<p>Tips & Notes</p>

013b26db-864e-4e5e-8bef-dabc0171e27c 200 6 5 fraction

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

Example of Non-linear Fit [Adapted from Smath source]. We will fit a model fnct using the CholeskyMatrixSolver

WS 1 time :

T 236.3 253.4 261.5 270.4 280.6 288.4 299.1 315.2 333.6 353.1 101 mat :

P 1510204060100200400760101 mat :

i 1 P length range y i el P i el log10 : for

y 0 0.699 1 1.301 1.6021 1.7782 2 2.301 2.6021 2.8808 101 mat

The data set looks like the boiling T, P of some chemical mixture ?

Just to show what equation we are fitting i.e: the model function to fit data points

P log10 b 1 el b 2 el T / + b 3 el T log10 * + b 4 el T 2^* + ≡

[1]

Memo 75 x x log10 x 2.236^51 mat

x φ 75 x x log10 x 2.236^41 mat :

n T length :

n 10

k x φ length :

k 4

NOTE of general interest --------------------------------------------------- The function φ(x) is a phantom function to generate the n_rows, k_cols to feed the numerical solver. This solver is of Legendre LS [Least Square], it is due to a rediscovered unpublished work from Cholesky. It can be used for fitting a polynomial of degree 'n' to a noisy data set of at least n+1 data points. Pure polynomial fit is a myth whereas in nature there are few "polynomial phenomenon". Rational fraction is a much different story and better fitter... About the CholeskyMatrixSolver --------------------------------------------- You can tweak the vector of initial guesses φ(x), but you don't have much leaway. It is not a "Minerr" solver like in Mathcad. Hower this example fits the model pretty well with little sweat ! The model is not a polynomial, the 3rd & 4th terms aren't monomials x^2, x^3

Unlike the other "Genfit models" this one is not based on the vector of deviation. Rather on the calculated values of one component. It does not need a vector of "β" guesses

Setup the LS matrix solver -------------

j 1 k range i 1 n range X i j el T i el φ j el : for for

X 75 236.3 2.3735 2.028 105^* 75 253.4 2.4038 2.3709 105^* 75 261.5 2.4175 2.5437 105^* 75 270.4 2.432 2.7414 105^* 75 280.6 2.4481 2.978 105^* 75 288.4 2.46 3.1663 105^* 75 299.1 2.4758 3.435 105^* 75 315.2 2.4986 3.8622 105^* 75 333.6 2.5232 4.3846 105^* 75 353.1 2.5479 4.9785 105^* 104 mat

Φ X transpose X * :

d X transpose y * :

X transpose 75757575757575757575 236.3 253.4 261.5 270.4 280.6 288.4 299.1 315.2 333.6 353.1 2.3735 2.4038 2.4175 2.432 2.4481 2.46 2.4758 2.4986 2.5232 2.5479 2.028 105^* 2.3709 105^* 2.5437 105^* 2.7414 105^* 2.978 105^* 3.1663 105^* 3.435 105^* 3.8622 105^* 4.3846 105^* 4.9785 105^* 410 mat

b Φ 1 -^eval d * :

b 3.6415109 - 0.2884685 - 140.2774592 4.1051667 105 -^* 41 mat

<p>Tips & Notes</p>

____________________ Tips & Notes _______________________

X transpose X * (1 -^eval X transpose * y * Smath eval  does not take chained arguments ... numerical error ≡

X transpose X * (1 -^X transpose * y * 3.6415 - 0.2885 - 140.2775 4.1049 105 -^* 41 mat

β X transpose X * (1 -^X transpose * y * : assigned calculations don't end ! ≡

Φ 1 -^d * 3.6415 - 0.2885 - 140.2775 4.1049 105 -^* 41 mat

Φ 1 -^918.266 102.031 37609.228 - 0.018 - 102.031 11.361 4180.564 - 0.002 - 37609.228 - 4180.564 - 1.54 106^* 0.748 0.018 - 0.002 - 0.748 3.656 107 -^* 44 mat

d 1212.309 4961.522 40.165 5.968 106^* 41 mat

_____________ ********** ____________

x Pcalc 10 b x φ *^eval :

Let recast the model function in [1] to eliminate the log10

i 1 n range Pc i el T i el Pcalc eval : for

T i el Pcalc eval

Δ Pc P - :

Δ Residuals @ experimental points ≡

Pc 0.9778 5.2669 10.2635 19.6919 37.9413 59.2061 101.9069 205.052 402.7108 756.2936 101 mat

P 1510204060100200400760101 mat

T 236.3 253.4 261.5 270.4 280.6 288.4 299.1 315.2 333.6 353.1 101 mat

Δ 0.0222 - 0.2669 0.2635 0.3081 - 2.0587 - 0.7939 - 1.9069 5.052 2.7108 3.7064 - 101 mat

353236 - 117

Prepare a populated export

m 135 :

dx 1 m 1 - / :

dx 0.0075

x fit 10 b x φ *^eval :

<= recast the scalar model

dx i * 235 + (i +

<= Xcol of 'i' rows

U i el fit

<= Ycol of 'i' rows

Uncontrolled wild plot, i.e: a scalar plot

x fit

data MyFit # 31 sys

data MyFit 21 sys

to add lines: coma after 'MyFit'

MyFit discretise x i 1 m 1 - range U i el dx i * 235 + (i + : for U 21 line discretise y i 1 m 1 - range FIT i el U i el fit : for FIT 21 line collect 2 vectors U FIT augment 61 line :

Disabled => reduce WScal's

The program returns directly the two side by side unested vectors U, FIT required to plot in the plotting argument.

Xcol Ycol # # # # 32 mat

T P char size color plotG n T length : plot T 1 el P 1 el char size color augment : i 2 n range plot plot T i el P i el char size color augment stack : for plot 41 line :

data T P o 5 blue plotG :

Tips, how to manage asymmetric plot ---------------------------------- 1. Once you are sure you will have a plot, disable AutoCalculations 2. Make a damned big plot from Insert menu 3. Click left and maintain in the upper right quadrant ... slide left/down to get rid of the X, Y axes, keep sliding up until you see the X values are in target. 4. @ this point, you have no plot because calculations are OFF ... press F9 ... bingo here is some plot components. 5. Now, scale both Y & X as you wish. ----------------------------------------------- If you want to see the two disabled plots, halt work sheet calculation enable the two plots, press F9.

data MyFit 2 1 sys

Compared to other Genfit models this work sheet is slow. The model contains the greedy X^Y.

Disabled => reduce WScal's

Complete plot: data points + the fitted model

data MyFit 2 1 sys

1 time WS - 601500 / sec *

eval('expression') - Converts analytic expression to the numeric notation. SMath uses its symbolic engine and most of the function can be obtained in either analytic or numeric notation. Those functions can be evaluated by either symbolic "→" or numeric equal to "=" operators. If the numeric "=" is used, the result is presented in floating point notation, but first the expression is calculated symbolicaly. Sometimes this might be very time consuming and the speed of calculation can be reduced drastically. If we need just numerical values, eval function could be used in order to avoid symbolic calculation, and the calculation speed could be increased significantly. -------------------- Convince yourself -----------------

x fit 10 b x φ *^eval ≡ 10 sec ≡

x fit 10 b x φ *^≡ 90 sec ≡

data x fit 21 sys Does NOT take mixed arguments i.e: data is NOT scalar in there 21 mat ≡

This work sheet is reconstructed from the Smath forum. Tutorial about the solver and plotting tips added as a primer convenience.

b 3.6415 - 0.2885 - 140.2775 4.1051 105 -^* 41 mat :

x φ 75 x x log10 x 2.236^41 mat :

10 b x φ *^10500000510924556111 x log10 * - (* 577 x * + (* - 41051 x 559250 /^* + 1000000000 /^

<= evaluate symbolic

x Fit 10500000510924556111 x log10 * - (* 577 x * + (* - 41051 x 559250 /^* + 1000000000 /^:

<= Copy/Paste for export

EQ.0

Assign the symbolic to the plot like in Mathcad

x Symbolic 10 b x φ *^: 10500000510924556111 x log10 * - (* 577 x * + (* - 41051 x 559250 /^* + 1000000000 /^

Uncontrolled wild plot, i.e: a scalar plot

x Symbolic

MyData T P augment :

MyData 236.3 1 253.4 5 261.5 10 270.4 20 280.6 40 288.4 60 299.1 100 315.2 200 333.6 400 353.1 760102 mat

----------- ooooooo Read + oooooooo ------------

P(x) is the Padé Normalised rational fraction [leading coef.in denom = 1] = 13 Aops [ArithmeticOperations] J_Frac(x) is P(x) converted in continued fraction = 8 Aops. Symbolic(x) ~ 115 Aops !!! @ lot worse residuals. Computation time is not 1/1 ... 115/13 ------------EQ.0 ~ 0.4 sec ... EQ.1 ~ 0.14 sec -----------

c 193.2367 1.904741 - 5.788059 103 -^* 5.014562 106 -^* - 5.065622 103 -^* - 6.553814 106 -^* 61 mat :

Padé/J_Frac Max/Min residuals @ data points 0.8 + 0.6 - 21 mat ≡

x P c 1 el c 2 el x * + c 3 el x 2^* + c 4 el x 3^* + 1 c 5 el x * + c 6 el x 2^* + / :

<= Padé rational fraction, Mathcad 11.2a GenfitMatrix ...

x J_Frac 0.76514 x * - 291.76 + 51628 x 481.74 - 12306 x 291.19 - / + / + :

<= ... converted J_Fraction, Mathcad/Maple

EQ.1

Uncontrolled wild plot, i.e: a scalar plot

x J_Frac x Symbolic 2 1 sys

Uncontrolled wild plot, i.e: a scalar plot

x P

This work sheet need be reconstructed wrt Printing Bounds

jmGiraud: 2015_07_09

****************

1 time WS - 2.1 sec *

--------------- More plotting stuff --------------

Alternate plots

Spline T P x cinterp 11 sys :

i 1 T length range Spline i 1 + el T i el P i el o 5 red 15 mat : for

Unranged scalar wild plot + data points

Spline

Spline 11 sys Plots by default the silent argument x ≡

Spline Plots both: data + default scalar Spline ≡

Define a scalar fnct in term of it's name algo.

x Spline Spline :

Unranged scalar wild plot + data points

x Spline

With so fwew data points, these two demos add negligible computation load.

315.2 Spline 1 el 200

300 Spline 1 el 104.2

pt 325 :

<= Interpolate & track

sizeX 5 :

sizeY 50 :

xy pt sizeX - pt Spline 1 el pt sizeX + pt Spline 1 el pt pt Spline 1 el pt pt Spline 1 el sizeY - pt pt Spline 1 el sizeY + 52 mat :

xy 320 291.9079 330 291.9079 325 291.9079 325 241.9079 325 341.9079 52 mat

data T P . 14 black plotG :

<= dot of size 14 = nice solid square

interp pt pt Spline 1 el augment :

pt pt Spline put result somewhere 1 = default matrix location 21 mat el augment

interp 325 291.9079 12 mat

Title 2561000 Hello! Jean François 14 black 15 mat :

region pt pt Spline 1 el . 35 #c0c0c0 15 mat :

Unranged scalar plot + data points + cursor

Spline xy data Title region 5 1 sys

show/hide plot arguments Spline xy data Title region 51 ys ≡