5c6596a9-5f24-483e-9dd4-d9aeb0f2ae9f 5 3 12 decimal

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

Linear and nonlinear data fitting

PRIMER D5D7D10

Pod 36.7 - 19.6 - 11.5 - 2.6 - 7.6 15.4 26.1 42.2 60.6 80.1 1510204060100200400760 . . . . . . . . . . 20202020202020202020410 mat transpose :

Vapor pressure = f(Temperature)

points Pod 3 col :

size Pod 4 col :

Temperature [C]

Pressure [atm]

t Pod 1 col :

p Pod 2 col :

p 10 b 1 el b 2 el T / + b 3 el T 10 log * + b 4 el T 2^* +^≡

b - unknown parameters T - temperature [K] p - vapor pressure [atm]

Proposed function for fitting the data:

Linear fitting

p 10 log b 1 el b 2 el T / + b 3 el T 10 log * + b 4 el T 2^* + ≡

Transforming the function in linear by parameters:

x φ 100 x 1.875^x log10 x 2^41 mat :

Temperature [K]

log-pressure

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

T t 273 + :

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

n T length :

n 10

k x φ length :

k 4

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

Cholesky  solver Num. j 1 k range i 1 n range X i j el T i el φ j el : for for Φ X transpose X * : d X transpose y * : b Φ 1 -^eval d * : 41 line

Φ X transpose X * :

d X transpose y * :

b 2.0594 - 0.0055 - 97.3429 0.0023 41 mat

b Φ 1 -^eval d * :

Parameters found

How to minimize this function (NLS problem)??

x f1 10 b x φ *^eval :

SSD p i el T i el f1 - (2^(i 1 n sum :

Sum of squared residuals obtained by linear fitting Could that be reduced by NL fitting?

SSD 53.455

Levenberg-Marquardt based procedure

x α f x y α ν maxiter LMA x α f x α J m α length : n x length : i 1 m range j 1 n range Jac j i el x j el α f (α i el diff : for for Jac 41 line : n T length : m α length : α S0 y i el x i el α f (- (2^(i 1 n sum : I m identity : λ 1 : k 1 maxiter range J x α f x α J : j 1 n range f j el x j el α f : for δ J transpose J * λ I * + (1 -^J transpose * y f - (* eval : α α δ + : α S1 y i el x i el α f (- (2^(i 1 n sum : α S1 α S0 < λ λ ν / : α S0 α S1 : 21 line λ λ ν * : if 61 line for α S1 α 21 mat 81 line :

Damping changing parameter, ν, maximal number of iteration, maxiter (change this if needed)

α 1 0.005 - 100 0.0025 41 mat :

Guess values - from linear fitting

T α f α 1 el α 2 el T / + α 3 el T log10 * + α 4 el T 2^* + :

ν 10 :

maxiter 20 :

coeffs T α f T p α ν maxiter LMA :

coeffs 2 el #

S res 1 el :

S #

NLS values

b res 2 el :

b #

Sum of squared residuals

S #

b S #

Reduced !

i 1 p length range prac i el T i el b f1 (: for

exp T p points size augment :

cal T prac augment :

exp cal 2 1 sys

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

prac #

s2 prac i el p i el - (2^(i 1 n sum n k - / :

s2 #

# exp