361385905029627809870913447162024482451019140625000000 x 118803865885992630123789058232268333642246684585229329980698145509706228956697952530724872876765000000 / - (* / - (* / + (* / -

Evaluation of the β's coefficients is manual only

β 6910000 / 34956429271172973125981961663569817959400000000 / 3884242410686621984633524407594160457479232708321748865746456809793813911512746984947076049325736868814442076513026305598933640000000 / 118803865885992630123789058232268333642246684585229329980698145509706228956697952530724872876765000000 / 3083178490897975000000000000000 / 4157756796633422298484854394201887133398247239152095833707774138850191201934400000000000000 / 130919076715954289329414323632153877191848493823541207335129720269004251407791456145265451421591701363455446843638681019259549059352875374021361385905029627809870913447162024482451019140625000000 / 71 mat :

β 0.0069 5.6688948 4.7723154 0.3600328 0.0616636 27.3364279 7.6949506 106 -^* 71 mat

β 0.0069 5.6688948 4.7723154 0.3600328 0.0616636 27.3364279 7.6949506 106 -^* 71 mat :

=> Transit =>

Desired  => visible and exportable β x J β 1 el β 5 el x β 2 el + β 6 el x β 3 el - β 7 el x β 4 el - / - / + / - :

J check glitches x 010 0.01 range (≡ 21 line x 010 0.0125 range : i 1 x rows range v i el β x i el J eval : for x v augment 41 line :

1. glitch
2. NO glitch x 010 0.01 range (≡ x 010 0.0125 range (≡ 21 sys

J data 3 1 sys

About J_Fraction & Rational Fraction

Both algorithms may suffer glitches. Past a visual good fit or from residuals, wrt the decimals of the β's and the discretised fineness that produces no glitch, make sure you don't interpolate in the region of possible glitch. J_Frac & RatFrac, "if" they glitch, they glitch in one region only. Residuals apply if there are more sample points than support points.

........ Cfr(K,X,x)............. converted ....... J(β,x) K 1 el x vx 1 el - K 2 el x vx 2 el - K 3 el x vx 3 el - K 4 el x vx 4 el - K 5 el x vx 5 el - K 6 el x vx 6 el - K 7 el / + / + / + / + / + / + β 1 el β 5 el x β 2 el + β 6 el x β 3 el - β 7 el x β 4 el - / - / + / - ≡