Instruction:
Can you generate a brief explanation for the following Python 3 code
def solve_discrete_lyapunov(A, B, max_it=50, method="doubling"):
    if method == "doubling":
        A, B = list(map(np.atleast_2d, [A, B]))
        alpha0 = A
        gamma0 = B

        diff = 5
        n_its = 1

        while diff > 1e-15:

            alpha1 = alpha0.dot(alpha0)
            gamma1 = gamma0 + np.dot(alpha0.dot(gamma0), alpha0.conjugate().T)

            diff = np.max(np.abs(gamma1 - gamma0))
            alpha0 = alpha1
            gamma0 = gamma1

            n_its += 1

            if n_its > max_it:
                msg = "Exceeded maximum iterations {}, check input matrics"
                raise ValueError(msg.format(n_its))

    elif method == "bartels-stewart":
        gamma1 = sp_solve_discrete_lyapunov(A, B)

    else:
        msg = "Check your method input. Should be doubling or bartels-stewart"
        raise ValueError(msg)

    return gamma1

Response: r Solve the discrete lyapunov problem.


Instruction:
Can you generate a brief explanation for the following Python 3 code
def solve_discrete_riccati(A, B, Q, R, N=None, tolerance=1e-10, max_iter=500,
                           method="doubling"):
    """
    Solves the discrete-time algebraic Riccati equation

    .. math::

        X = A'XA - (N + B'XA)'(B'XB + R)^{-1}(N + B'XA) + Q

    Computation is via a modified structured doubling algorithm, an
    explanation of which can be found in the reference below, if
    `method="doubling"` (default), and via a QZ decomposition method by
    calling `scipy.linalg.solve_discrete_are` if `method="qz"`.

    Parameters
    ----------
    A : array_like(float, ndim=2)
        k x k array.
    B : array_like(float, ndim=2)
        k x n array
    Q : array_like(float, ndim=2)
        k x k, should be symmetric and non-negative definite
    R : array_like(float, ndim=2)
        n x n, should be symmetric and positive definite
    N : array_like(float, ndim=2)
        n x k array
    tolerance : scalar(float), optional(default=1e-10)
        The tolerance level for convergence
    max_iter : scalar(int), optional(default=500)
        The maximum number of iterations allowed
    method : string, optional(default="doubling")
        Describes the solution method to use.  If it is "doubling" then
        uses the doubling algorithm to solve, if it is "qz" then it uses
        `scipy.linalg.solve_discrete_are` (in which case `tolerance` and
        `max_iter` are irrelevant).

    Returns
    -------
    X : array_like(float, ndim=2)
        The fixed point of the Riccati equation; a k x k array
        representing the approximate solution

    References
    ----------
    Chiang, Chun-Yueh, Hung-Yuan Fan, and Wen-Wei Lin. "STRUCTURED DOUBLING
    ALGORITHM FOR DISCRETE-TIME ALGEBRAIC RICCATI EQUATIONS WITH SINGULAR
    CONTROL WEIGHTING MATRICES." Taiwanese Journal of Mathematics 14, no. 3A
    (2010): pp-935.

    """
    methods = ['doubling', 'qz']
    if method not in methods:
        msg = "Check your method input. Should be {} or {}".format(*methods)
        raise ValueError(msg)

    # == Set up == #
    error = tolerance + 1
    fail_msg = "Convergence failed after {} iterations."

    # == Make sure that all array_likes are np arrays, two-dimensional == #
    A, B, Q, R = np.atleast_2d(A, B, Q, R)
    n, k = R.shape[0], Q.shape[0]
    I = np.identity(k)
    if N is None:
        N = np.zeros((n, k))
    else:
        N = np.atleast_2d(N)

    if method == 'qz':
        X = sp_solve_discrete_are(A, B, Q, R, s=N.T)
        return X

    # if method == 'doubling'
    # == Choose optimal value of gamma in R_hat = R + gamma B'B == #
    current_min = np.inf
    candidates = (0.01, 0.1, 0.25, 0.5, 1.0, 2.0, 10.0, 100.0, 10e5)
    BB = dot(B.T, B)
    BTA = dot(B.T, A)
    for gamma in candidates:
        Z = R + gamma * BB
        cn = np.linalg.cond(Z)
        if cn * EPS < 1:
            Q_tilde = - Q + dot(N.T, solve(Z, N + gamma * BTA)) + gamma * I
            G0 = dot(B, solve(Z, B.T))
            A0 = dot(I - gamma * G0, A) - dot(B, solve(Z, N))
            H0 = gamma * dot(A.T, A0) - Q_tilde
            f1 = np.linalg.cond(Z, np.inf)
            f2 = gamma * f1
            f3 = np.linalg.cond(I + dot(G0, H0))
            f_gamma = max(f1, f2, f3)
            if f_gamma < current_min:
                best_gamma = gamma
                current_min = f_gamma

    # == If no candidate successful then fail == #
    if current_min == np.inf:
        msg = "Unable to initialize routine due to ill conditioned arguments"
        raise ValueError(msg)

    gamma = best_gamma
    R_hat = R + gamma * BB

    # == Initial conditions == #
    Q_tilde = - Q + dot(N.T, solve(R_hat, N + gamma * BTA)) + gamma * I
    G0 = dot(B, solve(R_hat, B.T))
    A0 = dot(I - gamma * G0, A) - dot(B, solve(R_hat, N))
    H0 = gamma * dot(A.T, A0) - Q_tilde
    i = 1

    # == Main loop == #
    while error > tolerance:

        if i > max_iter:
            raise ValueError(fail_msg.format(i))

        else:
            A1 = dot(A0, solve(I + dot(G0, H0), A0))
            G1 = G0 + dot(dot(A0, G0), solve(I + dot(H0, G0), A0.T))
            H1 = H0 + dot(A0.T, solve(I + dot(H0, G0), dot(H0, A0)))

            error = np.max(np.abs(H1 - H0))
            A0 = A1
            G0 = G1
            H0 = H1
            i += 1

    return H1 + gamma * I

Response: Solve the discrete - time algebraic Riccati equation.


Instruction:
Can you tell what is the following Python 3 function doing
def next_k_array(a):
    # Logic taken from Algotirhm T in D. Knuth, The Art of Computer
    # Programming, Section 7.2.1.3 "Generating All Combinations".
    k = len(a)
    if k == 1 or a[0] + 1 < a[1]:
        a[0] += 1
        return a

    a[0] = 0
    i = 1
    x = a[i] + 1

    while i < k-1 and x == a[i+1]:
        i += 1
        a[i-1] = i - 1
        x = a[i] + 1
    a[i] = x

    return a

Response: Given an array a of k distinct nonnegative integers return the next k - array in the lexicographic order.


Instruction:
Explain what the following Python 3 code does
def k_array_rank(a):
    k = len(a)
    idx = int(a[0])  # Convert to Python int
    for i in range(1, k):
        idx += comb(a[i], i+1, exact=True)
    return idx

Response: Returns the ranking of the k distinct nonnegative integers in the lexicographic ordering of the elements of the elements in the array a.


Instruction:
Can you create a Python 3 function that
numbs jit version of k_array_rank.

Response:
def k_array_rank_jit(a):
    """
    Numba jit version of `k_array_rank`.

    Notes
    -----
    An incorrect value will be returned without warning or error if
    overflow occurs during the computation. It is the user's
    responsibility to ensure that the rank of the input array fits
    within the range of possible values of `np.intp`; a sufficient
    condition for it is `scipy.special.comb(a[-1]+1, len(a), exact=True)
    <= np.iinfo(np.intp).max`.

    """
    k = len(a)
    idx = a[0]
    for i in range(1, k):
        idx += comb_jit(a[i], i+1)
    return idx


Instruction:
Can you write a function in Python 3 where it
returns a function that solves the linear equation ax = b directly calling a Numba internal function.

Response:
def _numba_linalg_solve(a, b):
    """
    Solve the linear equation ax = b directly calling a Numba internal
    function. The data in `a` and `b` are interpreted in Fortran order,
    and dtype of `a` and `b` must be the same, one of {float32, float64,
    complex64, complex128}. `a` and `b` are modified in place, and the
    solution is stored in `b`. *No error check is made for the inputs.*

    Parameters
    ----------
    a : ndarray(ndim=2)
        2-dimensional ndarray of shape (n, n).

    b : ndarray(ndim=1 or 2)
        1-dimensional ndarray of shape (n,) or 2-dimensional ndarray of
        shape (n, nrhs).

    Returns
    -------
    r : scalar(int)
        r = 0 if successful.

    Notes
    -----
    From github.com/numba/numba/blob/master/numba/targets/linalg.py

    """
    numba_xgesv = _LAPACK().numba_xgesv(a.dtype)
    kind = ord(_blas_kinds[a.dtype])

    def _numba_linalg_solve_impl(a, b):  # pragma: no cover
        n = a.shape[-1]
        if b.ndim == 1:
            nrhs = 1
        else:  # b.ndim == 2
            nrhs = b.shape[-1]
        F_INT_nptype = np.int32
        ipiv = np.empty(n, dtype=F_INT_nptype)

        r = numba_xgesv(
            kind,         # kind
            n,            # n
            nrhs,         # nhrs
            a.ctypes,     # a
            n,            # lda
            ipiv.ctypes,  # ipiv
            b.ctypes,     # b
            n             # ldb
        )
        return r

    return _numba_linalg_solve_impl


Instruction:
Write a Python 3 script for
numbing jitted function that computes N choose k. Return `0` if the outcome exceeds the maximum value of `np.intp` or if N < 0, k < 0, or k > N. Parameters ---------- N : scalar(int) k : scalar(int) Returns ------- val : scalar(int)

Response:
def comb_jit(N, k):
    """
    Numba jitted function that computes N choose k. Return `0` if the
    outcome exceeds the maximum value of `np.intp` or if N < 0, k < 0,
    or k > N.

    Parameters
    ----------
    N : scalar(int)

    k : scalar(int)

    Returns
    -------
    val : scalar(int)

    """
    # From scipy.special._comb_int_long
    # github.com/scipy/scipy/blob/v1.0.0/scipy/special/_comb.pyx
    INTP_MAX = np.iinfo(np.intp).max
    if N < 0 or k < 0 or k > N:
        return 0
    if k == 0:
        return 1
    if k == 1:
        return N
    if N == INTP_MAX:
        return 0

    M = N + 1
    nterms = min(k, N - k)

    val = 1

    for j in range(1, nterms + 1):
        # Overflow check
        if val > INTP_MAX // (M - j):
            return 0

        val *= M - j
        val //= j

    return val


Instruction:
Explain what the following Python 3 code does
def _integrate_fixed_trajectory(self, h, T, step, relax):
        # initialize the solution using initial condition
        solution = np.hstack((self.t, self.y))

        while self.successful():

            self.integrate(self.t + h, step, relax)
            current_step = np.hstack((self.t, self.y))
            solution = np.vstack((solution, current_step))

            if (h > 0) and (self.t >= T):
                break
            elif (h < 0) and (self.t <= T):
                break
            else:
                continue

        return solution

Response: Generates a solution trajectory of fixed length.


Instruction:
Can you generate the documentation for the following Python 3 function
def _integrate_variable_trajectory(self, h, g, tol, step, relax):
        # initialize the solution using initial condition
        solution = np.hstack((self.t, self.y))

        while self.successful():

            self.integrate(self.t + h, step, relax)
            current_step = np.hstack((self.t, self.y))
            solution = np.vstack((solution, current_step))

            if g(self.t, self.y, *self.f_params) < tol:
                break
            else:
                continue

        return solution

Response: Generates a solution trajectory of variable length.


Instruction:
Here you have a function in Python 3, explain what it does
def _initialize_integrator(self, t0, y0, integrator, **kwargs):
        # set the initial condition
        self.set_initial_value(y0, t0)

        # select the integrator
        self.set_integrator(integrator, **kwargs)

Response: Initializes the integrator prior to integration.


Instruction:
Make a summary of the following Python 3 code
def compute_residual(self, traj, ti, k=3, ext=2):
        # B-spline approximations of the solution and its derivative
        soln = self.interpolate(traj, ti, k, 0, ext)
        deriv = self.interpolate(traj, ti, k, 1, ext)

        # rhs of ode evaluated along approximate solution
        T = ti.size
        rhs_ode = np.vstack(self.f(ti[i], soln[i, 1:], *self.f_params)
                            for i in range(T))
        rhs_ode = np.hstack((ti[:, np.newaxis], rhs_ode))

        # should be roughly zero everywhere (if approximation is any good!)
        residual = deriv - rhs_ode

        return residual

Response: r Compute the residual of the knot sequence of the current knot sequence.


Instruction:
Write a Python 3 function that can
calculate the Lorenz curve for a set of income and wealth.

Response:
def lorenz_curve(y):
    """
    Calculates the Lorenz Curve, a graphical representation of the distribution of income
    or wealth.

    It returns the cumulative share of people (x-axis) and the cumulative share of income earned

    Parameters
    ----------
    y : array_like(float or int, ndim=1)
        Array of income/wealth for each individual. Unordered or ordered is fine.

    Returns
    -------
    cum_people : array_like(float, ndim=1)
        Cumulative share of people for each person index (i/n)
    cum_income : array_like(float, ndim=1)
        Cumulative share of income for each person index

    References
    ----------
    .. [1] https://en.wikipedia.org/wiki/Lorenz_curve

    Examples
    --------
    >>> a_val, n = 3, 10_000
    >>> y = np.random.pareto(a_val, size=n)
    >>> f_vals, l_vals = lorenz(y)

    """

    n = len(y)
    y = np.sort(y)
    s = np.zeros(n + 1)
    s[1:] = np.cumsum(y)
    cum_people = np.zeros(n + 1)
    cum_income = np.zeros(n + 1)
    for i in range(1, n + 1):
        cum_people[i] = i / n
        cum_income[i] = s[i] / s[n]
    return cum_people, cum_income


Instruction:
Can you generate a brief explanation for the following Python 3 code
def gini_coefficient(y):
    n = len(y)
    i_sum = np.zeros(n)
    for i in prange(n):
        for j in range(n):
            i_sum[i] += abs(y[i] - y[j])
    return np.sum(i_sum) / (2 * n * np.sum(y))

Response: r Implements the Gini inequality index


Instruction:
Can you create a Python 3 function that
sets the internal _q and Q attributes.

Response:
def q(self, val):
        """
        Setter method for q.

        """
        self._q = np.asarray(val)
        self.Q = cumsum(val)


Instruction:
Explain what the following Python 3 code does
def draw(self, k=1, random_state=None):
        random_state = check_random_state(random_state)

        return self.Q.searchsorted(random_state.uniform(0, 1, size=k),
                                   side='right')

Response: Returns k draws from the probability set.


Instruction:
Write a Python 3 function that can
return a randomly sampled MarkovChain instance with n states where each state has k states with positive transition probability.

Response:
def random_markov_chain(n, k=None, sparse=False, random_state=None):
    """
    Return a randomly sampled MarkovChain instance with n states, where
    each state has k states with positive transition probability.

    Parameters
    ----------
    n : scalar(int)
        Number of states.

    k : scalar(int), optional(default=None)
        Number of states that may be reached from each state with
        positive probability. Set to n if not specified.

    sparse : bool, optional(default=False)
        Whether to store the transition probability matrix in sparse
        matrix form.

    random_state : int or np.random.RandomState, optional
        Random seed (integer) or np.random.RandomState instance to set
        the initial state of the random number generator for
        reproducibility. If None, a randomly initialized RandomState is
        used.

    Returns
    -------
    mc : MarkovChain

    Examples
    --------
    >>> mc = qe.markov.random_markov_chain(3, random_state=1234)
    >>> mc.P
    array([[ 0.19151945,  0.43058932,  0.37789123],
           [ 0.43772774,  0.34763084,  0.21464142],
           [ 0.27259261,  0.5073832 ,  0.22002419]])
    >>> mc = qe.markov.random_markov_chain(3, k=2, random_state=1234)
    >>> mc.P
    array([[ 0.19151945,  0.80848055,  0.        ],
           [ 0.        ,  0.62210877,  0.37789123],
           [ 0.56227226,  0.        ,  0.43772774]])

    """
    P = random_stochastic_matrix(n, k, sparse, format='csr',
                                 random_state=random_state)
    mc = MarkovChain(P)
    return mc


Instruction:
Given the following Python 3 function, write the documentation
def random_stochastic_matrix(n, k=None, sparse=False, format='csr',
                             random_state=None):
    """
    Return a randomly sampled n x n stochastic matrix with k nonzero
    entries for each row.

    Parameters
    ----------
    n : scalar(int)
        Number of states.

    k : scalar(int), optional(default=None)
        Number of nonzero entries in each row of the matrix. Set to n if
        not specified.

    sparse : bool, optional(default=False)
        Whether to generate the matrix in sparse matrix form.

    format : str, optional(default='csr')
        Sparse matrix format, str in {'bsr', 'csr', 'csc', 'coo', 'lil',
        'dia', 'dok'}. Relevant only when sparse=True.

    random_state : int or np.random.RandomState, optional
        Random seed (integer) or np.random.RandomState instance to set
        the initial state of the random number generator for
        reproducibility. If None, a randomly initialized RandomState is
        used.

    Returns
    -------
    P : numpy ndarray or scipy sparse matrix (float, ndim=2)
        Stochastic matrix.

    See also
    --------
    random_markov_chain : Return a random MarkovChain instance.

    """
    P = _random_stochastic_matrix(m=n, n=n, k=k, sparse=sparse, format=format,
                                  random_state=random_state)
    return P

Response: Returns a randomly sampled n x n stochastic matrix with k nonzero entries for each row.


Instruction:
Can you write a function in Python 3 where it
generates a random non - square stochastic matrix.

Response:
def _random_stochastic_matrix(m, n, k=None, sparse=False, format='csr',
                              random_state=None):
    """
    Generate a "non-square stochastic matrix" of shape (m, n), which
    contains as rows m probability vectors of length n with k nonzero
    entries.

    For other parameters, see `random_stochastic_matrix`.

    """
    if k is None:
        k = n
    # m prob vectors of dimension k, shape (m, k)
    probvecs = probvec(m, k, random_state=random_state)

    if k == n:
        P = probvecs
        if sparse:
            return scipy.sparse.coo_matrix(P).asformat(format)
        else:
            return P

    # if k < n:
    rows = np.repeat(np.arange(m), k)
    cols = \
        sample_without_replacement(
            n, k, num_trials=m, random_state=random_state
        ).ravel()
    data = probvecs.ravel()

    if sparse:
        P = scipy.sparse.coo_matrix((data, (rows, cols)), shape=(m, n))
        return P.asformat(format)
    else:
        P = np.zeros((m, n))
        P[rows, cols] = data
        return P


Instruction:
How would you explain what the following Python 3 function does
def random_discrete_dp(num_states, num_actions, beta=None,
               k=None, scale=1, sparse=False, sa_pair=False,
               random_state=None):
    """
    Generate a DiscreteDP randomly. The reward values are drawn from the
    normal distribution with mean 0 and standard deviation `scale`.

    Parameters
    ----------
    num_states : scalar(int)
        Number of states.

    num_actions : scalar(int)
        Number of actions.

    beta : scalar(float), optional(default=None)
        Discount factor. Randomly chosen from [0, 1) if not specified.

    k : scalar(int), optional(default=None)
        Number of possible next states for each state-action pair. Equal
        to `num_states` if not specified.

    scale : scalar(float), optional(default=1)
        Standard deviation of the normal distribution for the reward
        values.

    sparse : bool, optional(default=False)
        Whether to store the transition probability array in sparse
        matrix form.

    sa_pair : bool, optional(default=False)
        Whether to represent the data in the state-action pairs
        formulation. (If `sparse=True`, automatically set `True`.)

    random_state : int or np.random.RandomState, optional
        Random seed (integer) or np.random.RandomState instance to set
        the initial state of the random number generator for
        reproducibility. If None, a randomly initialized RandomState is
        used.

    Returns
    -------
    ddp : DiscreteDP
        An instance of DiscreteDP.

    """
    if sparse:
        sa_pair = True

    # Number of state-action pairs
    L = num_states * num_actions

    random_state = check_random_state(random_state)
    R = scale * random_state.randn(L)
    Q = _random_stochastic_matrix(L, num_states, k=k,
                                  sparse=sparse, format='csr',
                                  random_state=random_state)
    if beta is None:
        beta = random_state.random_sample()

    if sa_pair:
        s_indices, a_indices = sa_indices(num_states, num_actions)
    else:
        s_indices, a_indices = None, None
        R.shape = (num_states, num_actions)
        Q.shape = (num_states, num_actions, num_states)

    ddp = DiscreteDP(R, Q, beta, s_indices, a_indices)
    return ddp

Response: Generates a DiscreteDP from the given state - action pairs and random values.


Instruction:
Make a summary of the following Python 3 code
def robust_rule_simple(self, P_init=None, max_iter=80, tol=1e-8):
        # == Simplify names == #
        A, B, C, Q, R = self.A, self.B, self.C, self.Q, self.R
        beta, theta = self.beta, self.theta
        # == Set up loop == #
        P = np.zeros((self.n, self.n)) if P_init is None else P_init
        iterate, e = 0, tol + 1
        while iterate < max_iter and e > tol:
            F, new_P = self.b_operator(self.d_operator(P))
            e = np.sqrt(np.sum((new_P - P)**2))
            iterate += 1
            P = new_P
        I = np.identity(self.j)
        S1 = P.dot(C)
        S2 = C.T.dot(S1)
        K = inv(theta * I - S2).dot(S1.T).dot(A - B.dot(F))

        return F, K, P

Response: A simple algorithm for computing the robust policy F and the value function P for the current key - value entry. This method is used to compute the robust policy F and the value function P for the current key - value entry.


Instruction:
Can you tell what is the following Python 3 function doing
def F_to_K(self, F, method='doubling'):
        Q2 = self.beta * self.theta
        R2 = - self.R - dot(F.T, dot(self.Q, F))
        A2 = self.A - dot(self.B, F)
        B2 = self.C
        lq = LQ(Q2, R2, A2, B2, beta=self.beta)
        neg_P, neg_K, d = lq.stationary_values(method=method)

        return -neg_K, -neg_P

Response: Compute agent 2 s best cost minimizing response K given a given F.


Instruction:
Write a Python 3 function for
computing agent 1 s best value - maximizing response F given a given K.

Response:
def K_to_F(self, K, method='doubling'):
        """
        Compute agent 1's best value-maximizing response F, given K.

        Parameters
        ----------
        K : array_like(float, ndim=2)
            A j x n array
        method : str, optional(default='doubling')
            Solution method used in solving the associated Riccati
            equation, str in {'doubling', 'qz'}.

        Returns
        -------
        F : array_like(float, ndim=2)
            The policy function for a given K
        P : array_like(float, ndim=2)
            The value function for a given K

        """
        A1 = self.A + dot(self.C, K)
        B1 = self.B
        Q1 = self.Q
        R1 = self.R - self.beta * self.theta * dot(K.T, K)
        lq = LQ(Q1, R1, A1, B1, beta=self.beta)
        P, F, d = lq.stationary_values(method=method)

        return F, P


Instruction:
Write a Python 3 function that can
evaluate the policy function F and returns the matrix P_F and constant d_F associated with discounted cost F associated with discounted cost F.

Response:
def evaluate_F(self, F):
        """
        Given a fixed policy F, with the interpretation :math:`u = -F x`, this
        function computes the matrix :math:`P_F` and constant :math:`d_F`
        associated with discounted cost :math:`J_F(x) = x' P_F x + d_F`

        Parameters
        ----------
        F : array_like(float, ndim=2)
            The policy function, a k x n array

        Returns
        -------
        P_F : array_like(float, ndim=2)
            Matrix for discounted cost
        d_F : scalar(float)
            Constant for discounted cost
        K_F : array_like(float, ndim=2)
            Worst case policy
        O_F : array_like(float, ndim=2)
            Matrix for discounted entropy
        o_F : scalar(float)
            Constant for discounted entropy

        """
        # == Simplify names == #
        Q, R, A, B, C = self.Q, self.R, self.A, self.B, self.C
        beta, theta = self.beta, self.theta

        # == Solve for policies and costs using agent 2's problem == #
        K_F, P_F = self.F_to_K(F)
        I = np.identity(self.j)
        H = inv(I - C.T.dot(P_F.dot(C)) / theta)
        d_F = log(det(H))

        # == Compute O_F and o_F == #
        sig = -1.0 / theta
        AO = sqrt(beta) * (A - dot(B, F) + dot(C, K_F))
        O_F = solve_discrete_lyapunov(AO.T, beta * dot(K_F.T, K_F))
        ho = (trace(H - 1) - d_F) / 2.0
        tr = trace(dot(O_F, C.dot(H.dot(C.T))))
        o_F = (ho + beta * tr) / (1 - beta)

        return K_F, P_F, d_F, O_F, o_F


Instruction:
Can you tell what is the following Python 3 function doing
def process_input_line(self, line, store_history=True):
        #print "input='%s'"%self.input
        stdout = sys.stdout
        splitter = self.IP.input_splitter
        try:
            sys.stdout = self.cout
            splitter.push(line)
            more = splitter.push_accepts_more()
            if not more:
                source_raw = splitter.source_raw_reset()[1]
                self.IP.run_cell(source_raw, store_history=store_history)
        finally:
            sys.stdout = stdout

Response: process the input line capturing stdout


Instruction:
How would you explain what the following Python 3 function does
def process_image(self, decorator):
        savefig_dir = self.savefig_dir
        source_dir = self.source_dir
        saveargs = decorator.split(' ')
        filename = saveargs[1]
        # insert relative path to image file in source
        outfile = os.path.relpath(os.path.join(savefig_dir,filename),
                    source_dir)

        imagerows = ['.. image:: %s'%outfile]

        for kwarg in saveargs[2:]:
            arg, val = kwarg.split('=')
            arg = arg.strip()
            val = val.strip()
            imagerows.append('   :%s: %s'%(arg, val))

        image_file = os.path.basename(outfile) # only return file name
        image_directive = '\n'.join(imagerows)
        return image_file, image_directive

Response: process an image and return the filename and the image directive


Instruction:
How would you explain what the following Python 3 function does
def process_input(self, data, input_prompt, lineno):
        decorator, input, rest = data
        image_file = None
        image_directive = None
        #print 'INPUT:', data  # dbg
        is_verbatim = decorator=='@verbatim' or self.is_verbatim
        is_doctest = decorator=='@doctest' or self.is_doctest
        is_suppress = decorator=='@suppress' or self.is_suppress
        is_okexcept = decorator=='@okexcept' or self.is_okexcept
        is_savefig = decorator is not None and \
                     decorator.startswith('@savefig')

        input_lines = input.split('\n')

        self.datacontent = data

        continuation = '   %s:'%''.join(['.']*(len(str(lineno))+2))

        if is_savefig:
            image_file, image_directive = self.process_image(decorator)

        ret = []
        is_semicolon = False
        store_history = True

        for i, line in enumerate(input_lines):
            if line.endswith(';'):
                is_semicolon = True
            if is_semicolon or is_suppress:
                store_history = False

            if i==0:
                # process the first input line
                if is_verbatim:
                    self.process_input_line('')
                    self.IP.execution_count += 1 # increment it anyway
                else:
                    # only submit the line in non-verbatim mode
                    self.process_input_line(line, store_history=store_history)
                formatted_line = '%s %s'%(input_prompt, line)
            else:
                # process a continuation line
                if not is_verbatim:
                    self.process_input_line(line, store_history=store_history)

                formatted_line = '%s%s'%(continuation, line)

            if not is_suppress:
                ret.append(formatted_line)

        if not is_suppress:
            if len(rest.strip()):
                if is_verbatim:
                    # the "rest" is the standard output of the
                    # input, which needs to be added in
                    # verbatim mode
                    ret.append(rest)

        self.cout.seek(0)
        output = self.cout.read()
        if not is_suppress and not is_semicolon:
            ret.append(output.decode('utf-8'))

        if not is_okexcept and "Traceback" in output:
            sys.stdout.write(output)

        self.cout.truncate(0)
        return (ret, input_lines, output, is_doctest, image_file,
                    image_directive)

Response: Process the input block for INPUT token.


Instruction:
Write a Python 3 function that can
process data block for OUTPUT token.

Response:
def process_output(self, data, output_prompt,
                       input_lines, output, is_doctest, image_file):
        """Process data block for OUTPUT token."""
        if is_doctest:
            submitted = data.strip()
            found = output
            if found is not None:
                found = found.strip()

                # XXX - fperez: in 0.11, 'output' never comes with the prompt
                # in it, just the actual output text.  So I think all this code
                # can be nuked...

                # the above comment does not appear to be accurate... (minrk)

                ind = found.find(output_prompt)
                if ind<0:
                    e='output prompt="%s" does not match out line=%s' % \
                       (output_prompt, found)
                    raise RuntimeError(e)
                found = found[len(output_prompt):].strip()

                if found!=submitted:
                    e = ('doctest failure for input_lines="%s" with '
                         'found_output="%s" and submitted output="%s"' %
                         (input_lines, found, submitted) )
                    raise RuntimeError(e)


Instruction:
Can you generate the documentation for the following Python 3 function
def process_pure_python(self, content):
        output = []
        savefig = False # keep up with this to clear figure
        multiline = False # to handle line continuation
        fmtin = self.promptin

        for lineno, line in enumerate(content):

            line_stripped = line.strip()

            if not len(line):
                output.append(line) # preserve empty lines in output
                continue

            # handle decorators
            if line_stripped.startswith('@'):
                output.extend([line])
                if 'savefig' in line:
                    savefig = True # and need to clear figure
                continue

            # handle comments
            if line_stripped.startswith('#'):
                output.extend([line])
                continue

            # deal with multilines
            if not multiline: # not currently on a multiline

                if line_stripped.endswith('\\'): # now we are
                    multiline = True
                    cont_len = len(str(lineno)) + 2
                    line_to_process = line.strip('\\')
                    output.extend([u"%s %s" % (fmtin%lineno,line)])
                    continue
                else: # no we're still not
                    line_to_process = line.strip('\\')
            else: # we are currently on a multiline
                line_to_process += line.strip('\\')
                if line_stripped.endswith('\\'): # and we still are
                    continuation = '.' * cont_len
                    output.extend([(u'   %s: '+line_stripped) % continuation])
                    continue
                # else go ahead and run this multiline then carry on

            # get output of line
            self.process_input_line(unicode(line_to_process.strip()),
                                    store_history=False)
            out_line = self.cout.getvalue()
            self.clear_cout()

            # clear current figure if plotted
            if savefig:
                self.ensure_pyplot()
                self.process_input_line('plt.clf()', store_history=False)
                self.clear_cout()
                savefig = False

            # line numbers don't actually matter, they're replaced later
            if not multiline:
                in_line = u"%s %s" % (fmtin%lineno,line)

                output.extend([in_line])
            else:
                output.extend([(u'   %s: '+line_stripped) % continuation])
                multiline = False
            if len(out_line):
                output.extend([out_line])
            output.extend([u''])

        return output

Response: This function processes the pure python code.


Instruction:
Can you generate a brief explanation for the following Python 3 code
def process_pure_python2(self, content):
        output = []
        savefig = False # keep up with this to clear figure
        multiline = False # to handle line continuation
        multiline_start = None
        fmtin = self.promptin

        ct = 0

        # nuke empty lines
        content = [line for line in content if len(line.strip()) > 0]

        for lineno, line in enumerate(content):

            line_stripped = line.strip()
            if not len(line):
                output.append(line)
                continue

            # handle decorators
            if line_stripped.startswith('@'):
                output.extend([line])
                if 'savefig' in line:
                    savefig = True # and need to clear figure
                continue

            # handle comments
            if line_stripped.startswith('#'):
                output.extend([line])
                continue

            continuation  = u'   %s:'% ''.join(['.']*(len(str(ct))+2))
            if not multiline:
                modified = u"%s %s" % (fmtin % ct, line_stripped)
                output.append(modified)
                ct += 1
                try:
                    ast.parse(line_stripped)
                    output.append(u'')
                except Exception:
                    multiline = True
                    multiline_start = lineno
            else:
                modified = u'%s %s' % (continuation, line)
                output.append(modified)

                try:
                    ast.parse('\n'.join(content[multiline_start:lineno+1]))

                    if (lineno < len(content) - 1 and
                        _count_indent(content[multiline_start]) <
                        _count_indent(content[lineno + 1])):

                        continue

                    output.extend([continuation, u''])
                    multiline = False
                except Exception:
                    pass

            continue

        return output

Response: This function processes the pure python2 code.


Instruction:
Here you have a function in Python 3, explain what it does
def simulate_linear_model(A, x0, v, ts_length):
    A = np.asarray(A)
    n = A.shape[0]
    x = np.empty((n, ts_length))
    x[:, 0] = x0
    for t in range(ts_length-1):
        # x[:, t+1] = A.dot(x[:, t]) + v[:, t]
        for i in range(n):
            x[i, t+1] = v[i, t]                   # Shock
            for j in range(n):
                x[i, t+1] += A[i, j] * x[j, t]   # Dot Product
    return x

Response: r Simulates a vector linear system of A.


Instruction:
Can you generate the documentation for the following Python 3 function
def searchsorted(a, v):
    lo = -1
    hi = len(a)
    while(lo < hi-1):
        m = (lo + hi) // 2
        if v < a[m]:
            hi = m
        else:
            lo = m
    return hi

Response: Custom version of np. searchsorted.


Instruction:
Create a Python 3 function for
using a jitted version of the maximization routine from SciPy s fminbound method to perform the maximum function call.

Response:
def brent_max(func, a, b, args=(), xtol=1e-5, maxiter=500):
    """
    Uses a jitted version of the maximization routine from SciPy's fminbound.
    The algorithm is identical except that it's been switched to maximization
    rather than minimization, and the tests for convergence have been stripped
    out to allow for jit compilation.

    Note that the input function `func` must be jitted or the call will fail.

    Parameters
    ----------
    func : jitted function
    a : scalar
        Lower bound for search
    b : scalar
        Upper bound for search
    args : tuple, optional
        Extra arguments passed to the objective function.
    maxiter : int, optional
        Maximum number of iterations to perform.
    xtol : float, optional
        Absolute error in solution `xopt` acceptable for convergence.

    Returns
    -------
    xf : float
        The maximizer
    fval : float
        The maximum value attained
    info : tuple
        A tuple of the form (status_flag, num_iter).  Here status_flag
        indicates whether or not the maximum number of function calls was
        attained.  A value of 0 implies that the maximum was not hit.
        The value `num_iter` is the number of function calls.

    Example
    -------

    ```
        @njit
        def f(x):
            return -(x + 2.0)**2 + 1.0

        xf, fval, info = brent_max(f, -2, 2)
    ```

    """
    if not np.isfinite(a):
        raise ValueError("a must be finite.")

    if not np.isfinite(b):
        raise ValueError("b must be finite.")

    if not a < b:
        raise ValueError("a must be less than b.")

    maxfun = maxiter
    status_flag = 0

    sqrt_eps = np.sqrt(2.2e-16)
    golden_mean = 0.5 * (3.0 - np.sqrt(5.0))

    fulc = a + golden_mean * (b - a)
    nfc, xf = fulc, fulc
    rat = e = 0.0
    x = xf
    fx = -func(x, *args)
    num = 1

    ffulc = fnfc = fx
    xm = 0.5 * (a + b)
    tol1 = sqrt_eps * np.abs(xf) + xtol / 3.0
    tol2 = 2.0 * tol1

    while (np.abs(xf - xm) > (tol2 - 0.5 * (b - a))):
        golden = 1
        # Check for parabolic fit
        if np.abs(e) > tol1:
            golden = 0
            r = (xf - nfc) * (fx - ffulc)
            q = (xf - fulc) * (fx - fnfc)
            p = (xf - fulc) * q - (xf - nfc) * r
            q = 2.0 * (q - r)
            if q > 0.0:
                p = -p
            q = np.abs(q)
            r = e
            e = rat

            # Check for acceptability of parabola
            if ((np.abs(p) < np.abs(0.5*q*r)) and (p > q*(a - xf)) and
                    (p < q * (b - xf))):
                rat = (p + 0.0) / q
                x = xf + rat

                if ((x - a) < tol2) or ((b - x) < tol2):
                    si = np.sign(xm - xf) + ((xm - xf) == 0)
                    rat = tol1 * si
            else:      # do a golden section step
                golden = 1

        if golden:  # Do a golden-section step
            if xf >= xm:
                e = a - xf
            else:
                e = b - xf
            rat = golden_mean*e

        if rat == 0:
            si = np.sign(rat) + 1
        else:
            si = np.sign(rat)

        x = xf + si * np.maximum(np.abs(rat), tol1)
        fu = -func(x, *args)
        num += 1

        if fu <= fx:
            if x >= xf:
                a = xf
            else:
                b = xf
            fulc, ffulc = nfc, fnfc
            nfc, fnfc = xf, fx
            xf, fx = x, fu
        else:
            if x < xf:
                a = x
            else:
                b = x
            if (fu <= fnfc) or (nfc == xf):
                fulc, ffulc = nfc, fnfc
                nfc, fnfc = x, fu
            elif (fu <= ffulc) or (fulc == xf) or (fulc == nfc):
                fulc, ffulc = x, fu

        xm = 0.5 * (a + b)
        tol1 = sqrt_eps * np.abs(xf) + xtol / 3.0
        tol2 = 2.0 * tol1

        if num >= maxfun:
            status_flag = 1
            break

    fval = -fx
    info = status_flag, num

    return xf, fval, info


Instruction:
Make a summary of the following Python 3 code
def mean(self):
        "mean"
        n, a, b = self.n, self.a, self.b
        return n * a / (a + b)

Response: get the mean value of the record set


Instruction:
Can you tell what is the following Python 3 function doing
def skew(self):
        "skewness"
        n, a, b = self.n, self.a, self.b
        t1 = (a+b+2*n) * (b - a) / (a+b+2)
        t2 = sqrt((1+a+b) / (n*a*b * (n+a+b)))
        return t1 * t2

Response: skewness of the current set of entries


Instruction:
How would you explain what the following Python 3 function does
def _equilibrium_payoffs_abreu_sannikov(rpg, tol=1e-12, max_iter=500,
                                        u_init=np.zeros(2)):
    """
    Using 'abreu_sannikov' algorithm to compute the set of payoff pairs
    of all pure-strategy subgame-perfect equilibria with public randomization
    for any repeated two-player games with perfect monitoring and
    discounting, following Abreu and Sannikov (2014).

    Parameters
    ----------
    rpg : RepeatedGame
        Two player repeated game.

    tol : scalar(float), optional(default=1e-12)
        Tolerance for convergence checking.

    max_iter : scalar(int), optional(default=500)
        Maximum number of iterations.

    u_init : ndarray(float, ndim=1), optional(default=np.zeros(2))
        The initial guess of threat points.

    Returns
    -------
    hull : scipy.spatial.ConvexHull
        The convex hull of equilibrium payoff pairs.

    References
    ----------
    .. [1] Abreu, Dilip, and Yuliy Sannikov. "An algorithm for
       two‐player repeated games with perfect monitoring." Theoretical
       Economics 9.2 (2014): 313-338.
    """
    sg, delta = rpg.sg, rpg.delta

    if sg.N != 2:
        msg = "this algorithm only applies to repeated two-player games."
        raise NotImplementedError(msg)

    best_dev_gains = _best_dev_gains(rpg)
    IC = np.empty(2)
    action_profile_payoff = np.empty(2)
    # auxiliary array for checking if payoff is inside the convex hull
    # first two entries for payoff point, and the last entry is 1.
    extended_payoff = np.ones(3)
    # array to store new points of C in each intersection
    # at most 4 new points will be generated
    new_pts = np.empty((4, 2))
    # array to store the points of W
    # the length of v is limited by |A1|*|A2|*4
    W_new = np.empty((np.prod(sg.nums_actions)*4, 2))
    W_old = np.empty((np.prod(sg.nums_actions)*4, 2))
    # count the new points generated in each iteration
    n_new_pt = 0

    # copy the threat points
    u = np.copy(u_init)

    # initialization
    payoff_pts = \
        sg.payoff_profile_array.reshape(np.prod(sg.nums_actions), 2)
    W_new[:np.prod(sg.nums_actions)] = payoff_pts
    n_new_pt = np.prod(sg.nums_actions)

    n_iter = 0
    while True:
        W_old[:n_new_pt] = W_new[:n_new_pt]
        n_old_pt = n_new_pt
        hull = ConvexHull(W_old[:n_old_pt])

        W_new, n_new_pt = \
            _R(delta, sg.nums_actions, sg.payoff_arrays,
               best_dev_gains, hull.points, hull.vertices,
               hull.equations, u, IC, action_profile_payoff,
               extended_payoff, new_pts, W_new)

        n_iter += 1
        if n_iter >= max_iter:
            break

        # check convergence
        if n_new_pt == n_old_pt:
            if np.linalg.norm(W_new[:n_new_pt] - W_old[:n_new_pt]) < tol:
                break

        # update threat points
        _update_u(u, W_new[:n_new_pt])

    hull = ConvexHull(W_new[:n_new_pt])

    return hull

Response: This function computes the set of payoff pairs for all pure - strategy subgame - perfect equilibria with public randomization and public randomization of the two - player repeated game with perfect monitoring and public randomization of the two - player repeated game with public randomization.


Instruction:
Implement a Python 3 function for
calculating the normalized payoff gains from deviating from the current action to the best response for each player.

Response:
def _best_dev_gains(rpg):
    """
    Calculate the normalized payoff gains from deviating from the current
    action to the best response for each player.

    Parameters
    ----------
    rpg : RepeatedGame
        Two player repeated game.

    Returns
    -------
    best_dev_gains : tuple(ndarray(float, ndim=2))
        The normalized best deviation payoff gain arrays.
        best_dev_gains[i][ai, a-i] is normalized payoff gain
        player i can get if originally players are choosing
        ai and a-i, and player i deviates to the best response action.
    """
    sg, delta = rpg.sg, rpg.delta

    best_dev_gains = ((1-delta)/delta *
                      (np.max(sg.payoff_arrays[i], 0) - sg.payoff_arrays[i])
                      for i in range(2))

    return tuple(best_dev_gains)


Instruction:
Write a Python 3 script to
update the payoff convex hull by iterating all action pairs. Using the R operator proposed by Abreu and Sannikov 2014. Parameters ---------- delta : scalar(float) The common discount rate at which all players discount the future. nums_actions : tuple(int) Tuple of the numbers of actions, one for each player. payoff_arrays : tuple(ndarray(float, ndim=2)) Tuple of the payoff arrays, one for each player. best_dev_gains : tuple(ndarray(float, ndim=2)) Tuple of the normalized best deviation payoff gain arrays. best_dev_gains[i][ai, a-i] is payoff gain player i can get if originally players are choosing ai and a-i, and player i deviates to the best response action. points : ndarray(float, ndim=2) Coordinates of the points in the W, which construct a feasible payoff convex hull. vertices : ndarray(float, ndim=1) Indices of points forming the vertices of the convex hull, which are in counterclockwise order. equations : ndarray(float, ndim=2) [normal, offset] forming the hyperplane equation of the facet u : ndarray(float, ndim=1) The threat points. IC : ndarray(float, ndim=1) The minimum IC continuation values. action_profile_payoff : ndarray(float, ndim=1) Array of payoff for one action profile. extended_payoff : ndarray(float, ndim=2) The array [payoff0, payoff1, 1] for checking if [payoff0, payoff1] is in the feasible payoff convex hull. new_pts : ndarray(float, ndim=1) The 4 by 2 array for storing the generated potential extreme points of one action profile. One action profile can only generate at most 4 points. W_new : ndarray(float, ndim=2) Array for storing the coordinates of the generated potential extreme points that construct a new feasible payoff convex hull. tol: scalar(float), optional(default=1e-10) The tolerance for checking if two values are equal. Returns ------- W_new : ndarray(float, ndim=2) The coordinates of the generated potential extreme points that construct a new feasible payoff convex hull. n_new_pt : scalar(int) The number of points in W_new that construct the feasible payoff convex hull.

Response:
def _R(delta, nums_actions, payoff_arrays, best_dev_gains, points,
       vertices, equations, u, IC, action_profile_payoff,
       extended_payoff, new_pts, W_new, tol=1e-10):
    """
    Updating the payoff convex hull by iterating all action pairs.
    Using the R operator proposed by Abreu and Sannikov 2014.

    Parameters
    ----------
    delta : scalar(float)
            The common discount rate at which all players discount
            the future.

    nums_actions : tuple(int)
        Tuple of the numbers of actions, one for each player.

    payoff_arrays : tuple(ndarray(float, ndim=2))
        Tuple of the payoff arrays, one for each player.

    best_dev_gains : tuple(ndarray(float, ndim=2))
        Tuple of the normalized best deviation payoff gain arrays.
        best_dev_gains[i][ai, a-i] is payoff gain player i
        can get if originally players are choosing ai and a-i,
        and player i deviates to the best response action.

    points : ndarray(float, ndim=2)
        Coordinates of the points in the W, which construct a
        feasible payoff convex hull.

    vertices : ndarray(float, ndim=1)
        Indices of points forming the vertices of the convex hull,
        which are in counterclockwise order.

    equations : ndarray(float, ndim=2)
        [normal, offset] forming the hyperplane equation of the facet

    u : ndarray(float, ndim=1)
        The threat points.

    IC : ndarray(float, ndim=1)
        The minimum IC continuation values.

    action_profile_payoff : ndarray(float, ndim=1)
        Array of payoff for one action profile.

    extended_payoff : ndarray(float, ndim=2)
        The array [payoff0, payoff1, 1] for checking if
        [payoff0, payoff1] is in the feasible payoff convex hull.

    new_pts : ndarray(float, ndim=1)
        The 4 by 2 array for storing the generated potential
        extreme points of one action profile. One action profile
        can only generate at most 4 points.

    W_new : ndarray(float, ndim=2)
        Array for storing the coordinates of the generated potential
        extreme points that construct a new feasible payoff convex hull.

    tol: scalar(float), optional(default=1e-10)
        The tolerance for checking if two values are equal.

    Returns
    -------
    W_new : ndarray(float, ndim=2)
        The coordinates of the generated potential extreme points
        that construct a new feasible payoff convex hull.

    n_new_pt : scalar(int)
        The number of points in W_new that construct the feasible
        payoff convex hull.
    """
    n_new_pt = 0
    for a0 in range(nums_actions[0]):
        for a1 in range(nums_actions[1]):
            action_profile_payoff[0] = payoff_arrays[0][a0, a1]
            action_profile_payoff[1] = payoff_arrays[1][a1, a0]
            IC[0] = u[0] + best_dev_gains[0][a0, a1]
            IC[1] = u[1] + best_dev_gains[1][a1, a0]

            # check if payoff is larger than IC
            if (action_profile_payoff >= IC).all():
                # check if payoff is inside the convex hull
                extended_payoff[:2] = action_profile_payoff
                if (np.dot(equations, extended_payoff) <= tol).all():
                    W_new[n_new_pt] = action_profile_payoff
                    n_new_pt += 1
                    continue

            new_pts, n = _find_C(new_pts, points, vertices, equations,
                                 extended_payoff, IC, tol)

            for i in range(n):
                W_new[n_new_pt] = \
                    delta * new_pts[i] + (1-delta) * action_profile_payoff
                n_new_pt += 1

    return W_new, n_new_pt


Instruction:
Here you have a function in Python 3, explain what it does
def _find_C(C, points, vertices, equations, extended_payoff, IC, tol):
    n = 0
    weights = np.empty(2)
    # vertices is ordered counterclockwise
    for i in range(len(vertices)-1):
        n = _intersect(C, n, weights, IC,
                       points[vertices[i]],
                       points[vertices[i+1]], tol)

    n = _intersect(C, n, weights, IC,
                   points[vertices[-1]],
                   points[vertices[0]], tol)

    # check the case that IC is an interior point of the convex hull
    extended_payoff[:2] = IC
    if (np.dot(equations, extended_payoff) <= tol).all():
        C[n, :] = IC
        n += 1

    return C, n

Response: This function finds all the intersection points between the current convex hull and the two IC constraints.


Instruction:
Implement a function in Python 3 to
find the intersection points of a half-closed simplex (pt0, pt1] and IC constraints. Parameters ---------- C : ndarray(float, ndim=2) The 4 by 2 array for storing the generated points of one action profile. One action profile can only generate at most 4 points. n : scalar(int) The number of intersection points that have been found. weights : ndarray(float, ndim=1) The size 2 array for storing the weights when calculate the intersection point of simplex and IC constraints. IC : ndarray(float, ndim=1) The minimum IC continuation values. pt0 : ndarray(float, ndim=1) Coordinates of the starting point of the simplex. pt1 : ndarray(float, ndim=1) Coordinates of the ending point of the simplex. tol : scalar(float) The tolerance for checking if two values are equal. Returns ------- n : scalar(int) The updated number of found intersection points.

Response:
def _intersect(C, n, weights, IC, pt0, pt1, tol):
    """
    Find the intersection points of a half-closed simplex
    (pt0, pt1] and IC constraints.

    Parameters
    ----------
    C : ndarray(float, ndim=2)
        The 4 by 2 array for storing the generated points of
        one action profile. One action profile can only
        generate at most 4 points.

    n : scalar(int)
        The number of intersection points that have been found.

    weights : ndarray(float, ndim=1)
        The size 2 array for storing the weights when calculate
        the intersection point of simplex and IC constraints.

    IC : ndarray(float, ndim=1)
        The minimum IC continuation values.

    pt0 : ndarray(float, ndim=1)
        Coordinates of the starting point of the simplex.

    pt1 : ndarray(float, ndim=1)
        Coordinates of the ending point of the simplex.

    tol : scalar(float)
        The tolerance for checking if two values are equal.

    Returns
    -------
    n : scalar(int)
        The updated number of found intersection points.
    """
    for i in range(2):
        if (abs(pt0[i] - pt1[i]) < tol):
            if (abs(pt1[i] - IC[i]) < tol):
                x = pt1[1-i]
            else:
                continue
        else:
            weights[i] = (pt0[i] - IC[i]) / (pt0[i] - pt1[i])
            # pt0 is not included to avoid duplication
            # weights in (0, 1]
            if (0 < weights[i] <= 1):
                x = (1 - weights[i]) * pt0[1-i] + weights[i] * pt1[1-i]
            else:
                continue
        # x has to be strictly higher than IC[1-j]
        # if it is equal, then it means IC is one of the vertex
        # it will be added to C in below
        if x - IC[1-i] > tol:
            C[n, i] = IC[i]
            C[n, 1-i] = x
            n += 1
        elif x - IC[1-i] > -tol:
            # to avoid duplication when IC is a vertex
            break

    return n


Instruction:
Write a Python 3 script for
updating the threat points u by the minimum of feasible payoffs.

Response:
def _update_u(u, W):
    """
    Update the threat points if it not feasible in the new W,
    by the minimum of new feasible payoffs.

    Parameters
    ----------
    u : ndarray(float, ndim=1)
        The threat points.

    W : ndarray(float, ndim=1)
        The points that construct the feasible payoff convex hull.

    Returns
    -------
    u : ndarray(float, ndim=1)
        The updated threat points.
    """
    for i in range(2):
        W_min = W[:, i].min()
        if u[i] < W_min:
            u[i] = W_min

    return u


Instruction:
Can you generate the documentation for the following Python 3 function
def equilibrium_payoffs(self, method=None, options=None):
        if method is None:
            method = 'abreu_sannikov'

        if options is None:
            options = {}

        if method in ('abreu_sannikov', 'AS'):
            return _equilibrium_payoffs_abreu_sannikov(self, **options)
        else:
            msg = f"method {method} not supported."
            raise NotImplementedError(msg)

Response:
Compute the set of payoff pairs of all pure - strategy subgame - perfect and public randomization for any repeated two - player games
            with perfect monitoring and discounting.


Instruction:
Make a summary of the following Python 3 code
def random_game(nums_actions, random_state=None):
    N = len(nums_actions)
    if N == 0:
        raise ValueError('nums_actions must be non-empty')

    random_state = check_random_state(random_state)
    players = [
        Player(random_state.random_sample(nums_actions[i:]+nums_actions[:i]))
        for i in range(N)
    ]
    g = NormalFormGame(players)
    return g

Response: Returns a random NormalFormGame instance where the payoffs are drawn independently from the uniform distribution on [ 0 1 ).


Instruction:
Make a summary of the following Python 3 code
def covariance_game(nums_actions, rho, random_state=None):
    N = len(nums_actions)
    if N <= 1:
        raise ValueError('length of nums_actions must be at least 2')
    if not (-1 / (N - 1) <= rho <= 1):
        lb = '-1' if N == 2 else '-1/{0}'.format(N-1)
        raise ValueError('rho must be in [{0}, 1]'.format(lb))

    mean = np.zeros(N)
    cov = np.empty((N, N))
    cov.fill(rho)
    cov[range(N), range(N)] = 1

    random_state = check_random_state(random_state)
    payoff_profile_array = \
        random_state.multivariate_normal(mean, cov, nums_actions)
    g = NormalFormGame(payoff_profile_array)
    return g

Response: Returns a random NormalFormGame instance where the payoff profiles are drawn independently from the standard multi - normal with the covariance of any pair of payoffs equal to rho.


Instruction:
Given the following Python 3 function, write the documentation
def random_pure_actions(nums_actions, random_state=None):
    random_state = check_random_state(random_state)
    action_profile = tuple(
        [random_state.randint(num_actions) for num_actions in nums_actions]
    )
    return action_profile

Response: Returns a tuple of random pure actions ( integers.


Instruction:
How would you explain what the following Python 3 function does
def random_mixed_actions(nums_actions, random_state=None):
    random_state = check_random_state(random_state)
    action_profile = tuple(
        [probvec(1, num_actions, random_state).ravel()
         for num_actions in nums_actions]
    )
    return action_profile

Response: Returns a tuple of random mixed actions.


Instruction:
Can you generate the documentation for the following Python 3 function
def fetch_nb_dependencies(files, repo=REPO, raw=RAW, branch=BRANCH, folder=FOLDER, overwrite=False, verbose=True):
    import requests

    #-Generate Common Data Structure-#
    if type(files) == list:
        files = {"" : files}

    status = []

    #-Obtain each requested file-#
    for directory in files.keys():
        if directory != "":
            if verbose: print("Parsing directory: %s"%directory)
        for fl in files[directory]:
            if directory != "":
                fl = directory+"/"+fl
            #-Check for Local Copy of File (Default Behaviour is to Skip)-#
            if not overwrite:
                if os.path.isfile(fl):
                    if verbose: print(
                        "A file named %s already exists in the specified directory ... skipping download." % fl)
                    status.append(False)
                    continue
            else:
                if verbose: print("Overwriting file %s ..."%fl)
            if verbose: print("Fetching file: %s"%fl)
            #-Get file in OS agnostic way using requests-#
            url = "/".join([repo, raw, branch, folder, fl])
            r = requests.get(url)
            with open(fl, "wb") as fl:
                fl.write(r.content)
            status.append(True)

    return status

Response: This function fetches raw files from QuantEcon. notebooks or other Github repository and returns a list of files in the specified branch.


Instruction:
Can you generate a brief explanation for the following Python 3 code
def _results(r):
    x, funcalls, iterations, flag = r
    return results(x, funcalls, iterations, flag == 0)

Response: r Select from a tuple of root funccalls iterations and flag


Instruction:
Given the following Python 3 function, write the documentation
def newton(func, x0, fprime, args=(), tol=1.48e-8, maxiter=50,
           disp=True):
    """
    Find a zero from the Newton-Raphson method using the jitted version of
    Scipy's newton for scalars. Note that this does not provide an alternative
    method such as secant. Thus, it is important that `fprime` can be provided.

    Note that `func` and `fprime` must be jitted via Numba.
    They are recommended to be `njit` for performance.

    Parameters
    ----------
    func : callable and jitted
        The function whose zero is wanted. It must be a function of a
        single variable of the form f(x,a,b,c...), where a,b,c... are extra
        arguments that can be passed in the `args` parameter.
    x0 : float
        An initial estimate of the zero that should be somewhere near the
        actual zero.
    fprime : callable and jitted
        The derivative of the function (when available and convenient).
    args : tuple, optional(default=())
        Extra arguments to be used in the function call.
    tol : float, optional(default=1.48e-8)
        The allowable error of the zero value.
    maxiter : int, optional(default=50)
        Maximum number of iterations.
    disp : bool, optional(default=True)
        If True, raise a RuntimeError if the algorithm didn't converge

    Returns
    -------
    results : namedtuple
        A namedtuple containing the following items:
        ::

            root - Estimated location where function is zero.
            function_calls - Number of times the function was called.
            iterations - Number of iterations needed to find the root.
            converged - True if the routine converged

    """

    if tol <= 0:
        raise ValueError("tol is too small <= 0")
    if maxiter < 1:
        raise ValueError("maxiter must be greater than 0")

    # Convert to float (don't use float(x0); this works also for complex x0)
    p0 = 1.0 * x0
    funcalls = 0
    status = _ECONVERR

    # Newton-Raphson method
    for itr in range(maxiter):
        # first evaluate fval
        fval = func(p0, *args)
        funcalls += 1
        # If fval is 0, a root has been found, then terminate
        if fval == 0:
            status = _ECONVERGED
            p = p0
            itr -= 1
            break
        fder = fprime(p0, *args)
        funcalls += 1
        # derivative is zero, not converged
        if fder == 0:
            p = p0
            break
        newton_step = fval / fder
        # Newton step
        p = p0 - newton_step
        if abs(p - p0) < tol:
            status = _ECONVERGED
            break
        p0 = p

    if disp and status == _ECONVERR:
        msg = "Failed to converge"
        raise RuntimeError(msg)

    return _results((p, funcalls, itr + 1, status))

Response: This function is used to find a zero from the Newton - Raphson method using the jitted version of the Scala algorithm.


Instruction:
Given the following Python 3 function, write the documentation
def newton_secant(func, x0, args=(), tol=1.48e-8, maxiter=50,
                  disp=True):
    """
    Find a zero from the secant method using the jitted version of
    Scipy's secant method.

    Note that `func` must be jitted via Numba.

    Parameters
    ----------
    func : callable and jitted
        The function whose zero is wanted. It must be a function of a
        single variable of the form f(x,a,b,c...), where a,b,c... are extra
        arguments that can be passed in the `args` parameter.
    x0 : float
        An initial estimate of the zero that should be somewhere near the
        actual zero.
    args : tuple, optional(default=())
        Extra arguments to be used in the function call.
    tol : float, optional(default=1.48e-8)
        The allowable error of the zero value.
    maxiter : int, optional(default=50)
        Maximum number of iterations.
    disp : bool, optional(default=True)
        If True, raise a RuntimeError if the algorithm didn't converge.

    Returns
    -------
    results : namedtuple
        A namedtuple containing the following items:
        ::

            root - Estimated location where function is zero.
            function_calls - Number of times the function was called.
            iterations - Number of iterations needed to find the root.
            converged - True if the routine converged
    """

    if tol <= 0:
        raise ValueError("tol is too small <= 0")
    if maxiter < 1:
        raise ValueError("maxiter must be greater than 0")

    # Convert to float (don't use float(x0); this works also for complex x0)
    p0 = 1.0 * x0
    funcalls = 0
    status = _ECONVERR

    # Secant method
    if x0 >= 0:
        p1 = x0 * (1 + 1e-4) + 1e-4
    else:
        p1 = x0 * (1 + 1e-4) - 1e-4
        q0 = func(p0, *args)
    funcalls += 1
    q1 = func(p1, *args)
    funcalls += 1
    for itr in range(maxiter):
        if q1 == q0:
            p = (p1 + p0) / 2.0
            status = _ECONVERGED
            break
        else:
            p = p1 - q1 * (p1 - p0) / (q1 - q0)
        if np.abs(p - p1) < tol:
            status = _ECONVERGED
            break
        p0 = p1
        q0 = q1
        p1 = p
        q1 = func(p1, *args)
        funcalls += 1

    if disp and status == _ECONVERR:
        msg = "Failed to converge"
        raise RuntimeError(msg)

    return _results((p, funcalls, itr + 1, status))

Response: This function returns a newton version of the current state of the newton s secant method.


Instruction:
Write a Python 3 function for
finding root of a function within an interval adapted from Scipy's bisect. Basic bisection routine to find a zero of the function `f` between the arguments `a` and `b`. `f(a)` and `f(b)` cannot have the same signs. `f` must be jitted via numba. Parameters ---------- f : jitted and callable Python function returning a number. `f` must be continuous. a : number One end of the bracketing interval [a,b]. b : number The other end of the bracketing interval [a,b]. args : tuple, optional(default=()) Extra arguments to be used in the function call. xtol : number, optional(default=2e-12) The computed root ``x0`` will satisfy ``np.allclose(x, x0, atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The parameter must be nonnegative. rtol : number, optional(default=4*np.finfo(float).eps) The computed root ``x0`` will satisfy ``np.allclose(x, x0, atol=xtol, rtol=rtol)``, where ``x`` is the exact root. maxiter : number, optional(default=100) Maximum number of iterations. disp : bool, optional(default=True) If True, raise a RuntimeError if the algorithm didn't converge. Returns ------- results : namedtuple

Response:
def bisect(f, a, b, args=(), xtol=_xtol,
           rtol=_rtol, maxiter=_iter, disp=True):
    """
    Find root of a function within an interval adapted from Scipy's bisect.

    Basic bisection routine to find a zero of the function `f` between the
    arguments `a` and `b`. `f(a)` and `f(b)` cannot have the same signs.

    `f` must be jitted via numba.

    Parameters
    ----------
    f : jitted and callable
        Python function returning a number.  `f` must be continuous.
    a : number
        One end of the bracketing interval [a,b].
    b : number
        The other end of the bracketing interval [a,b].
    args : tuple, optional(default=())
        Extra arguments to be used in the function call.
    xtol : number, optional(default=2e-12)
        The computed root ``x0`` will satisfy ``np.allclose(x, x0,
        atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The
        parameter must be nonnegative.
    rtol : number, optional(default=4*np.finfo(float).eps)
        The computed root ``x0`` will satisfy ``np.allclose(x, x0,
        atol=xtol, rtol=rtol)``, where ``x`` is the exact root.
    maxiter : number, optional(default=100)
        Maximum number of iterations.
    disp : bool, optional(default=True)
        If True, raise a RuntimeError if the algorithm didn't converge.

    Returns
    -------
    results : namedtuple

    """

    if xtol <= 0:
        raise ValueError("xtol is too small (<= 0)")

    if maxiter < 1:
        raise ValueError("maxiter must be greater than 0")

    # Convert to float
    xa = a * 1.0
    xb = b * 1.0

    fa = f(xa, *args)
    fb = f(xb, *args)
    funcalls = 2
    root, status = _bisect_interval(xa, xb, fa, fb)

    # Check for sign error and early termination
    if status == _ECONVERGED:
        itr = 0
    else:
        # Perform bisection
        dm = xb - xa
        for itr in range(maxiter):
            dm *= 0.5
            xm = xa + dm
            fm = f(xm, *args)
            funcalls += 1

            if fm * fa >= 0:
                xa = xm

            if fm == 0 or abs(dm) < xtol + rtol * abs(xm):
                root = xm
                status = _ECONVERGED
                itr += 1
                break

    if disp and status == _ECONVERR:
        raise RuntimeError("Failed to converge")

    return _results((root, funcalls, itr, status))


Instruction:
Can you generate a brief explanation for the following Python 3 code
def brentq(f, a, b, args=(), xtol=_xtol,
           rtol=_rtol, maxiter=_iter, disp=True):
    """
    Find a root of a function in a bracketing interval using Brent's method
    adapted from Scipy's brentq.

    Uses the classic Brent's method to find a zero of the function `f` on
    the sign changing interval [a , b].

    `f` must be jitted via numba.

    Parameters
    ----------
    f : jitted and callable
        Python function returning a number.  `f` must be continuous.
    a : number
        One end of the bracketing interval [a,b].
    b : number
        The other end of the bracketing interval [a,b].
    args : tuple, optional(default=())
        Extra arguments to be used in the function call.
    xtol : number, optional(default=2e-12)
        The computed root ``x0`` will satisfy ``np.allclose(x, x0,
        atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The
        parameter must be nonnegative.
    rtol : number, optional(default=4*np.finfo(float).eps)
        The computed root ``x0`` will satisfy ``np.allclose(x, x0,
        atol=xtol, rtol=rtol)``, where ``x`` is the exact root.
    maxiter : number, optional(default=100)
        Maximum number of iterations.
    disp : bool, optional(default=True)
        If True, raise a RuntimeError if the algorithm didn't converge.

    Returns
    -------
    results : namedtuple

    """
    if xtol <= 0:
        raise ValueError("xtol is too small (<= 0)")
    if maxiter < 1:
        raise ValueError("maxiter must be greater than 0")

    # Convert to float
    xpre = a * 1.0
    xcur = b * 1.0

    fpre = f(xpre, *args)
    fcur = f(xcur, *args)
    funcalls = 2

    root, status = _bisect_interval(xpre, xcur, fpre, fcur)

    # Check for sign error and early termination
    if status == _ECONVERGED:
        itr = 0
    else:
        # Perform Brent's method
        for itr in range(maxiter):

            if fpre * fcur < 0:
                xblk = xpre
                fblk = fpre
                spre = scur = xcur - xpre
            if abs(fblk) < abs(fcur):
                xpre = xcur
                xcur = xblk
                xblk = xpre

                fpre = fcur
                fcur = fblk
                fblk = fpre

            delta = (xtol + rtol * abs(xcur)) / 2
            sbis = (xblk - xcur) / 2

            # Root found
            if fcur == 0 or abs(sbis) < delta:
                status = _ECONVERGED
                root = xcur
                itr += 1
                break

            if abs(spre) > delta and abs(fcur) < abs(fpre):
                if xpre == xblk:
                    # interpolate
                    stry = -fcur * (xcur - xpre) / (fcur - fpre)
                else:
                    # extrapolate
                    dpre = (fpre - fcur) / (xpre - xcur)
                    dblk = (fblk - fcur) / (xblk - xcur)
                    stry = -fcur * (fblk * dblk - fpre * dpre) / \
                        (dblk * dpre * (fblk - fpre))

                if (2 * abs(stry) < min(abs(spre), 3 * abs(sbis) - delta)):
                    # good short step
                    spre = scur
                    scur = stry
                else:
                    # bisect
                    spre = sbis
                    scur = sbis
            else:
                # bisect
                spre = sbis
                scur = sbis

            xpre = xcur
            fpre = fcur
            if (abs(scur) > delta):
                xcur += scur
            else:
                xcur += (delta if sbis > 0 else -delta)
            fcur = f(xcur, *args)
            funcalls += 1

    if disp and status == _ECONVERR:
        raise RuntimeError("Failed to converge")

    return _results((root, funcalls, itr, status))

Response: This function returns a root of a function in a bracketing interval using the classic Brent s method.


Instruction:
Can you generate the documentation for the following Python 3 function
def _compute_fixed_point_ig(T, v, max_iter, verbose, print_skip, is_approx_fp,
                            *args, **kwargs):
    """
    Implement the imitation game algorithm by McLennan and Tourky (2006)
    for computing an approximate fixed point of `T`.

    Parameters
    ----------
    is_approx_fp : callable
        A callable with signature `is_approx_fp(v)` which determines
        whether `v` is an approximate fixed point with a bool return
        value (i.e., True or False)

    For the other parameters, see Parameters in compute_fixed_point.

    Returns
    -------
    x_new : scalar(float) or ndarray(float)
        Approximate fixed point.

    converged : bool
        Whether the routine has converged.

    iterate : scalar(int)
        Number of iterations.

    """
    if verbose == 2:
        start_time = time.time()
        _print_after_skip(print_skip, it=None)

    x_new = v
    y_new = T(x_new, *args, **kwargs)
    iterate = 1
    converged = is_approx_fp(x_new)

    if converged or iterate >= max_iter:
        if verbose == 2:
            error = np.max(np.abs(y_new - x_new))
            etime = time.time() - start_time
            print_skip = 1
            _print_after_skip(print_skip, iterate, error, etime)
        if verbose >= 1:
            if not converged:
                warnings.warn(_non_convergence_msg, RuntimeWarning)
            elif verbose == 2:
                print(_convergence_msg.format(iterate=iterate))
        return x_new, converged, iterate

    if verbose == 2:
        error = np.max(np.abs(y_new - x_new))
        etime = time.time() - start_time
        _print_after_skip(print_skip, iterate, error, etime)

    # Length of the arrays to store the computed sequences of x and y.
    # If exceeded, reset to min(max_iter, buff_size*2).
    buff_size = 2**8
    buff_size = min(max_iter, buff_size)

    shape = (buff_size,) + np.asarray(x_new).shape
    X, Y = np.empty(shape), np.empty(shape)
    X[0], Y[0] = x_new, y_new
    x_new = Y[0]

    tableaux = tuple(np.empty((buff_size, buff_size*2+1)) for i in range(2))
    bases = tuple(np.empty(buff_size, dtype=int) for i in range(2))
    max_piv = 10**6  # Max number of pivoting steps in _lemke_howson_tbl

    while True:
        y_new = T(x_new, *args, **kwargs)
        iterate += 1
        converged = is_approx_fp(x_new)

        if converged or iterate >= max_iter:
            break

        if verbose == 2:
            error = np.max(np.abs(y_new - x_new))
            etime = time.time() - start_time
            _print_after_skip(print_skip, iterate, error, etime)

        try:
            X[iterate-1] = x_new
            Y[iterate-1] = y_new
        except IndexError:
            buff_size = min(max_iter, buff_size*2)
            shape = (buff_size,) + X.shape[1:]
            X_tmp, Y_tmp = X, Y
            X, Y = np.empty(shape), np.empty(shape)
            X[:X_tmp.shape[0]], Y[:Y_tmp.shape[0]] = X_tmp, Y_tmp
            X[iterate-1], Y[iterate-1] = x_new, y_new

            tableaux = tuple(np.empty((buff_size, buff_size*2+1))
                             for i in range(2))
            bases = tuple(np.empty(buff_size, dtype=int) for i in range(2))

        m = iterate
        tableaux_curr = tuple(tableau[:m, :2*m+1] for tableau in tableaux)
        bases_curr = tuple(basis[:m] for basis in bases)
        _initialize_tableaux_ig(X[:m], Y[:m], tableaux_curr, bases_curr)
        converged, num_iter = _lemke_howson_tbl(
            tableaux_curr, bases_curr, init_pivot=m-1, max_iter=max_piv
        )
        _, rho = _get_mixed_actions(tableaux_curr, bases_curr)

        if Y.ndim <= 2:
            x_new = rho.dot(Y[:m])
        else:
            shape_Y = Y.shape
            Y_2d = Y.reshape(shape_Y[0], np.prod(shape_Y[1:]))
            x_new = rho.dot(Y_2d[:m]).reshape(shape_Y[1:])

    if verbose == 2:
        error = np.max(np.abs(y_new - x_new))
        etime = time.time() - start_time
        print_skip = 1
        _print_after_skip(print_skip, iterate, error, etime)
    if verbose >= 1:
        if not converged:
            warnings.warn(_non_convergence_msg, RuntimeWarning)
        elif verbose == 2:
            print(_convergence_msg.format(iterate=iterate))

    return x_new, converged, iterate

Response: This function computes an approximate fixed point of T.


Instruction:
Create a Python 3 function to
initialize the tableaux and base arrays in place for the geometric imitation game as defined by McLennan and Tourky 2006.

Response:
def _initialize_tableaux_ig(X, Y, tableaux, bases):
    """
    Given sequences `X` and `Y` of ndarrays, initialize the tableau and
    basis arrays in place for the "geometric" imitation game as defined
    in McLennan and Tourky (2006), to be passed to `_lemke_howson_tbl`.

    Parameters
    ----------
    X, Y : ndarray(float)
        Arrays of the same shape (m, n).

    tableaux : tuple(ndarray(float, ndim=2))
        Tuple of two arrays to be used to store the tableaux, of shape
        (2m, 2m). Modified in place.

    bases : tuple(ndarray(int, ndim=1))
        Tuple of two arrays to be used to store the bases, of shape
        (m,). Modified in place.

    Returns
    -------
    tableaux : tuple(ndarray(float, ndim=2))
        View to `tableaux`.

    bases : tuple(ndarray(int, ndim=1))
        View to `bases`.

    """
    m = X.shape[0]
    min_ = np.zeros(m)

    # Mover
    for i in range(m):
        for j in range(2*m):
            if j == i or j == i + m:
                tableaux[0][i, j] = 1
            else:
                tableaux[0][i, j] = 0
        # Right hand side
        tableaux[0][i, 2*m] = 1

    # Imitator
    for i in range(m):
        # Slack variables
        for j in range(m):
            if j == i:
                tableaux[1][i, j] = 1
            else:
                tableaux[1][i, j] = 0
        # Payoff variables
        for j in range(m):
            d = X[i] - Y[j]
            tableaux[1][i, m+j] = _square_sum(d) * (-1)
            if tableaux[1][i, m+j] < min_[j]:
                min_[j] = tableaux[1][i, m+j]
        # Right hand side
        tableaux[1][i, 2*m] = 1
    # Shift the payoff values
    for i in range(m):
        for j in range(m):
            tableaux[1][i, m+j] -= min_[j]
            tableaux[1][i, m+j] += 1

    for pl, start in enumerate([m, 0]):
        for i in range(m):
            bases[pl][i] = start + i

    return tableaux, bases


Instruction:
How would you explain what the following Python 3 function does
def get_participants(self, namespace, room):
        for sid, active in six.iteritems(self.rooms[namespace][room].copy()):
            yield sid

Response: Return an iterable with the active participants in a room.


Instruction:
Explain what the following Python 3 code does
def connect(self, sid, namespace):
        self.enter_room(sid, namespace, None)
        self.enter_room(sid, namespace, sid)

Response: Register a client connection to a namespace.


Instruction:
Write a Python 3 function for
putting the client in the to - be - disconnected list.

Response:
def pre_disconnect(self, sid, namespace):
        """Put the client in the to-be-disconnected list.

        This allows the client data structures to be present while the
        disconnect handler is invoked, but still recognize the fact that the
        client is soon going away.
        """
        if namespace not in self.pending_disconnect:
            self.pending_disconnect[namespace] = []
        self.pending_disconnect[namespace].append(sid)


Instruction:
Here you have a function in Python 3, explain what it does
def disconnect(self, sid, namespace):
        if namespace not in self.rooms:
            return
        rooms = []
        for room_name, room in six.iteritems(self.rooms[namespace].copy()):
            if sid in room:
                rooms.append(room_name)
        for room in rooms:
            self.leave_room(sid, namespace, room)
        if sid in self.callbacks and namespace in self.callbacks[sid]:
            del self.callbacks[sid][namespace]
            if len(self.callbacks[sid]) == 0:
                del self.callbacks[sid]
        if namespace in self.pending_disconnect and \
                sid in self.pending_disconnect[namespace]:
            self.pending_disconnect[namespace].remove(sid)
            if len(self.pending_disconnect[namespace]) == 0:
                del self.pending_disconnect[namespace]

Response: Register a client disconnect from a namespace.


Instruction:
Can you write a function in Python 3 where it
adds a client to a room.

Response:
def enter_room(self, sid, namespace, room):
        """Add a client to a room."""
        if namespace not in self.rooms:
            self.rooms[namespace] = {}
        if room not in self.rooms[namespace]:
            self.rooms[namespace][room] = {}
        self.rooms[namespace][room][sid] = True


Instruction:
Can you generate the documentation for the following Python 3 function
def leave_room(self, sid, namespace, room):
        try:
            del self.rooms[namespace][room][sid]
            if len(self.rooms[namespace][room]) == 0:
                del self.rooms[namespace][room]
                if len(self.rooms[namespace]) == 0:
                    del self.rooms[namespace]
        except KeyError:
            pass

Response: Remove a client from a room.


Instruction:
Given the following Python 3 function, write the documentation
def close_room(self, room, namespace):
        try:
            for sid in self.get_participants(namespace, room):
                self.leave_room(sid, namespace, room)
        except KeyError:
            pass

Response: Remove all participants from a room.


Instruction:
Explain what the following Python 3 code does
def get_rooms(self, sid, namespace):
        r = []
        try:
            for room_name, room in six.iteritems(self.rooms[namespace]):
                if room_name is not None and sid in room and room[sid]:
                    r.append(room_name)
        except KeyError:
            pass
        return r

Response: Return the rooms a client is in.


Instruction:
Here you have a function in Python 3, explain what it does
def trigger_callback(self, sid, namespace, id, data):
        callback = None
        try:
            callback = self.callbacks[sid][namespace][id]
        except KeyError:
            # if we get an unknown callback we just ignore it
            self._get_logger().warning('Unknown callback received, ignoring.')
        else:
            del self.callbacks[sid][namespace][id]
        if callback is not None:
            callback(*data)

Response: Invoke an application callback.


Instruction:
Here you have a function in Python 3, explain what it does
def _generate_ack_id(self, sid, namespace, callback):
        namespace = namespace or '/'
        if sid not in self.callbacks:
            self.callbacks[sid] = {}
        if namespace not in self.callbacks[sid]:
            self.callbacks[sid][namespace] = {0: itertools.count(1)}
        id = six.next(self.callbacks[sid][namespace][0])
        self.callbacks[sid][namespace][id] = callback
        return id

Response: Generate a unique identifier for an ACK packet.


Instruction:
Implement a function in Python 3 to
get the appropriate logger for this instance.

Response:
def _get_logger(self):
        """Get the appropriate logger

        Prevents uninitialized servers in write-only mode from failing.
        """

        if self.logger:
            return self.logger
        elif self.server:
            return self.server.logger
        else:
            return default_logger


Instruction:
Can you implement a function in Python 3 that
registers an event handler for the specified event.

Response:
def on(self, event, handler=None, namespace=None):
        """Register an event handler.

        :param event: The event name. It can be any string. The event names
                      ``'connect'``, ``'message'`` and ``'disconnect'`` are
                      reserved and should not be used.
        :param handler: The function that should be invoked to handle the
                        event. When this parameter is not given, the method
                        acts as a decorator for the handler function.
        :param namespace: The Socket.IO namespace for the event. If this
                          argument is omitted the handler is associated with
                          the default namespace.

        Example usage::

            # as a decorator:
            @socket_io.on('connect', namespace='/chat')
            def connect_handler(sid, environ):
                print('Connection request')
                if environ['REMOTE_ADDR'] in blacklisted:
                    return False  # reject

            # as a method:
            def message_handler(sid, msg):
                print('Received message: ', msg)
                eio.send(sid, 'response')
            socket_io.on('message', namespace='/chat', message_handler)

        The handler function receives the ``sid`` (session ID) for the
        client as first argument. The ``'connect'`` event handler receives the
        WSGI environment as a second argument, and can return ``False`` to
        reject the connection. The ``'message'`` handler and handlers for
        custom event names receive the message payload as a second argument.
        Any values returned from a message handler will be passed to the
        client's acknowledgement callback function if it exists. The
        ``'disconnect'`` handler does not take a second argument.
        """
        namespace = namespace or '/'

        def set_handler(handler):
            if namespace not in self.handlers:
                self.handlers[namespace] = {}
            self.handlers[namespace][event] = handler
            return handler

        if handler is None:
            return set_handler
        set_handler(handler)


Instruction:
How would you code a function in Python 3 to
register a namespace handler object.

Response:
def register_namespace(self, namespace_handler):
        """Register a namespace handler object.

        :param namespace_handler: An instance of a :class:`Namespace`
                                  subclass that handles all the event traffic
                                  for a namespace.
        """
        if not isinstance(namespace_handler, namespace.Namespace):
            raise ValueError('Not a namespace instance')
        if self.is_asyncio_based() != namespace_handler.is_asyncio_based():
            raise ValueError('Not a valid namespace class for this server')
        namespace_handler._set_server(self)
        self.namespace_handlers[namespace_handler.namespace] = \
            namespace_handler


Instruction:
How would you code a function in Python 3 to
enter a room. This function adds the client to a room. The :func:`emit` and :func:`send` functions can optionally broadcast events to all the clients in a room. :param sid: Session ID of the client. :param room: Room name. If the room does not exist it is created. :param namespace: The Socket.IO namespace for the event. If this argument is omitted the default namespace is used.

Response:
def enter_room(self, sid, room, namespace=None):
        """Enter a room.

        This function adds the client to a room. The :func:`emit` and
        :func:`send` functions can optionally broadcast events to all the
        clients in a room.

        :param sid: Session ID of the client.
        :param room: Room name. If the room does not exist it is created.
        :param namespace: The Socket.IO namespace for the event. If this
                          argument is omitted the default namespace is used.
        """
        namespace = namespace or '/'
        self.logger.info('%s is entering room %s [%s]', sid, room, namespace)
        self.manager.enter_room(sid, namespace, room)


Instruction:
How would you explain what the following Python 3 function does
def leave_room(self, sid, room, namespace=None):
        namespace = namespace or '/'
        self.logger.info('%s is leaving room %s [%s]', sid, room, namespace)
        self.manager.leave_room(sid, namespace, room)

Response: This function removes the client from a room.


Instruction:
Explain what the following Python 3 code does
def rooms(self, sid, namespace=None):
        namespace = namespace or '/'
        return self.manager.get_rooms(sid, namespace)

Response: Return the rooms a client is in.


Instruction:
Given the following Python 3 function, write the documentation
def get_session(self, sid, namespace=None):
        namespace = namespace or '/'
        eio_session = self.eio.get_session(sid)
        return eio_session.setdefault(namespace, {})

Response: Return the user session for a client.


Instruction:
Can you generate the documentation for the following Python 3 function
def save_session(self, sid, session, namespace=None):
        namespace = namespace or '/'
        eio_session = self.eio.get_session(sid)
        eio_session[namespace] = session

Response: Store the user session for a client.


Instruction:
Create a Python 3 function for
returning the user session dictionary for a client with context manager syntax.

Response:
def session(self, sid, namespace=None):
        """Return the user session for a client with context manager syntax.

        :param sid: The session id of the client.

        This is a context manager that returns the user session dictionary for
        the client. Any changes that are made to this dictionary inside the
        context manager block are saved back to the session. Example usage::

            @sio.on('connect')
            def on_connect(sid, environ):
                username = authenticate_user(environ)
                if not username:
                    return False
                with sio.session(sid) as session:
                    session['username'] = username

            @sio.on('message')
            def on_message(sid, msg):
                with sio.session(sid) as session:
                    print('received message from ', session['username'])
        """
        class _session_context_manager(object):
            def __init__(self, server, sid, namespace):
                self.server = server
                self.sid = sid
                self.namespace = namespace
                self.session = None

            def __enter__(self):
                self.session = self.server.get_session(sid,
                                                       namespace=namespace)
                return self.session

            def __exit__(self, *args):
                self.server.save_session(sid, self.session,
                                         namespace=namespace)

        return _session_context_manager(self, sid, namespace)


Instruction:
How would you explain what the following Python 3 function does
def start_background_task(self, target, *args, **kwargs):
        return self.eio.start_background_task(target, *args, **kwargs)

Response: Start a background task using the appropriate async model.


Instruction:
How would you implement a function in Python 3 that
sends a message to a client.

Response:
def _emit_internal(self, sid, event, data, namespace=None, id=None):
        """Send a message to a client."""
        if six.PY2 and not self.binary:
            binary = False  # pragma: nocover
        else:
            binary = None
        # tuples are expanded to multiple arguments, everything else is sent
        # as a single argument
        if isinstance(data, tuple):
            data = list(data)
        else:
            data = [data]
        self._send_packet(sid, packet.Packet(packet.EVENT, namespace=namespace,
                                             data=[event] + data, id=id,
                                             binary=binary))


Instruction:
Create a Python 3 function for
sending a Socket. IO packet to a client.

Response:
def _send_packet(self, sid, pkt):
        """Send a Socket.IO packet to a client."""
        encoded_packet = pkt.encode()
        if isinstance(encoded_packet, list):
            binary = False
            for ep in encoded_packet:
                self.eio.send(sid, ep, binary=binary)
                binary = True
        else:
            self.eio.send(sid, encoded_packet, binary=False)


Instruction:
Can you tell what is the following Python 3 function doing
def _handle_connect(self, sid, namespace):
        namespace = namespace or '/'
        self.manager.connect(sid, namespace)
        if self.always_connect:
            self._send_packet(sid, packet.Packet(packet.CONNECT,
                                                 namespace=namespace))
        fail_reason = None
        try:
            success = self._trigger_event('connect', namespace, sid,
                                          self.environ[sid])
        except exceptions.ConnectionRefusedError as exc:
            fail_reason = exc.error_args
            success = False

        if success is False:
            if self.always_connect:
                self.manager.pre_disconnect(sid, namespace)
                self._send_packet(sid, packet.Packet(
                    packet.DISCONNECT, data=fail_reason, namespace=namespace))
            self.manager.disconnect(sid, namespace)
            if not self.always_connect:
                self._send_packet(sid, packet.Packet(
                    packet.ERROR, data=fail_reason, namespace=namespace))
            if sid in self.environ:  # pragma: no cover
                del self.environ[sid]
            return False
        elif not self.always_connect:
            self._send_packet(sid, packet.Packet(packet.CONNECT,
                                                 namespace=namespace))

Response: Handle a client connection request.


Instruction:
Can you tell what is the following Python 3 function doing
def _handle_disconnect(self, sid, namespace):
        namespace = namespace or '/'
        if namespace == '/':
            namespace_list = list(self.manager.get_namespaces())
        else:
            namespace_list = [namespace]
        for n in namespace_list:
            if n != '/' and self.manager.is_connected(sid, n):
                self._trigger_event('disconnect', n, sid)
                self.manager.disconnect(sid, n)
        if namespace == '/' and self.manager.is_connected(sid, namespace):
            self._trigger_event('disconnect', '/', sid)
            self.manager.disconnect(sid, '/')

Response: Handle a client disconnect.


Instruction:
Create a Python 3 function for
handling an incoming client event.

Response:
def _handle_event(self, sid, namespace, id, data):
        """Handle an incoming client event."""
        namespace = namespace or '/'
        self.logger.info('received event "%s" from %s [%s]', data[0], sid,
                         namespace)
        if self.async_handlers:
            self.start_background_task(self._handle_event_internal, self, sid,
                                       data, namespace, id)
        else:
            self._handle_event_internal(self, sid, data, namespace, id)


Instruction:
Write a Python 3 script for
invoking an application event handler.

Response:
def _trigger_event(self, event, namespace, *args):
        """Invoke an application event handler."""
        # first see if we have an explicit handler for the event
        if namespace in self.handlers and event in self.handlers[namespace]:
            return self.handlers[namespace][event](*args)

        # or else, forward the event to a namespace handler if one exists
        elif namespace in self.namespace_handlers:
            return self.namespace_handlers[namespace].trigger_event(
                event, *args)


Instruction:
Here you have a function in Python 3, explain what it does
def _handle_eio_connect(self, sid, environ):
        if not self.manager_initialized:
            self.manager_initialized = True
            self.manager.initialize()
        self.environ[sid] = environ
        return self._handle_connect(sid, '/')

Response: Handle the Engine. IO connection event.


Instruction:
How would you implement a function in Python 3 that
dispatches Engine. IO messages.

Response:
def _handle_eio_message(self, sid, data):
        """Dispatch Engine.IO messages."""
        if sid in self._binary_packet:
            pkt = self._binary_packet[sid]
            if pkt.add_attachment(data):
                del self._binary_packet[sid]
                if pkt.packet_type == packet.BINARY_EVENT:
                    self._handle_event(sid, pkt.namespace, pkt.id, pkt.data)
                else:
                    self._handle_ack(sid, pkt.namespace, pkt.id, pkt.data)
        else:
            pkt = packet.Packet(encoded_packet=data)
            if pkt.packet_type == packet.CONNECT:
                self._handle_connect(sid, pkt.namespace)
            elif pkt.packet_type == packet.DISCONNECT:
                self._handle_disconnect(sid, pkt.namespace)
            elif pkt.packet_type == packet.EVENT:
                self._handle_event(sid, pkt.namespace, pkt.id, pkt.data)
            elif pkt.packet_type == packet.ACK:
                self._handle_ack(sid, pkt.namespace, pkt.id, pkt.data)
            elif pkt.packet_type == packet.BINARY_EVENT or \
                    pkt.packet_type == packet.BINARY_ACK:
                self._binary_packet[sid] = pkt
            elif pkt.packet_type == packet.ERROR:
                raise ValueError('Unexpected ERROR packet.')
            else:
                raise ValueError('Unknown packet type.')


Instruction:
Can you generate a brief explanation for the following Python 3 code
def _handle_eio_disconnect(self, sid):
        self._handle_disconnect(sid, '/')
        if sid in self.environ:
            del self.environ[sid]

Response: Handle Engine. IO disconnect event.