low_rank_toolbox.cssp.sqdeim

Strong QDEIM - Strong Rank-Revealing QR based DEIM.

Author: Benjamin Carrel, University of Geneva, 2024

Functions

sQDEIM(U[, eta, return_projector, ...])

sQDEIM - Strong RRQR based DEIM of U (size n x k)
low_rank_toolbox.cssp.sqdeim.sQDEIM(U, eta=2, return_projector=False, return_inverse=False, **extra_args)[source]

sQDEIM - Strong RRQR based DEIM of U (size n x k)

Key advantage: the selection of the indexes is guaranteed to satisfy the condition: sigma_{min}(U[p, :])^{-1} <= sqrt(1 + eta * r (n-k))

By default, eta = 2

Parameters:

  • U (ndarray) – Orthonormal matrix of size n x k

  • eta (float) – Bounding factor for R_11^{-1} R_12, must be >= 1

  • mode (str) – Specifies the truncation criterion. Must be ‘rank’ or ‘tol’.

  • param (int or float) –

    The parameter for the chosen mode.

    • If mode is ‘rank’, param is the desired rank k.

    • If mode is ‘tol’, param is the error tolerance.

  • return_projector (bool) – If True, return also the matrix U @ inv(U[S, :])

  • return_inverse (bool) – If True, return also the matrix inv(U[S, :])

Return type:

ndarray | tuple

Returns:

  • p (list) – Selection of m row indices with guaranteed upper bound: norm(inv(U[S,:])) <= sqrt(n-k+1) * O(2^m).

  • P_U (ndarray (n x k) (optional)) – Matrix U @ inv(U[S, :])

  • inv_U (ndarray (k x k) (optional)) – Matrix inv(U[S, :])