I’ve added two new matrix functions to NTPoly. First, the Moore-Penrose pseudoinverse which can be used to approximate the inverse of a singular matrix. The second is the Polar Decomposition of a matrix.

Both of these functions can be implemented using already existing routines in NTPoly. The Moore-Penrose pseudoinverse can be computed using Hotelling’s method, we just change the criteria for convergence. The following paper by Li Weiguo gives a nice overview:

Weiguo, Li, Li Juan, and Qiao Tiantian. “A family of iterative methods for computing Moore–Penrose inverse of a matrix.” Linear Algebra and Its Applications 438.1 (2013): 47-56.

The Polar Decomposition can be computed using a Newton-Schultz iteration, identical to the matrix sign function. However, in this case it is necessary to transpose the matrix in the iteration. The Polar Decomposition is closely related to the matrix sign function, so I’ve bundled them into the same module.

Higham, Nicholas J. “The matrix sign decomposition and its relation to the polar decomposition.” Linear Algebra and its Applications 212 (1994): 3-20.