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&[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 11 x / x 10 log x 2^41 mat :

Temperature [K]

log-pressure

i 1 p length range y i el p i el 10 log : for

T t 273 + :

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 eval : for for

Φ X transpose X * eval :

d X transpose y * :

b Φ 1 -^d * eval :

b 216.205 9295.479 - 75.568 - 4.438 105 -^* 41 mat

Parameters found

How to minimize this function (NLS problem)??

T b f1 10 b 1 el b 2 el T / + b 3 el T 10 log * + b 4 el T 2^* +^:

b S p i el T i el b f1 - (2^(i 1 n sum 11 line :

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

b S 141.5349

Levenberg-Marquardt based procedure

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

Guess values - from linear fitting

b 216.2054 9295.4795 - 75.5675 - 4.4384 105 -^* 41 mat

b S 141.5349

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

ν 10 :

maxiter 20 :

res T# b# f1 T p b ν maxiter LMA :

res 7.5793 198.3165 8486.2423 - 69.4179 - 4.3854 105 -^* 41 mat 21 mat

S res 1 el :

S 7.5793

b res 2 el :

b 198.3165 8486.2423 - 69.4179 - 4.3854 105 -^* 41 mat

NLS values

Sum of squared residuals

S 7.5793

b S 7.5793

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 1.2338 5.9652 11.1498 20.6071 38.4599 59.0191 100.2436 201.088 399.4368 760.0888 101 mat

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

s2 1.3