Numpy matrix determinant precision problems

Question

I am trying to write a script on python to determine a matrix determinant using Gauss method. It works correctly, but the precision isn't enough for me. My code is:

import scipy.linalg as sla
import numpy as np
def my_det(X):
    n = len(X)
    s = 0
    if n != len(X[0]):
        return ValueError
    for i in range(0, n):
        maxElement = abs(X[i][i])
        maxRow = i
        for k in range(i+1, n):
            if abs(X[k][i]) > maxElement:
                maxElement = abs(X[k][i])
                maxRow = k
        if maxRow != i:
            s += 1
        for k in range(i, n):
            X[i][k], X[maxRow][k] = X[maxRow][k], X[i][k]
        for k in range(i+1, n):
            c = -X[k][i]/X[i][i]
            for j in range(i, n):
                if i == j:
                    X[k][j] = 0
                else:
                    X[k][j] += c * X[i][j]
    det = (-1)**s
    for i in range(n):
        det *= X[i][i]
    return det

And I have tests for this code:

for x in range(10):
X = np.random.rand(3,3)
if np.abs(my_det(X) - sla.det(X)) > 1e-6:
    print('FAILED')

My function fails all tests. I tried Decimals, but it didn't help. What's wrong?

Answer 1

The reason why the code fails the test condition, abs(my_det(X) - sla.det(X)) < 1e-6, is not due to lack of precision but rather the change in sign brought about the unintended side-effect of my_det mutating X:

X[i][k], X[maxRow][k] = X[maxRow][k], X[i][k]

This row swapping changes the sign of the determinant. The code uses s to adjust for the change in sign, but X itself is altered in a way which changes the sign of the determinant.

So the X passed to my_det is not the same as the X which is subsequently passed to sla.det. Here is an example where the alteration of X changes the sign of the determinant:

In [55]: X = np.random.rand(3, 3); X
Out[55]: 
array([[ 0.38062719,  0.41892961,  0.88277747],
       [ 0.39881724,  0.00188804,  0.79258322],
       [ 0.40195279,  0.3950311 ,  0.32771527]])

In [56]: my_det(X)
Out[56]: 0.098180005266934267

In [57]: X
Out[57]: 
array([[ 0.40195279,  0.3950311 ,  0.32771527],
       [ 0.        , -0.39006151,  0.46742438],
       [ 0.        ,  0.        ,  0.62620267]])

In [58]: sla.det(X)
Out[58]: -0.09818000526693427

---

You can fix the problem by making a copy of X inside my_det:

def my_det(X):
    X = np.array(X, copy=True)  # copy=True is the default; shown here for emphasis
    ...

Thus, subsequent changes to X within my_det no longer affect the X outside of my_det.

---

import scipy.linalg as sla
import numpy as np


def my_det(X):
    X = np.array(X, dtype='float64', copy=True)
    n = len(X)
    s = 0
    if n != len(X[0]):
        return ValueError
    for i in range(0, n):
        maxElement = abs(X[i, i])
        maxRow = i
        for k in range(i + 1, n):
            if abs(X[k, i]) > maxElement:
                maxElement = abs(X[k, i])
                maxRow = k
        if maxRow != i:
            s += 1
        for k in range(i, n):
            X[i, k], X[maxRow, k] = X[maxRow, k], X[i, k]
        for k in range(i + 1, n):
            c = -X[k, i] / X[i, i]
            for j in range(i, n):
                if i == j:
                    X[k, j] = 0
                else:
                    X[k, j] += c * X[i, j]
    det = (-1)**s
    for i in range(n):
        det *= X[i, i]
    return det


for i in range(10):
    X = np.random.rand(3, 3)
    diff = abs(my_det(X) - sla.det(X))
    if diff > 1e-6:
        print('{} FAILED: {:0.8f}'.format(i, diff))

---

Also note that dtype matters:

In [88]: my_det(np.arange(9).reshape(3,3))
Out[88]: 6

while the correct answer is

In [89]: my_det(np.arange(9).reshape(3,3).astype(float))
Out[89]: 0.0

Since my_det uses division (in c = -X[k, i] / X[i, i]), we need X to have floating point dtype so that the / performs floating point division, not integer division. So to fix this, use X = np.asarray(X, dtype='float64') to ensure that X has dtype float64:

def my_det(X):
    X = np.array(X, dtype='float64', copy=True)
    ...

With this change,

In [91]: my_det(np.arange(9).reshape(3,3))
Out[91]: 0.0

now gives the correct answer.