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scipy.linalg.interpolative.

# interp_decomp

scipy.linalg.interpolative.interp_decomp(A, eps_or_k, rand=True, rng=None)

  Compute ID of a matrix.

  An ID of a matrix A is a factorization defined by a rank k, a column index array idx, and interpolation coefficients proj such that:

      numpy.dot(A[:,idx[:k]], proj) = A[:,idx[k:]]

  The original matrix can then be reconstructed as:

      numpy.hstack([A[:,idx[:k]],
                                  numpy.dot(A[:,idx[:k]], proj)]
                              )[:,numpy.argsort(idx)]

  or via the routine `reconstruct_matrix_from_id`. This can equivalently be written as:

      numpy.dot(A[:,idx[:k]],
                          numpy.hstack([numpy.eye(k), proj])
                        )[:,np.argsort(idx)]

  in terms of the skeleton and interpolation matrices:

      B = A[:,idx[:k]]

  and:

      P = numpy.hstack([numpy.eye(k), proj])[:,np.argsort(idx)]

  respectively. See also `reconstruct_interp_matrix` and `reconstruct_skel_matrix`.

  The ID can be computed to any relative precision or rank (depending on the value of eps_or_k). If a precision is specified (eps_or_k < 1), then this function has the output signature:

      k, idx, proj = interp_decomp(A, eps_or_k)

  Otherwise, if a rank is specified (eps_or_k >= 1), then the output signature is:

      idx, proj = interp_decomp(A, eps_or_k)

  Parameters:
    A `numpy.ndarray` or `scipy.sparse.linalg.LinearOperator` with rmatvec

      Matrix to be factored

    eps_or_k float or int

      Relative error (if `eps_or_k < 1`) or rank (if `eps_or_k >= 1`) of approximation.

    rand bool, optional

      Whether to use random sampling if A is of type `numpy.ndarray` (randomized algorithms are always used if A is of type `scipy.sparse.linalg.LinearOperator`).

    rng `numpy.random.Generator`, optional

      Pseudorandom number generator state. When rng is None, a new `numpy.random.Generator` is created using entropy from the operating system. Types other than `numpy.random.Generator` are passed to `numpy.random.default_rng` to instantiate a `Generator`. If `rand` is `False`, the argument is ignored.

  Returns:
    k int

      Rank required to achieve specified relative precision if `eps_or_k < 1`.

    idx `numpy.ndarray`

      Column index array.

    proj `numpy.ndarray`

      Interpolation coefficients.