human
Complete the below
def corr_nearest_factor(corr, rank, ctol=1e-06, lam_min=1e-30, lam_max=1e+30, maxiter=1000):

    ''''Find the nearest correlation matrix with factor structure to a
    given square matrix.
    Parameters
    corr : square array
    The target matrix (to which the nearest correlation matrix is
    sought).  Must be square, but need not be positive
    semidefinite.
    rank : positive integer
    The rank of the factor structure of the solution, i.e., the
    number of linearly independent columns of X.
    ctol : positive real
    Convergence criterion.
    lam_min : float
    Tuning parameter for spectral projected gradient optimization
    (smallest allowed step in the search direction).
    lam_max : float
    Tuning parameter for spectral projected gradient optimization
    (largest allowed step in the search direction).
    maxiter : integer
    Maximum number of iterations in spectral projected gradient
    optimization.
    Returns
    rslt : Bunch
    rslt.corr is a FactoredPSDMatrix defining the estimated
    correlation structure.  Other fields of `rslt` contain
    returned values from spg_optim.
    Notes
    A correlation matrix has factor structure if it can be written in
    the form I + XX\' - diag(XX\'), where X is n x k with linearly
    independent columns, and with each row having sum of squares at
    most equal to 1.  The approximation is made in terms of the
    Frobenius norm.
    This routine is useful when one has an approximate correlation
    matrix that is not positive semidefinite, and there is need to
    estimate the inverse, square root, or inverse square root of the
    population correlation matrix.  The factor structure allows these
    tasks to be done without constructing any n x n matrices.
    This is a non-convex problem with no known gauranteed globally
    convergent algorithm for computing the solution.  Borsdof, Higham
    and Raydan (2010) compared several methods for this problem and
    found the spectral projected gradient (SPG) method (used here) to
    perform best.
    The input matrix `corr` can be a dense numpy array or any scipy
    sparse matrix.  The...'''
assistant
#Here is Python program to find the nearest correlation matrix with factor structure to a
.
    (p, _) = corr.shape
    (u, s, vt) = svds(corr, rank)
    X = (u * np.sqrt(s))
    nm = np.sqrt((X ** 2).sum(1))
    ii = np.flatnonzero((nm > 1e-05))
    X[ii, :] /= nm[ii][:, None]
    corr1 = corr.copy()
    if (type(corr1) == np.ndarray):
        np.fill_diagonal(corr1, 0)
    elif sparse.issparse(corr1):
        corr1.setdiag(np.zeros(corr1.shape[0]))
        corr1.eliminate_zeros()
        corr1.sort_indices()
    else:
        raise ValueError('Matrix    type    not    supported')
    def grad(X):
        gr = np.dot(X, np.dot(X.T, X))
        if (type(corr1) == np.ndarray):
            gr -= np.dot(corr1, X)
        else:
            gr -= corr1.dot(X)
        gr -= ((X * X).sum(1)[:, None] * X)
        return (4 * gr)
    def func(X):
        if (type(corr1) == np.ndarray):
            M = np.dot(X, X.T)
            np.fill_diagonal(M, 0)
            M -= corr1
            fval = (M * M).sum()
            return fval
        else:
            fval = 0.0
            max_ws = 1000000.0
            bs = int((max_ws / X.shape[0]))
            ir = 0
            while (ir < X.shape[0]):
                ir2 = min((ir + bs), X.shape[0])
                u = np.dot(X[ir:ir2, :], X.T)
                ii = np.arange(u.shape[0])
                u[(ii, (ir + ii))] = 0
                u -= np.asarray(corr1[ir:ir2, :].todense())
                fval += (u * u).sum()
                ir += bs
            return fval
    rslt = _spg_optim(func, grad, X, _project_correlation_factors)
    root = rslt.params
    diag = (1 - (root ** 2).sum(1))
    soln = FactoredPSDMatrix(diag, root)
    rslt.corr = soln
    del rslt.params
    return rslt