Optimize Common Parameters Across Several Equations Using Least Squares In Python

I would like to optimize common parameters among several equations, but i dont know how to fit them simultaneously. The problem is essentially like this, with four equations to so

Solution 1:

This is a system of 4 equations with 4 unknowns. It can be solved algebraically.

solving for a, b, and c and plugging into the first equation. I'm assuming the (*100) is in the denominator of the equations for a, b, and c.

12 x 100 x T+15 x 100 x T+37 x 100 x T+1750=TT=3.66

Then, plug this value of T into the equations for a, b, and c

Solution 2:

It seems your equations actually have 4 parameters, a, b, c and T, so you have a set of 4 linear equations with 4 parameters:

a + b + c - T = -1750
100 * a - 12 * Y = 0
100 * b - 15 * Y = 0
100 * c - 37 * Y = 0

You can solve this using the coefficients in a matrix:

import numpy as np

a = np.array([[1., 1., 1., -1.],
              [100., 0, 0, -12.],
              [0, 100., 0, -15.],
              [0, 0, 100., -37.]])
b = np.array([-1750., 0, 0, 0])

If there is an analytical solution, you can use

res = np.linalg.solve(a, b)
# res: [  583.33333333   729.16666667  1798.61111111  4861.11111111]

Otherwise (or for a more general case) you can approximate a solution using least-squared algorithm

res, err, _, __ = np.linalg.lstsq(a, b)# res: [  583.33333333   729.16666667  1798.61111111  4861.11111111]