Matrix Square Root Newton Step

Problem

Perform one Newton iteration for the matrix square root of A, starting from X:

X_new = 0.5 * (X + A * inv(X))

A and X are 2x2 matrices. Return None if X is singular. For a good starting X this doubles the number of correct digits.

Hint

This is the matrix analogue of the scalar Newton iteration x <- (x + a/x) / 2.

case 1

in: [[[4,0],[0,9]],[[2,0],[0,3]]]

expected: [[2,0],[0,3]]

case 2

in: [[[4,0],[0,9]],[[1,0],[0,1]]]

expected: [[2.5,0],[0,5]]

case 3

in: [[[5,4],[4,5]],[[2,0],[0,2]]]

expected: [[2.25,1],[1,2.25]]

case 4

in: [[[1,0],[0,1]],[[1,1],[1,1]]]

expected: null

Starter codePython

def sqrt_newton_step(A, X):
    # Your code here
    pass

4 test cases · run your code against them in the editor