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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def approximate_spectral_radius(A, tol=0.01, maxiter=15, restart=5, symmetric=None, initial_guess=None, return_vector=False):
 """Approximate the spectral radius of a matrix. Parameters A : {dense or sparse matrix} E.g. csr_matrix, csc_matrix, ndarray, etc. tol : {scalar} Relative tolerance of approximation, i.e., the error divided by the approximate spectral radius is compared to tol. maxiter : {integer} Maximum number of iterations to perform restart : {integer} Number of restarted Arnoldi processes. For example, a value of 0 will run Arnoldi once, for maxiter iterations, and a value of 1 will restart Arnoldi once, using the maximal eigenvector from the first Arnoldi process as the initial guess. symmetric : {boolean} True - if A is symmetric Lanczos iteration is used (more efficient) False - if A is non-symmetric Arnoldi iteration is used (less efficient) initial_guess : {array|None} If n x 1 array, then use as initial guess for Arnoldi/Lanczos. If None, then use a random initial guess. return_vector : {boolean} True - return an approximate dominant eigenvector, in addition to the spectral radius. False - Do not return the approximate dominant eigenvector Returns ------- An approximation to the spectral radius of A, and if return_vector=True, then also return the approximate dominant eigenvector Notes ----- The spectral radius is approximated by looking at the Ritz eigenvalues. Arnoldi iteration (or Lanczos) is used to project the matrix A onto a Krylov subspace: H = Q* A Q. The eigenvalues of H (i.e. the Ritz eigenvalues) should represent the eigenvalues of A in the sense that the minimum and maximum values are usually well matched (for the symmetric case it is true since the eigenvalues are real). References .. [1] Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, editors. "Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide", SIAM, Philadelphia, 2000. Examples -------- 1.0 1.0 """

    if not hasattr(A, 'rho') or return_vector:
        # somehow more restart causes a nonsymmetric case to fail...look at
        # this what about A.dtype=int?  convert somehow?

        # The use of the restart vector v0 requires that the full Krylov
        # subspace V be stored.  So, set symmetric to False.
        symmetric = False

        if maxiter < 1:
            raise ValueError('expected maxiter > 0')
        if restart < 0:
            raise ValueError('expected restart >= 0')
        if A.dtype == int:
            raise ValueError('expected A to be float (complex or real)')
        if A.shape[0] != A.shape[1]:
            raise ValueError('expected square A')

        if initial_guess is None:
            v0 = sp.rand(A.shape[1], 1)
            if A.dtype == complex:
                v0 = v0 + 1.0j * sp.rand(A.shape[1], 1)
        else:
            if initial_guess.shape[0] != A.shape[0]:
                raise ValueError('initial_guess and A must have same shape')
            if (len(initial_guess.shape) > 1) and (initial_guess.shape[1] > 1):
                raise ValueError('initial_guess must be an (n,1) or\
                                  (n,) vector')
            v0 = initial_guess.reshape(-1, 1)
            v0 = np.array(v0, dtype=A.dtype)

        for j in range(restart+1):
            [evect, ev, H, V, breakdown_flag] =\
                _approximate_eigenvalues(A, tol, maxiter,
                                         symmetric, initial_guess=v0)
            # Calculate error in dominant eigenvector
            nvecs = ev.shape[0]
            max_index = np.abs(ev).argmax()
            error = H[nvecs, nvecs-1]*evect[-1, max_index]

            # error is a fast way of calculating the following line
            # error2 = ( A - ev[max_index]*sp.mat(
            #           sp.eye(A.shape[0],A.shape[1])) )*\
            #           ( sp.mat(sp.hstack(V[:-1]))*\
            #           evect[:,max_index].reshape(-1,1) )
            # print str(error) + "    " + str(sp.linalg.norm(e2))

            if (np.abs(error)/np.abs(ev[max_index]) < tol) or\
               breakdown_flag:
                # halt if below relative tolerance
                v0 = np.dot(np.hstack(V[:-1]),
                            evect[:, max_index].reshape(-1, 1))
                break
            else:
                v0 = np.dot(np.hstack(V[:-1]),
                            evect[:, max_index].reshape(-1, 1))
        # end j-loop

        rho = np.abs(ev[max_index])
        if sparse.isspmatrix(A):
            A.rho = rho

        if return_vector:
            return (rho, v0)
        else:
            return rho

    else:
        return A.rho