Matrix multiplication, solve Ax = b solve for x

Question

So I was given a homework assignment that requires solving the coefficients of cubic splines. Now I clearly understand how to do the math on paper as well as with MatLab, I want to solve the problem with Python. Given an equation Ax = b where I know the values of A and b, I want to be able to solve for x with Python and I am having trouble finding a good resource to do such a thing.

Ex.

A = |1 0 0|
    |1 4 1|
    |0 0 1|

x = Unknown 3x1 matrix

b = |0 |
    |24| 
    |0 |

Solve for x

Answer 1

In a general case, use solve:

>>> import numpy as np
>>> from scipy.linalg import solve
>>> 
>>> A = np.random.random((3, 3))
>>> b = np.random.random(3)
>>> 
>>> x = solve(A, b)
>>> x
array([ 0.98323512,  0.0205734 ,  0.06424613])
>>> 
>>> np.dot(A, x) - b
array([ 0.,  0.,  0.])

If your problem is banded (which cubic splines it often are), then there's http://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.solve_banded.html

To comment on some of the comments to the question: better not use inv for solving linear systems. numpy.lstsq is a bit different, it's more useful for fitting.

As this is homework, you're really better off at least reading up on ways of solving tridiagonal linear systems.

Answer 2

Numpy is the main package for scientific computing in Python. If you are windows user then download it here: http://www.lfd.uci.edu/~gohlke/pythonlibs/#numpy else follow these instructions: http://www.scipy.org/install.html.

import numpy
A = [[1,0,0],[1,4,1],[0,0,1]]
b = [0,24,0]
x = numpy.linalg.lstsq(A,b)

Answer 3

In addition to the code of Zhenya, you might also find it intuitive to use the np.dot function:

import numpy as np
A = [[1,0,0],
    [1,1,1],
    [6,7,0]]
b = [0,24,0]
# Now simply solve for x
x = np.dot(np.linalg.inv(A), b) 
#np.linalg.inv(A)  is simply the inverse of A, np.dot is the dot product
print x

Out[27]: array([  0.,   0.,  24.])