<p> plotG(vx,vy,char,size,color) </p>

0828326c-78fc-4e85-86ca-773b1434197a 40 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 :

This section generates the continued fraction on the length of the support points. Copy/paste in new sheet where you will fill K's & X's from source/project.

length of support points j 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 / + / + / + / + / + / + / +

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 Support collects index list select list : 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 :

This section will extract the K's and recovers the X's from the user set of support points as selected in "list". These two sections can be collapsed

vector of Thiele X's & K's X Y 0 Thiele

Thiele triangular table X Y 1 Thiele

==================================== Example ======================================

This section demonstrates how to use this document wrt "continued fraction(s)" from a pure invented function.

x f 11 x + / 1 + :

x f

data x 0.5 - 50 range : i 1 x rows range y i el x i el f eval : for x y augment 31 line :

1. list of support points
2. vectors for Thiele list 123471016 mat : XY data list Support : 21 sys

XY 0.5 - 302 0.5 1.6667 1 1.5 2.5 1.2857 4 1.2 62 mat

support points X XY 1 col : Y XY 2 col : support X Y . 12 black plotG : 31 sys

continued fraction extracted X's & K's Thiele X Y 0 Thiele : 0.5 - 30 0.5 - 0.5 2 - 1 3.3333 1013^* - 2.5 9.4249 1015 -^* - 4 2.6906 1014^* - 62 mat

f x 0.5 - 5 0.45 - range : i 1 x rows range y i el x i el f eval : for x y augment 31 line :

support f 2 1 sys

Thiele 0.5 - 30 0.5 - 0.5 2 - 1 3.3333 1013^* - 2.5 9.4249 1015 -^* - 4 2.6906 1014^* - 62 mat :

truncate/transit

X Thiele 1 col : K Thiele 2 col : j Thiele rows : 31 line

There are two good reasons to truncate/transit: 1. physical data sets from published source are not accurate, they are just "data" that don't need 15 decimals. 2. if you want your project be recompacted in the more convenient J_Fraction, maple(convert(Cfr(K,X,x),confrac,x))will choke at full native K's extracted coefficients. NEXT => we recover the Cfr(K,X,x)from the symbolic wrt X's, K's

Thiele on length of support points 'j' 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 / + / + / + :

f K X x Cfr support 3 1 sys

x Δ x f K X x Cfr - :

0.25 - Δ 1.5 Δ 3.5 Δ 31 mat 2.5 1015 -^* 1.4 1014 -^* 6.2 1014 -^* 31 mat

Define Symbolic ... Optimiz = NONE x collect K X x Cfr confrac x convert maple :

x collect 2 x * 66666000000001 / - 44443555560003999960000000024444355556000133332000000001 / + 296287407496284889111110000066666000000000296287407496309333066668000199998000000001 x 133331999999999133332000000002 / + (* / +

x Δ x f 2 x * 66666000000001 / 1 + 1 x 1 + / + (- :

K X x Cfr x diff

K X x Cfr x 0.5 4 int 4.704

0.25 - Δ 1.5 Δ 3.5 Δ 31 mat 7.5 1015 -^* 4.5 1014 -^* - 1.1 1013 -^* - 31 mat

Advanced Engineering example

step 0 => reduce the data Y

t0 1 time :

WUKALOWITSCH [1958] data 10 999.65 50 988.01 100 958.35 150 917.01 200 864.67 250 798.89 300 712.14 325 654.32 350 574.69 360 527.84 370 450.03 374322122 mat :

TK 10450 10.214777 - 100 4.025304 - 150 6406.546109 200 0.023623 250 3.923232 105^* - 300 0.000572 325 9.306911 105^* - 350 0.000314 360 76987.173789 - 370 0.000284 - 374 35984.735997 - 122 mat :

X data 1 col : Y data 2 col : 21 line

X kg m 3^* ≡ Y °C ≡ 21 line

reduce i 1 data rows range R i el 1 999.9 Y i el - / : for R 21 line :

T TK 1 col : ρ TK 2 col : 21 line

t r ρ 1 el x T 1 el - ρ 2 el x T 2 el - ρ 3 el x T 3 el - ρ 4 el x T 4 el - ρ 5 el x T 5 el - ρ 6 el x T 6 el - ρ 7 el x T 7 el - ρ 8 el x T 8 el - ρ 9 el x T 9 el - ρ 10 el x T 10 el + ρ 11 el x T 11 el + ρ 12 el / + / + / + / + / + / + / + / + / + / + / + :

step 3 => Recast/Reverse Engineer the "reduced"

recast as function of 'x' x kgm3 999.9 1 x r / - :

Optimiz => numeric W t 0.01 3741 range : i 1 t rows range ρ i el t i el kgm3 eval : 11 line for t ρ augment 31 line :

Notes: 1. the triple point of water 273.16 K, 0.01 °C 2. max density @ approx: 4 °C 3. "surfusion" water droplet may be as low as - 10°C 4. any & all kg/m^3 formulas are approximation. 5. this Thiele is of exceptional quality [Jean Giraud]. 6. Thiele is exact @ the support points

step 4 =>Plot discrete, Interpolate kg/m^3 vs °C

XY X Y . 10 black plotG :

Water masse volumique ... kg/m^3 vs °C critical range

322 W XY 3 1 sys

Interpolate in-situ 100 kgm3 200 kgm3 325 kgm3 31 mat 958.35 864.67 654.34 31 mat

1 time t0 - 1 s *

x kgm3 x 0.9 - kgm3

Optimiz => numeric v t 0.01 3741 range : i 1 t rows range specificvolume i el t i el kgm3 eval (1 -^: 11 line for t specificvolume augment 31 line :

support pts specific vol xy i 1 Y rows range spec i el 1 Y i el / : for X spec augment 21 line :

xx xy 1 col : yy xy 2 col : 21 line

uv xx yy . 10 black plotG :

uv v 2 1 sys

1 time t0 - 2 s *

jmGiraud:revisited 2016/06/02

Example from solving a DE

data 0 1.8248 0.5 1.4711 1 1.1667 1.5 0.9105 2 0.6998 2.5 0.5306 3 0.3974 3.5 0.2947 4 0.2167 4.5 0.1583 5 0.1151 5.5 0.0833 6 0.0602 6.5 0.0434 7 0.0312 7.5 0.0224 8 0.0161 8.5 0.0116 9 0.0083 9.5 0.0059 10 0.0043 212 mat :

1. list of support points
2. vectors for Thiele list 14812162016 mat : XY data list Support : 21 sys

XY 0 1.8248 1.5 0.9105 3.5 0.2947 5.5 0.0833 7.5 0.0224 9.5 0.0059 62 mat

The data set is clean from a DE Pharmacokinetics solution.

support points X XY 1 col : Y XY 2 col : support X Y + 15 blue plotG : 31 sys

X Y 0 Thiele 0 1.8248 1.5 1.6406 - 3.5 3.092 - 5.5 4.3836 7.5 1.6154 9.5 25.4537 - 62 mat

Thiele 0 1.8248 1.5 1.6406 - 3.5 3.092 - 5.5 4.3836 7.5 1.6154 9.5 25.4537 - 62 mat :

X Thiele 1 col : K Thiele 2 col : j Thiele rows : 31 line

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 / + / + / + / + / + :

Define Symbolic ... Optimiz = NONE x collect K X x Cfr confrac x convert maple :

x collect 10000 x * 227107 / - 354073830321629257887947245000 / + 4436354723767371525780477407292840790174851075000000000 x 1200511860845187069837575459000883939221345043565839259056542954411805325000000 / + 47365622891938288553820165038618001990915075900263988279336842568318033368452590145858536367075767198443661393155410551757812500000 x 134783498605681294750088376856231189912445366753918055671121875000 / - (* / + (* / -

β 10000227107 / 354073830321629257887947245000 / 1200511860845187069837575459000883939221345043565839259056542954411805325000000 / 134783498605681294750088376856231189912445366753918055671121875000 / 4436354723767371525780477407292840790174851075000000000 / 47365622891938288553820165038618001990915075900263988279336842568318033368452590145858536367075767198443661393155410551757812500000 / 61 mat : 0.044 1.373 3.4793 0.7097 15.1494 26.2655 61 mat

Cfr(K,X,x) converted => J(β,x) β x J β 1 el x * - β 2 el + β 5 el x β 3 el + β 6 el x β 4 el - / + / - :

data data 1 col data 2 col . 12 black plotG :

β x J x 010 int 3.5661

x D β x J x diff :

x f 1.333 :

x c 10 x ≤ (x 10 ≤ (& cases :

β x J x f - 0 ≡ x 01 solve 0.7238

J x 010 0.1 range : i 1 x rows range J i el β x i el J eval : for x J augment 31 line :

data J support 0.724 1.333 + 20 magenta 1 5 mat 4 1 sys x D x c *