python中的反转矩阵稍微关闭

Question

I'm trying to use this http://www.irma-international.org/viewtitle/41011/ algorithm to invert a nxn matrix.

I ran the function on this matrix

[[1.0, -0.5],

 [-0.4444444444444444, 1.0]]

and got the output

[[ 1.36734694,  0.64285714]

 [ 0.57142857,  1.28571429]]

the correct output is meant to be

[[ 1.28571429,  0.64285714]

 [ 0.57142857,  1.28571429]]

My function:

def inverse(m):
    n = len(m)
    P = -1
    D =  1
    mI = m
    while True:
        P += 1
        if m[P][P] == 0:
           raise Exception("Not Invertible")
        else:
            D = D * m[P][P]
            for j in range(n):
                if j != P:
                    mI[P][j] =  m[P][j] / m[P][P]
            for i in range(n):
                if i != P:
                    mI[i][P] = -m[i][P] / m[P][P]
            for i in range(n):
                for j in range(n):
                    if i != P and j != P:
                        mI[i][j] = m[i][j] + (m[P][j] * mI[i][P])
            mI[P][P] = 1 / m[P][P]
            if P == n - 1: # All elements have been looped through
                break

    return mI

Where am I making my mistake?

Answer 1

https://repl.it/repls/PowerfulOriginalOpensoundsystem

Output

inverse: [[Decimal('1.285714285714285693893862813'), Decimal('0.6428571428571428469469314065')], [Decimal('0.5714285714285713877877256260'), Decimal('1.285714285714285693893862813')]] numpy: [[ 1.28571429 0.64285714] [ 0.57142857 1.28571429]]

from decimal import Decimal
import numpy as np

def inverse(m):
    m = [[Decimal(n) for n in a] for a in m]
    n = len(m)
    P = -1
    D =  Decimal(1)
    mI = [[Decimal(0) for n in a] for a in m]
    while True:
        P += 1
        if m[P][P] == 0:
           raise Exception("Not Invertible")
        else:
            D = D * m[P][P]
            for j in range(n):
                if j != P:
                    mI[P][j] =  m[P][j] / m[P][P]
            for i in range(n):
                if i != P:
                    mI[i][P] = -m[i][P] / m[P][P]
            for i in range(n):
                for j in range(n):
                    if i != P and j != P:
                      mI[i][j] = m[i][j] + (m[P][j] * mI[i][P])
            mI[P][P] = 1 / m[P][P]
            m = [[Decimal(n) for n in a] for a in mI]
            mI = [[Decimal(0) for n in a] for a in m]
            if P == n - 1: # All elements have been looped through
                break

    return m

m = [[1.0, -0.5],

 [-0.4444444444444444, 1.0]]

print(inverse(m))
print(np.linalg.inv(np.array(m)))

My thought process:

At first, I thought you might have lurking floating point roundoff errors. This turned out to not be true. That's what the Decimal jazz is for.

Your bug is here

mI = m # this just creates a pointer that points to the SAME list as m

and here

for i in range(n):
                for j in range(n):
                    if i != P and j != P:
                        mI[i][j] = m[i][j] + (m[P][j] * mI[i][P])
            mI[P][P] = 1 / m[P][P] 
            # you are not copying mI to m for the next iteration
            # you are also not zeroing mI 
            if P == n - 1: # All elements have been looped through
                break

    return mI

In adherence to the algorithm, every iteration creates a NEW a' matrix, it does not continue to modify the same old a. I inferred this to mean that in the loop invariant, a becomes a'. Works for your test case, turns out to be true.