Solve linear equation with 2 unkown and 3 equations in numpy with np.linalg.solve-CodePudding

3 euqations with two unknow have 3 solutions: One solution, infinte solutions, no solution. How would you write this in Numpy to get the solutions? I tried it the way you would do it with 3 unknowns:

import numpy as np

a = np.array([-9,-8, 14])

A = np.array([[ 1, 2, -2],
              [-3,-1, 4],
              ])
x = np.linalg.solve(A, a)
print(x)

But gives an Error, as A is not square. Sadly if I remove the last column of a and A, although i get an answer the system might still have no solution as it might not fit in the third equation.

CodePudding user response：

You do all this using the lstsq method. For example,

a = np.array([-9,-8, 14])
A = np.array([[ 1, 2, -2],
              [-3,-1, 4],
              ])
x,err,rk = np.linalg.lstsq(A.T, a)[:3]
print(x)
print(err)
print(rk)

yields the output

[-3.  2.]
[9.98402083e-31]
2

From the fact that the error is zero (up to numerical precision), you know that this solution is exact, which is to say that A.T@x should exactly equal a. So, the system has at least one solution.

From the fact that the rank is 2 (which matches the number of columns in A.T), we deduce that A.T has a trivial nullspace, which means that any solutions are unique.