Plot Utilities

b5534cf9-4c0a-46ba-bfd0-ac96149d5453 17 15 12 decimal

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Plot Utilities

vx vy char size clr plotG n vx length : plot vx 1 el vy 1 el char size clr augment : i 2 n range plot plot vx i el vy i el char size clr augment stack : for plot 41 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 :

Bar plot unicolor style, bar width user. data dt BarSize v data 2 col : t v bar t dt - 0 t dt - v t dt + v t dt + 042 mat : j 1 data rows range j 1 ≡ B data j 1 el data j 2 el bar : B B data j 1 el data j 2 el bar stack : if 11 line for B 41 line :

ListCollect

data list ListCollect 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 :

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 :

Levenberg Marquardt's second algorithm . . . Carlos 20200115

x y β Tol βs LMSA Eps Tol : eta1 Eps sqrt : eta2 eta1 : h eta1 : stop 0 : 15 mat m x rows : n β rows : 12 mat β fx i 1 m range ff i el x i el ft y i el eval - : for ff 21 line : β derfh i 1 m range j 1 n range β j el β j el h + : jjh1 β fx i el : β j el β j el 2 h * - : jjh2 β fx i el : jjh0 i j el jjh1 jjh2 - 2 h * / : β j el β j el h + : 61 line for 11 line for jjh0 21 line : A n n Zeros : g n Zeros : J m n Zero : 13 mat Fn 1 Zero : dL 1 Zero : dF 1 Zero : F 1 Zero : 14 mat f β fx : J β derfh : 12 mat A J transpose J * : g J transpose f * : ng g normi : 13 mat F f transpose f * 2 / : mu eta1 A Diag max * : nu 2 : 13 mat stop ¬ ng eta2 ≤ stop 1 : p A mu n identity * + (1 -^g - (* : break try np β normi : nx eta2 β normi + : np eta2 nx * ≤ stop 2 : stop ¬ xnew β p + : fn xnew fx : Jn xnew derfh : Fn fn transpose fn * 2 / : dL p transpose mu p * g - (* 2 / : dF F Fn - : dL 1 el 0 > (dF 1 el 0 > (& β xnew : F Fn : J Jn : f fn : A J transpose J * : g J transpose f * : ng g normi : mu mu 13 / 12 dF 1 el dL 1 el / * 1 - (3^- Max * : nu 2 : 91 line mu mu nu * : nu 2 nu * : 21 line if 71 line if if 41 line if 11 line while βs β 15 round vectorize : 121 line :

m Clear 1 :

3 appVersion 0.98.6179 ≡ Timing 4 min ≡ 21 line

D2 ... D3

0.38 -

D2 00 0.021 2.22 0.04 3.375 0.059 3.789 0.079 3.782 0.099 3.538 0.12 3.164 0.141 2.754 0.164 2.322 0.188 1.919 0.213 1.552 0.237 1.253 0.263 0.991 0.29 0.77 0.318 0.595 0.346 0.456 0.377 0.341 0.406 0.257 0.437 0.192 0.47 0.14 0.509 0.096 0.545 0.068 0.591 0.044 0.645 0.026 0.918 0.002 1.125 0 1.154 0272 mat :

D3 0 10.861 0.021 6.056 0.04 3.327 0.059 1.687 0.079 0.706 0.099 0.133 0.12 0.187 - 0.141 0.338 - 0.164 0.395 - 0.188 0.394 - 0.213 0.362 - 0.237 0.32 - 0.263 0.272 - 0.29 0.226 - 0.318 0.185 - 0.346 0.149 - 0.377 0.118 - 0.406 0.093 - 0.437 0.073 - 0.47 0.056 - 0.509 0.041 - 0.545 0.031 - 0.591 0.022 - 0.645 0.014 - 0.918 0.002 - 1.125 0 1.154 0272 mat :

Error stem plot for QuickPlot unicolor data Error stemX 1 data rows range : i 1 stemX rows range stem i el data i 1 el data i 2 el data i 1 el 022 mat : for i 1 stem rows 1 - (range stem i el stem i el stem i 1 + (1 el 1 el 012 mat stack : for V stem 1 el : j 2 stem rows 1 - (range V V stem j el stack : for V V data data rows row stack : 71 line :

XY D2 :

x β Clear 1

model to D2(x) x ft x β 1 el β 2 el x β 4 el / - exp * + β 3 el x β 5 el / - exp * + (* :

1. selct the support points
2. collect for al_nleqsolve list 1514222715 mat : xy XY list ListCollect : 21 sys

raw XY o 5 black plot : support xy . 14 blue plot : 21 line

support raw 2 1 sys

xy 00 0.079 3.782 0.29 0.77 0.545 0.068 1.154 052 mat

fails 21 min x XY 1 col : y XY 2 col : 21 sys

succeeds 1.8 min x xy 1 col : y xy 2 col : 21 sys

vector of guesses ... β 1101 - 0.1 5 - 15 mat transpose :

Tol 106 -^:

t0 1 time :

α x y β Tol βs LMSA : 0.6716 142.3882 0.5829 - 0.0724 8.147 - 51 mat

1 time t0 - min 1.8

x β Clear 1

x 1012 -^0.51 0.82 1.02 1.23 1.43 1.85 2.05 2.36 4.52 9.22 111 mat :

y 0.78 0.8 0.92 1.14 1.26 1.28 1.22 1.16 1.12 1.08 1.08 111 mat :

algo style L_M OK SS 7251

Fetched by ... LMSA(x,y,β,Tol,βs)... L-M module x ft β 2 el β 1 el β 2 el - 1 x β 3 el / (β 4 el^+ / + β 1 el β 2 el - 1 x β 3 el / (β 4 el^+ (2^/ x β 3 el / (β 4 el^* β 4 el * - :

UV x y augment :

1. selct the support points
2. collect for al_nleqsolve list 136914 mat : new UV list ListCollect : 21 sys

raw UV o 5 black plot : support new . 14 blue plot : 21 line

support raw 2 1 sys

new 11012 -^* 0.78 0.82 0.92 1.43 1.28 2.36 1.12 42 mat

x new 1 col : y new 2 col : 21 sys

β 0.5 11414 mat transpose :

Tol 103 -^:

SS 6179 fails x y β Tol βs LMSA #

SS 7251 0.7773 1.0727 1.2781 4.8009 41 mat

========================== Piecewise LM =========================

x 1012 -^0.51 0.82 1.02 1.23 1.43 1.85 2.05 2.36 4.52 9.22 111 mat :

y 0.78 0.8 0.92 1.14 1.26 1.28 1.22 1.16 1.12 1.08 1.08 111 mat :

β Clear 1

β β 1 el β 2 el β 3 el β 4 el 41 mat :

copy/paste in  ... ia(x,β):=   module collapsed β 2 el β 1 el β 2 el - 1 x β 3 el / (β 4 el^+ / + β 1 el β 2 el - 1 x β 3 el / (β 4 el^+ (2^/ x β 3 el / (β 4 el^* β 4 el * -

n β length : m x length : 21 line

initialise the solver init 111114 mat transpose :

typical for this example tol 0.0001 :

t0 1 time :

L-M Components => algo solves 'β'

x β ia β 2 el β 1 el β 2 el - 1 x β 3 el / (β 4 el^+ / + β 1 el β 2 el - 1 x β 3 el / (β 4 el^+ (2^/ x β 3 el / (β 4 el^* β 4 el * - : ................................... i 1 m range ΦΦ i el x i el β ia y i el - : for β Φ ΦΦ : 21 line ................................... i 1 m range j 1 n range β j el β j el h + : jjh1 x i el β ia y i el - : β j el β j el 2 h * - : jjh2 x i el β ia y i el - : jjh0 i j el jjh1 jjh2 - 2 h * / : β j el β j el h + : 61 line for 11 line for ................................. β derfhφ jjh0 : 71 line

h tol : μ 0.01 : μ1 100 : 31 sys

cont 0 : dnor 1 : update 1 : 31 line dnor tol > update 1 ≡ f init Φ : J init derfhφ : jtj J transpose J * : res f norme (2^: 41 line if 11 line a jtj μ n identity * + : s1 a 1 -^J transpose f * (* : b1 init s1 - (: f1 b1 Φ : SSD f1 norme (2^μ1 / : 51 line SSD res < init b1 : f f1 : 12 mat dnor s1 normi init normi / : cont cont 1 + : Sol cont 1 el cont : z 1 n range Sol cont z 1 + el init z el : 11 line for Sol cont n 2 + el SSD μ1 * : Sol cont n 3 + el μ : Sol cont n 4 + el dnor : μ μ 10 / : update 1 : 101 line μ μ 10 * : update 0 : 21 line if 31 line while 21 line

assign 'β' from solved 'init' β init :

recover @ 'tol' β β 4 round vectorize :

β 0.7773 1.0727 1.2781 4.8009 41 mat

1 time t0 - 22 s *

========= Chemist is reflexive ... solves β4 only =========

Levenberg Marquardt's second algorithm . . . Carlos 20200115

x y β Tol βs LMSA Eps Tol : eta1 Eps sqrt : eta2 eta1 : h eta1 : stop 0 : 15 mat m x rows : n β rows : 12 mat β fx i 1 m range ff i el x i el ft y i el eval - : for ff 21 line : β derfh i 1 m range j 1 n range β j el β j el h + : jjh1 β fx i el : β j el β j el 2 h * - : jjh2 β fx i el : jjh0 i j el jjh1 jjh2 - 2 h * / : β j el β j el h + : 61 line for 11 line for jjh0 21 line : A n n Zeros : g n Zeros : J m n Zero : 13 mat Fn 1 Zero : dL 1 Zero : dF 1 Zero : F 1 Zero : 14 mat f β fx : J β derfh : 12 mat A J transpose J * : g J transpose f * : ng g normi : 13 mat F f transpose f * 2 / : mu eta1 A Diag max * : nu 2 : 13 mat stop ¬ ng eta2 ≤ stop 1 : p A mu n identity * + (1 -^g - (* : break try np β normi : nx eta2 β normi + : np eta2 nx * ≤ stop 2 : stop ¬ xnew β p + : fn xnew fx : Jn xnew derfh : Fn fn transpose fn * 2 / : dL p transpose mu p * g - (* 2 / : dF F Fn - : dL 1 el 0 > (dF 1 el 0 > (& β xnew : F Fn : J Jn : f fn : A J transpose J * : g J transpose f * : ng g normi : mu mu 13 / 12 dF 1 el dL 1 el / * 1 - (3^- Max * : nu 2 : 91 line mu mu nu * : nu 2 nu * : 21 line if 71 line if if 41 line if 11 line while βs β : 121 line :

m Clear 1 :

XY 2 0.8 2.4 0.8 2.8 0.8 2.91 0.8 2.92 0.797 2.93 0.759 2.94 0.664 2.95 0.568 2.96 0.506 2.97 0.472 2.98 0.456 2.99 0.448 3 0.444 3.2 0.444 3.6 0.444 4 0.444 162 mat :

x β Clear 1

escape divide 0 [x=2] xo 2.00000001 :

x XY 1 col : y XY 2 col : 12 mat

Fetched by ... LMSA(x,y,β,Tol,βs)... L-M module x ft 1 β 1 el x xo - (β 2 el x xo - (β 3 el x * -^*^* β 4 el + / eval :

excat fit 1117 1.25 41 mat

β 1117 1.1 14 mat transpose :

Tol 103 -^:

t0 1 time :

Uncertainty x y β Tol βs LMSA 1.00 1.00 17.00 1.25 41 mat

1.002721 0.996774 16.999227 1.248666 41 mat

1 time t0 - 44 s *

Q U 2.8 3.2 2.801 range : j 1 U rows range vy j el U j el ft eval : for U vy augment 31 line :

xy XY . 12 black plot : undocumented from Originator 21 line

Transmute model fit ... for undocumented reaction/reactants

xy Q 2.944 0.622 + 100 blue 1 5 mat 2.944 0.622 o 7 black 1 5 mat 4 1 sys

β 1117 1.25 14 mat transpose :

xo 2 :

x β f 1 β 1 el x xo - (β 2 el x xo - (β 3 el x * -^*^* β 4 el + / :

Improve root accuracy

Locate the point of inflexion diff((f(x,β)),x,2) scaled 0.01

x β f x diff 0.01 x β f x 2 diff * 2.9395 20 2.9395 20 - 2 2 mat 3 1 sys

x f x β f x 2 diff :

a 2.93 : b 2.95 : xn b a + (2 / : 31 line

ε 1012 -^:

Dichotomy n 0 : xn f abs ε > xn f a f * 0 ≤ b xn : a xn : if xn b a + (2 / : n n 1 + : 31 line while 31 line

xn 2.9395326457237

xn f 7.1 1014 -^* -

n 90

2.939533 β f 0.669072