Function lstsq

numpy/linalg/_linalg.py:2418–2563

r""" Return the least-squares solution to a linear matrix equation. Computes the vector `x` that approximately solves the equation ``a @ x = b``. The equation may be under-, well-, or over-determined (i.e., the number of linearly independent rows of `a` can be less than, equal t

(a, b, rcond=None)

@array_function_dispatch(_lstsq_dispatcher)
def lstsq(a, b, rcond=None):
    r"""
    Return the least-squares solution to a linear matrix equation.

    Computes the vector `x` that approximately solves the equation
    ``a @ x = b``. The equation may be under-, well-, or over-determined
    (i.e., the number of linearly independent rows of `a` can be less than,
    equal to, or greater than its number of linearly independent columns).
    If `a` is square and of full rank, then `x` (but for round-off error)
    is the "exact" solution of the equation. Else, `x` minimizes the
    Euclidean 2-norm :math:`||b - ax||`. If there are multiple minimizing
    solutions, the one with the smallest 2-norm :math:`||x||` is returned.

    Parameters
    ----------
    a : (M, N) array_like
        "Coefficient" matrix.
    b : {(M,), (M, K)} array_like
        Ordinate or "dependent variable" values. If `b` is two-dimensional,
        the least-squares solution is calculated for each of the `K` columns
        of `b`.
    rcond : float, optional
        Cut-off ratio for small singular values of `a`.
        For the purposes of rank determination, singular values are treated
        as zero if they are smaller than `rcond` times the largest singular
        value of `a`.
        The default uses the machine precision times ``max(M, N)``.  Passing
        ``-1`` will use machine precision.

        .. versionchanged:: 2.0
            Previously, the default was ``-1``, but a warning was given that
            this would change.

    Returns
    -------
    x : {(N,), (N, K)} ndarray
        Least-squares solution. If `b` is two-dimensional,
        the solutions are in the `K` columns of `x`.
    residuals : {(1,), (K,), (0,)} ndarray
        Sums of squared residuals: Squared Euclidean 2-norm for each column in
        ``b - a @ x``.
        If the rank of `a` is < N or M <= N, this is an empty array.
        If `b` is 1-dimensional, this is a (1,) shape array.
        Otherwise the shape is (K,).
    rank : int
        Rank of matrix `a`.
    s : (min(M, N),) ndarray
        Singular values of `a`.

    Raises
    ------
    LinAlgError
        If computation does not converge.

    See Also
    --------
    scipy.linalg.lstsq : Similar function in SciPy.