Solving a Linear Systes of Equations Using python

Solving linear systems of equations is straightforward using the scipy command linalg.solve. This command expects an input matrix and a right-hand-side vector. The solution vector is then computed. An option for entering a symmetrix matrix is offered which can speed up the processing when applicable. As an example, suppose it is desired to solve the following simultaneous equations:

We could find the solution vector using a matrix inverse:

However, it is better to use the linalg.solve command which can be faster and more numerically stable. In this case it however gives the same answer as shown in the following example:

>>> import numpy as np
>>> from scipy import linalg
>>> A = np.array([[1,2],[3,4]])
>>> A
array([[1, 2],
      [3, 4]])
>>> b = np.array([[5],[6]])
>>> b
array([[5],
      [6]])
>>> linalg.inv(A).dot(b) #slow
array([[-4. ],
      [ 4.5]]
>>> A.dot(linalg.inv(A).dot(b))-b #check
array([[  8.88178420e-16],
      [  2.66453526e-15]])
>>> np.linalg.solve(A,b) #fast
array([[-4. ],
      [ 4.5]])
>>> A.dot(np.linalg.solve(A,b))-b #check
array([[ 0.],
      [ 0.]])