I need help obtaining the values c1,c2.c3 from a least squares fit of the data.

I need help obtaining the values c1,c2.c3 from a least squares fit of the data.

[James Tursa]

What have you done so far? What specific problems are you having with your code?

[Lovie Reid]

Ive plug in

K = [76.5 80.4 83.3 85.7 88.1 90.2 92.0 93.6 95.1 96.5 98.9];

MPa = [0.08 0.12 0.16 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60];

y=log(MPa/3.77);

beyond this im confused as how to properly write out z

Answer 1

[James Tursa]

It appears to me that your equation is linear in c1, c2, c3. Just make a matrix equation out of this and then apply your least squares technique to the matrix. That is, you know everything in that equation except the c1, c2, c3. So just calculate all the values for each set of data and that will give you a matrix equation of the form:

b = A*c

where

each elemet of b is calculated from your data (the ln(Psat/Pc) values)

each row of A is calculated from your data corresponding to the element of b (the x, x^2, ln(Tsat/Tc) values)

c is simply the unknown column vector [c1;c2;c3]

Then use your least squares technique to solve for c.

[Lovie Reid]

okay so far I've plugged in

K = [76.5 80.4 83.3 85.7 88.1 90.2 92.0 93.6 95.1 96.5 98.9 101 103 106.3 109.2 111.8 114.2 116.4 118.4 122.9 127.0];

MPa = [0.08 0.12 0.16 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.60 0.70 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.50 3.00];

y = log(MPa./3.77);

x = (1./132.5 - 1./K);

z = [x x.^2 log(K./132.5) ones(size(x))];

Am I on the right track with this?

[James Tursa]

Make K and MPa column vectors instead of row vectors.

And good programming practice is to avoid "magic" numbers in your code like the 3.77 and the 132.5 values. Instead, use named variables with comments. E.g.,

Pc = 3.77; % Critical Pressure (MPa)
Tc = 132.5; % Critical Temperature (K)

Then use Pc and Tc downstream in your code. It seems a simple thing, but it really helps with the readability and maintainability of your code. And put comments with description and units on every line that has values.

What is the ones(size(x)) for?

[James Tursa]

Lovie Reid Answer moved here:

I've removed the ones(size(x)) I was just using other examples to help, but would this be right for the least squares

a = polyfit(x,y,2);

[James Tursa]

polyfit(x,y,2) would be used if you had a set of x,y coordinates and you wanted to fit that with a 2nd degree polynomial. But you don't have that situation. You have a linear matrix equation with three variables that you are trying to fit. It is linear in the c1,c2,c3 terms. Using your variable names, you have this matrix equation (assuming you make y and x column vectors as I suggested):

y = z * c

You have three unknowns, the c = [c1;c2;c3], and 21 equations. An overdetermined linear system of equations where you know y and z. You are being asked to produce the least squares solution for c.