r"""Function generating the length-squared (LS) probability distributions for sampling matrix A. Args: m (int): number of rows of matrix

r"""Function used to generate matrix C by performing LS sampling of rows and columns of matrix A. Args: A (array[complex]): rectangular,

r""" Function to reconstruct right-singular vector of matrix A Args: l (int): singular vector index A (array[complex]): rectangul

r""" Function printing out numerical results and running times Args: r (int): number of sampled rows from matrix A c (int): numbe

r""" Function to perform LS sampling of the solution vector :math: '\bm{x}' Args: A (array[complex]): rectangular, in general, complex ma

r""" Function to compute the approximated value for a specific entry of the solution vector :math: '\widetilde{x}_\mathrm{comp}' for the system of

r""" Function to estimate the coefficients :math: '\lambda_l = \langle A_\mathrm{user}^\mathrm{T}, v^l \rangle' Args: A (array[complex]):

r""" Function to compute the matrix element :math: 'A_{\mathrm{user}, \mathrm{comp}}' of the preference matrix A Args: A (array[complex])

r""" Function to estimate the coefficients :math: '\lambda_l = \langle v^l \vert A^\dagger \vert b \rangle' Args: A (array[complex]): rec

r""" Function to solve the the linear system of equations :math:'A \bm{x} = b' using the quantum-inspired algorithm Args: A (array[co

r""" Function to solve the the linear system of equations :math:'A \bm{x} = b' using FKV algorithm and a direct calculation of the coefficients :m

r""" Function to optimize the portfolio allocation vector for different assets for a given expected return Args: A (array[complex]):

r""" Function to compute missing entries of preference matrix row :math: 'A_{\mathrm{user}.}' for the user "user" Args: A (array[complex]

r""" Function to reconstruct right-singular vector of matrix A Args: l (int): singular vector index A (array[complex]): rectangul