import cvxpy as cp
import math
import numpy as np

from collections import OrderedDict
from functools import partial
from multiprocessing import Pool, cpu_count
from scipy.optimize import minimize_scalar
from tqdm.auto import tqdm


def p_num_samples(epsilon, delta, n_x=3, const=None):
    """Compute the number of samples needed to satisfy the specified probabilistic guarantees for the p-norm ball reachable set estimate

    :param epsilon: The accuracy parameter
    :type epsilon: float
    :param delta: The confidence parameter
    :type delta: float
    :param n_x: The state dimension, defaults to 3
    :type n_x: int
    :param const: The constraints placed on the parameters A and b, defaults to None
    :type const: string, optional
    :return: The number of samples needed to satisfy the specified probabilistic guarantees
    :rtype: int
    """
    if const is None:
        n_theta = 0.5 * (n_x ** 2 + 3 * n_x)
    elif const == "diagonal":
        n_theta = 2 * n_x
    elif const == "scalar":
        n_theta = 1
    N = math.ceil(math.e * (math.log(1 / delta) + n_theta) / (epsilon * (math.e - 1)))
    return N


def solve_p_norm(sample, n_x=3, p=2, const=None):
    """Solves the scenario relaxation problem for the given sample with p-Norm Balls

    :param sample: Sample from dynamical system (num_samples, n_x)
    :type sample: numpy.ndarray
    :param n_x: The state dimension, defaults to 3
    :type n_x: int, optional
    :param p: The order of p-norm, defaults to 2
    :type p: int, optional
    :param const: The constraints placed on the parameters A and b, defaults to None
    :type const: string, optional
    :return: The values of matrix A and vector b corresponding to the optimal p-Norm Ball, as well as the status of the optimizer.
    :rtype: tuple
    """
    if const is None:
        A = cp.Variable((n_x, n_x), symmetric=True)
        b = cp.Variable((n_x, 1))
    elif const == "diagonal":
        a = cp.Variable((n_x, 1))
        A = cp.diag(a)
        b = cp.Variable((n_x, 1))
    elif const == "scalar":
        sigma = cp.Variable()
        A = sigma * np.identity(n_x)
        b = np.zeros((n_x, 1))

    obj = cp.Minimize(-cp.log_det(A))
    constraints = [cp.pnorm(A @ r.reshape(n_x, 1) - b, p=p) <= 1 for r in sample]
    prob = cp.Problem(obj, constraints)
    prob.solve()

    if const != "scalar":
        return A.value, b.value, prob.status
    else:
        return A, b, prob.status


def multi_p_norm(samples, p=2, const=None):
    """Computes a the p-norm ball reachable set estimates across a series of timesteps

    :param samples: The samples from a dynamic system across time, an array of shape (num_samples, timesteps, state_dim)
    :type samples: numpy.ndarray
    :param p: The order of p-norm, defaults to 2
    :type p: int, optional
    :param const: The constraints placed on the parameters A and b, defaults to None
    :type const: string, optional
    :raises ValueError: [description]
    :return: [description]
    :rtype: [type]
    """
    if len(samples.shape) != 3:
        raise ValueError("Samples must be of shape (num_samples, timesteps, state_dim")
    n_x = samples.shape[2]
    keys = ("A", "b", "status")
    solve_p_norm_map = partial(solve_p_norm, n_x=n_x, p=p, const=const)
    p = Pool(cpu_count())
    solutions = [
        dict(zip(keys, sol))
        for sol in tqdm(
            p.imap(solve_p_norm_map, samples.swapaxes(0, 1)), total=samples.shape[1]
        )
    ]

    return solutions


def p_norm_cont(arr, axis, default_val, n_x, A_val, b_val, p, minimum=True):
    """Solve for the optimal value that satisfies the p-Norm Ball conditions at the specified axis

    :param arr: Array of shape (n_x - 1,) containing the independent variables of the p-norm condition
    :type arr: numpy.ndarray
    :param axis: The axis of the dependent variable for which to solve for (i.e. z -> axis=2).
    :type axis: int
    :param default_val: The value to return if no solution for the dependent variable is found that satisfies the p-norm conditions
    :type default_val: float
    :param n_x: The state dimension
    :type n_x: int
    :param A_val: The matrix of shape (n_x, n_x) corresponding to the optimal p-norm ball
    :type A_val: numpy.ndarray
    :param b_val: The vector of shape (n_x, 1) corresponding to the optimal p-norm ball
    :type b_val: numpy.ndarray
    :param p: The order of p-norm
    :type p: int
    :param minimum: True if optimizing for the minimal value of the dependent variable that satisfies the p-norm conditions, defaults to True
    :type minimum: bool, optional
    :return: The value at the specified axis which corresponds the the optimal value of the (n_x, 1) vector that satisfies the p-Norm Ball conditions at the specified axis
    :rtype: float
    """
    vec = cp.Variable((n_x, 1))
    other_dims = list(range(n_x))
    other_dims.remove(axis)
    constraints = [vec[i][0] == arr[j] for i, j in zip(other_dims, range(n_x - 1))]
    constraints.append(cp.pnorm(A_val @ vec - b_val, p=p) <= 1)
    if minimum:
        obj = cp.Minimize(vec[axis])
    else:
        obj = cp.Maximize(vec[axis])
    prob = cp.Problem(obj, constraints)

    try:
        prob.solve()
    except:
        return default_val

    if prob.status != "optimal":
        return default_val

    return vec.value[axis]


def p_norm_cont_proj(arr, axis, default_val, n_x, A_val, b_val, p):
    """Minimizes the p-Norm value with respect to value at the specified axis.

    :param arr: Array of shape (n_x - 1,) containing the independent variables of the p-norm condition.
    :type arr: numpy.ndarray
    :param axis: The axis of the dependent variable for which to solve for (i.e. z -> axis=2).
    :type axis: int
    :param default_val: The value to return if no solution for the dependent variable is found that satisfies the p-norm conditions.
    :type default_val: float
    :param n_x: The state dimension.
    :type n_x: int
    :param A_val: The matrix of shape (n_x, n_x) corresponding to the optimal p-norm ball.
    :type A_val: numpy.ndarray
    :param b_val: The vector of shape (n_x, 1) corresponding to the optimal p-norm ball.
    :type b_val: numpy.ndarray
    :param p: The order of p-norm.
    :type p: int
    :return: The value at the specified axis which corresponds the the minimum p-Norm value of the (n_x, 1)  vector.
    :rtype: float
    """
    vec = np.zeros((n_x))
    other_dims = list(range(n_x))
    other_dims.remove(axis)
    for i, j in zip(other_dims, range(n_x - 1)):
        vec[i] = arr[j]

    def f(x):
        vec[axis] = x
        return np.linalg.norm(A_val @ vec.reshape((n_x, 1)) - b_val, ord=p)

    res = minimize_scalar(f)
    vec[axis] = res.x

    if np.linalg.norm(A_val @ vec.reshape((n_x, 1)) - b_val, ord=p) <= 1:
        return res.x
    else:
        return default_val


def p_compute_contour_2D(sample, A_val, b_val, cont_axis=2, n_x=3, p=2, grid_n=200):
    """Computes the 3D contour for 2 dimensions based on sample data and the A_val, and b_val corresponding to the optimal p-norm ball.

    :param sample: Sample from dynamical system (num_samples, n_x)
    :type sample: numpy.ndarray
    :param A_val: The matrix of shape (n_x, n_x) corresponding to the optimal p-norm ball
    :type A_val: numpy.ndarray
    :param b_val: The vector of shape (n_x, 1) corresponding to the optimal p-norm ball
    :type b_val: numpy.ndarray
    :param cont_axis: The axis for which the contours are to be solved for, defaults to 2
    :type cont_axis: int, optional
    :param n_x: The state dimension, defaults to 3
    :type n_x: int, optional
    :param p: The order of p-norm, defaults to 2
    :type p: int, optional
    :param grid_n: The side length of the cube of points to be used for computing contours, defaults to 200
    :type grid_n: int, optional
    :return: The meshgrid, corresponding computed contour, and the extremum values for the chosen axis
    :rtype: tuple
    """
    x_min, x_max = sample[:, 0].min(), sample[:, 0].max()
    y_min, y_max = sample[:, 1].min(), sample[:, 1].max()
    z_min, z_max = sample[:, 2].min(), sample[:, 2].max()

    x = np.linspace(
        x_min - 0.4 * (x_max - x_min), x_max + 0.4 * (x_max - x_min), grid_n
    )
    y = np.linspace(
        y_min - 0.4 * (y_max - y_min), y_max + 0.4 * (y_max - y_min), grid_n
    )
    z = np.linspace(
        x_min - 0.4 * (z_max - z_min), z_max + 0.4 * (z_max - z_min), grid_n
    )

    if cont_axis == 2:
        d0, d1 = np.meshgrid(x, y)
        c_min, c_max = z_min, z_max
    elif cont_axis == 1:
        d0, d1 = np.meshgrid(x, z)
        c_min, c_max = y_min, y_max
    elif cont_axis == 0:
        d0, d1 = np.meshgrid(y, z)
        c_min, c_max = x_min, x_max

    d2 = np.array([d0.flatten(), d1.flatten()]).T

    solve_cont_d2 = partial(
        p_norm_cont_proj,
        axis=cont_axis,
        default_val=c_max + 1,
        n_x=n_x,
        A_val=A_val,
        b_val=b_val,
        p=p,
    )
    cont = np.fromiter(map(solve_cont_d2, d2), dtype=np.float64).reshape(grid_n, grid_n)

    return d0, d1, cont, c_min, c_max


def p_compute_contour_3D(
    sample, A_val, b_val, cont_axis=2, n_x=3, p=2, grid_n=200, stretch=0.4
):
    """Computes the 3D contour for 3 dimensions based on sample data and the A_val, and b_val corresponding to the optimal p-norm ball.

    :param sample: Sample from dynamical system (num_samples, n_x)
    :type sample: numpy.ndarray
    :param A_val: The matrix of shape (n_x, n_x) corresponding to the optimal p-norm ball
    :type A_val: numpy.ndarray
    :param b_val: The vector of shape (n_x, 1) corresponding to the optimal p-norm ball
    :type b_val: numpy.ndarray
    :param cont_axis: The axis for which the contours are to be solved for, defaults to 2
    :type cont_axis: int, optional
    :param n_x: The state dimension, defaults to 3
    :type n_x: int, optional
    :param p: The order of p-norm, defaults to 2
    :type p: int, optional
    :param grid_n: The side length of the cube of points to be used for computing contours, defaults to 200
    :type grid_n: int, optional
    :param stretch: The factor by which to stretch the grid used to compute the contour, defaults to 0.4
    :type stretch: float, optional
    :param minimum: True if optimizing for the minimal value of the dependent variable that satisfies the p-norm conditions, defaults to True
    :type minimum: bool, optional
    :return: The meshgrid, corresponding computed contour, and the extremum values for the chosen axis
    :rtype: tuple
    """
    x_min, x_max = sample[:, 0].min(), sample[:, 0].max()
    y_min, y_max = sample[:, 1].min(), sample[:, 1].max()
    z_min, z_max = sample[:, 2].min(), sample[:, 2].max()

    x = np.linspace(
        x_min - stretch * (x_max - x_min), x_max + stretch * (x_max - x_min), grid_n
    )
    y = np.linspace(
        y_min - stretch * (y_max - y_min), y_max + stretch * (y_max - y_min), grid_n
    )
    z = np.linspace(
        x_min - stretch * (z_max - z_min), z_max + stretch * (z_max - z_min), grid_n
    )

    if cont_axis == 2:
        d0, d1 = np.meshgrid(x, y)
        c_min, c_max = z_min, z_max
    elif cont_axis == 1:
        d0, d1 = np.meshgrid(x, z)
        c_min, c_max = y_min, y_max
    elif cont_axis == 0:
        d0, d1 = np.meshgrid(y, z)
        c_min, c_max = x_min, x_max

    d2 = np.array([d0.flatten(), d1.flatten()]).T

    solve_cont_d2_min = partial(
        p_norm_cont,
        axis=cont_axis,
        default_val=c_max + 1,
        n_x=n_x,
        A_val=A_val,
        b_val=b_val,
        p=p,
        minimum=True,
    )

    solve_cont_d2_max = partial(
        p_norm_cont,
        axis=cont_axis,
        default_val=c_min - 1,
        n_x=n_x,
        A_val=A_val,
        b_val=b_val,
        p=p,
        minimum=False,
    )

    cont_min = np.fromiter(map(solve_cont_d2_min, d2), dtype=np.float64).reshape(
        grid_n, grid_n
    )
    cont_max = np.fromiter(map(solve_cont_d2_max, d2), dtype=np.float64).reshape(
        grid_n, grid_n
    )

    return d0, d1, cont_min, cont_max, c_min, c_max


def p_compute_vals(sample, A_val, b_val, p=2, grid_n=200):
    """Computes the values within a p-norm ball in 1 dimension

    :param sample: The sample from a specific time step, an array of shape (num_samples,)
    :type sample: numpy.ndarray
    :param A_val: The matrix of shape (1, 1) corresponding to the optimal p-norm ball
    :type A_val: numpy.ndarray
    :param b_val: The vector of shape (1, 1) corresponding to the optimal p-norm ball
    :type b_val: numpy.ndarray
    :param p: The order of p-norm, defaults to 2
    :type p: int, optional
    :param grid_n: The number of points to test for the p-norm ball estimation at each a given time step, defaults to 200
    :type grid_n: int, optional
    :return: The values within the p-norm ball
    :rtype: list
    """
    # assuming sample is (num_samples,) shaped array
    y_min, y_max = sample.min(), sample.max()
    y = np.linspace(
        y_min - 0.4 * (y_max - y_min), y_max + 0.4 * (y_max - y_min), grid_n
    )

    vals = []
    for v in y:
        try:
            if np.linalg.norm(A_val @ np.array([[v]]) - b_val, ord=p) <= 1:
                vals.append(v)
        except ValueError:
            pass
    return vals


def p_get_dict(i, samples, solution_list, items, grid_n=50):
    """Generates a dictionary where keys are the passed in labels and values are the output of p_compute_contour_3D, the contour information for a p-norm ball reachable set estimate at a given time

    :param i: The index
    :type i: int
    :param samples: Array of shape (num_samples, time, 3)
    :type samples: numpy.ndarray
    :param solution_list: List of solutions where each solution is a dictionary with keys ("A", "b", "status"), defaults to None
    :type solution_list: list, optional
    :param items: A list of specific indices corresponding to the times at which the p-norm ball reachable set estimate will be chosen
    :type items: list
    :param grid_n: The side length of the cube of points to be used for computing contours, defaults to 50
    :type grid_n: int, optional
    :return: A dictionary where keys are the passed in labels and values are the output of p_compute_contour_3D, the contour information for a p-norm ball reachable set estimate at a given time
    :rtype: dict
    """
    labels = ["xv", "yv", "z_cont", "z_cont2", "z_min", "z_max"]
    index = items[i]
    A_i, b_i = solution_list[index]["A"], solution_list[index]["b"]
    return dict(
        zip(labels, p_compute_contour_3D(samples[:, index, :], A_i, b_i, grid_n=grid_n))
    )


def p_dict_list(
    samples, solution_list, num_parts, num_indices=20, logspace=True, grid_n=50
):
    """Generates a list of dictionaries, each containing the contour information for a p-norm ball reachable set estimate at a given time

    :param samples: Array of shape (num_samples, time, 3)
    :type samples: numpy.ndarray
    :param solution_list: List of solutions where each solution is a dictionary with keys ("A", "b", "status"), defaults to None
    :type solution_list: list, optional
    :param num_parts: The number of total timesteps for the samples
    :type num_parts: int
    :param num_indices: The number of indices corresponding to the number of reachable set estimates made at different times, defaults to 20
    :type num_indices: int, optional
    :param logspace: If True, a logarithmic scale is used for choosing times to compute reachable set estimates, defaults to True
    :type logspace: bool, optional
    :param grid_n: The side length of the cube of points to be used for computing contours, defaults to 50
    :type grid_n: int, optional
    :return: A list of dictionaries, each containing the contour information for a p-norm ball reachable set estimate at a given time
    :rtype: list
    """
    if logspace:
        log_ceil = np.log10(num_parts)
        items = list(
            OrderedDict.fromkeys(
                [int(i) - 1 for i in np.logspace(0, log_ceil, num_indices)]
            )
        )
    else:
        items = [int(i) for i in np.linspace(0, num_parts, num_indices)]

    get_dict = partial(
        p_get_dict,
        samples=samples,
        solution_list=solution_list,
        items=items,
        grid_n=grid_n,
    )

    p = Pool(cpu_count())
    dict_list = [
        d for d in tqdm(p.imap(get_dict, np.arange(len(items))), total=len(items))
    ]

    return dict_list


def p_emp_estimate(samples, A_val, b_val, n_x=3, p=2):
    """Computes the ratio of samples within the estimated reachable set for the p-norm ball reachable set estimation

    :param samples: Sample from dynamical system (num_samples, n_x)
    :type samples: numpy.ndarray
    :param A_val: The matrix of shape (n_x, n_x) corresponding to the optimal p-norm ball
    :type A_val: numpy.ndarray
    :param b_val: The vector of shape (n_x, 1) corresponding to the optimal p-norm ball
    :type b_val: numpy.ndarray
    :param n_x: The state dimension, defaults to 3
    :type n_x: int, optional
    :param p: The order of p-norm, defaults to 2
    :type p: int, optional
    :return: The ratio of samples within the estimated reachability set
    :rtype: float
    """
    num_samples = samples.shape[0]
    count = 0
    for sample in samples:
        vec = sample.reshape(n_x, 1)
        if np.linalg.norm(A_val @ vec - b_val, ord=p) <= 1:
            count += 1
    return count / num_samples


def p_get_reachable_2D(samples, A_val, b_val, p=2, grid_n=50):
    """Obtains the reachable set estimate for a set of samples at a give timestep. A utility function for plotting the reachable set estimate across all timesteps for two state variables.

    :param samples: Samples of shape (num_samples, 2)
    :type samples: numpy.ndarray
    :param A_val: Matrix corresponding to the reachable set estimate at a given timestep for two state variables, a (2, 2) array
    :type A_val: numpy.ndarray
    :param b_val: Vector corresponding to the reachable set estimate at a given timestep for two state variables, a (2, 1) array
    :type b_val: numpy.ndarray
    :param p: The order of the p-norm ball, defaults to 2
    :type p: int, optional
    :param grid_n: The side length of the square of points to be used for checking for points inside the p-norm ball, defaults to 50
    :type grid_n: int, optional
    :return: The x and y values included in the p-norm ball
    :rtype: tuple
    """
    x_min, x_max = samples[:, 0].min(), samples[:, 0].max()
    y_min, y_max = samples[:, 1].min(), samples[:, 1].max()

    x = np.linspace(
        x_min - 0.4 * (x_max - x_min), x_max + 0.4 * (x_max - x_min), grid_n
    )
    y = np.linspace(
        y_min - 0.4 * (y_max - y_min), y_max + 0.4 * (y_max - y_min), grid_n
    )

    d0, d1 = np.meshgrid(x, y)
    d2 = np.array([d0.flatten(), d1.flatten()]).T

    xs, ys = [], []
    for pair in d2:
        if np.linalg.norm(A_val @ pair.reshape((2, 1)) - b_val, ord=p) <= 1:
            xs.append(pair[0])
            ys.append(pair[1])

    return xs, ys

def p_get_reachable_3D(samples, A_val, b_val, p=2, grid_n=25):
    """Obtains the reachable set estimate for a set of samples at a give timestep. A utility function for plotting the reachable set estimate across all timesteps for three state variables.

    :param samples: Samples of shape (num_samples, 3)
    :type samples: numpy.ndarray
    :param A_val: Matrix corresponding to the reachable set estimate at a given timestep for two state variables, a (3, 3) array
    :type A_val: numpy.ndarray
    :param b_val: Vector corresponding to the reachable set estimate at a given timestep for two state variables, a (3, 1) array
    :type b_val: numpy.ndarray
    :param p: The order of the p-norm ball, defaults to 2
    :type p: int, optional
    :param grid_n: The side length of the square of points to be used for checking for points inside the p-norm ball, defaults to 50
    :type grid_n: int, optional
    :return: The x and y values included in the p-norm ball
    :rtype: tuple
    """
    x_min, x_max = samples[:, 0].min(), samples[:, 0].max()
    y_min, y_max = samples[:, 1].min(), samples[:, 1].max()
    z_min, z_max = samples[:, 2].min(), samples[:, 2].max()

    x = np.linspace(
        x_min - 0.4 * (x_max - x_min), x_max + 0.4 * (x_max - x_min), grid_n
    )
    y = np.linspace(
        y_min - 0.4 * (y_max - y_min), y_max + 0.4 * (y_max - y_min), grid_n
    )
    z = np.linspace(
        z_min - 0.4 * (z_max - z_min), z_max + 0.4 * (z_max - z_min), grid_n
    )
    
    d0, d1, d2 = np.meshgrid(x, y, z)
    d3 = np.array([d0.flatten(), d1.flatten(), d2.flatten()]).T
    
    xs, ys, zs = [], [], []
    for trio in d3:
        if np.linalg.norm(A_val @ trio.reshape((3, 1)) - b_val, ord=p) <= 1:
            xs.append(trio[0])
            ys.append(trio[1])
            zs.append(trio[2])
    
    return xs, ys, zs

import cvxpy as cp
import math
import numpy as np

from collections import OrderedDict
from functools import partial
from multiprocessing import Pool, cpu_count
from scipy.optimize import minimize_scalar
from tqdm.auto import tqdm


def p_num_samples(epsilon, delta, n_x=3, const=None):
    """Compute the number of samples needed to satisfy the specified probabilistic guarantees for the p-norm ball reachable set estimate

    :param epsilon: The accuracy parameter
    :type epsilon: float
    :param delta: The confidence parameter
    :type delta: float
    :param n_x: The state dimension, defaults to 3
    :type n_x: int
    :param const: The constraints placed on the parameters A and b, defaults to None
    :type const: string, optional
    :return: The number of samples needed to satisfy the specified probabilistic guarantees
    :rtype: int
    """
    if const is None:
        n_theta = 0.5 * (n_x ** 2 + 3 * n_x)
    elif const == "diagonal":
        n_theta = 2 * n_x
    elif const == "scalar":
        n_theta = 1
    N = math.ceil(math.e * (math.log(1 / delta) + n_theta) / (epsilon * (math.e - 1)))
    return N


def solve_p_norm(sample, n_x=3, p=2, const=None):
    """Solves the scenario relaxation problem for the given sample with p-Norm Balls

    :param sample: Sample from dynamical system (num_samples, n_x)
    :type sample: numpy.ndarray
    :param n_x: The state dimension, defaults to 3
    :type n_x: int, optional
    :param p: The order of p-norm, defaults to 2
    :type p: int, optional
    :param const: The constraints placed on the parameters A and b, defaults to None
    :type const: string, optional
    :return: The values of matrix A and vector b corresponding to the optimal p-Norm Ball, as well as the status of the optimizer.
    :rtype: tuple
    """
    if const is None:
        A = cp.Variable((n_x, n_x), symmetric=True)
        b = cp.Variable((n_x, 1))
    elif const == "diagonal":
        a = cp.Variable((n_x, 1))
        A = cp.diag(a)
        b = cp.Variable((n_x, 1))
    elif const == "scalar":
        sigma = cp.Variable()
        A = sigma * np.identity(n_x)
        b = np.zeros((n_x, 1))

    obj = cp.Minimize(-cp.log_det(A))
    constraints = [cp.pnorm(A @ r.reshape(n_x, 1) - b, p=p) <= 1 for r in sample]
    prob = cp.Problem(obj, constraints)
    prob.solve()

    if const != "scalar":
        return A.value, b.value, prob.status
    else:
        return A, b, prob.status


def multi_p_norm(samples, p=2, const=None):
    """Computes a the p-norm ball reachable set estimates across a series of timesteps

    :param samples: The samples from a dynamic system across time, an array of shape (num_samples, timesteps, state_dim)
    :type samples: numpy.ndarray
    :param p: The order of p-norm, defaults to 2
    :type p: int, optional
    :param const: The constraints placed on the parameters A and b, defaults to None
    :type const: string, optional
    :raises ValueError: [description]
    :return: [description]
    :rtype: [type]
    """
    if len(samples.shape) != 3:
        raise ValueError("Samples must be of shape (num_samples, timesteps, state_dim")
    n_x = samples.shape[2]
    keys = ("A", "b", "status")
    solve_p_norm_map = partial(solve_p_norm, n_x=n_x, p=p, const=const)
    p = Pool(cpu_count())
    solutions = [
        dict(zip(keys, sol))
        for sol in tqdm(
            p.imap(solve_p_norm_map, samples.swapaxes(0, 1)), total=samples.shape[1]
        )
    ]

    return solutions


def p_norm_cont(arr, axis, default_val, n_x, A_val, b_val, p, minimum=True):
    """Solve for the optimal value that satisfies the p-Norm Ball conditions at the specified axis

    :param arr: Array of shape (n_x - 1,) containing the independent variables of the p-norm condition
    :type arr: numpy.ndarray
    :param axis: The axis of the dependent variable for which to solve for (i.e. z -> axis=2).
    :type axis: int
    :param default_val: The value to return if no solution for the dependent variable is found that satisfies the p-norm conditions
    :type default_val: float
    :param n_x: The state dimension
    :type n_x: int
    :param A_val: The matrix of shape (n_x, n_x) corresponding to the optimal p-norm ball
    :type A_val: numpy.ndarray
    :param b_val: The vector of shape (n_x, 1) corresponding to the optimal p-norm ball
    :type b_val: numpy.ndarray
    :param p: The order of p-norm
    :type p: int
    :param minimum: True if optimizing for the minimal value of the dependent variable that satisfies the p-norm conditions, defaults to True
    :type minimum: bool, optional
    :return: The value at the specified axis which corresponds the the optimal value of the (n_x, 1) vector that satisfies the p-Norm Ball conditions at the specified axis
    :rtype: float
    """
    vec = cp.Variable((n_x, 1))
    other_dims = list(range(n_x))
    other_dims.remove(axis)
    constraints = [vec[i][0] == arr[j] for i, j in zip(other_dims, range(n_x - 1))]
    constraints.append(cp.pnorm(A_val @ vec - b_val, p=p) <= 1)
    if minimum:
        obj = cp.Minimize(vec[axis])
    else:
        obj = cp.Maximize(vec[axis])
    prob = cp.Problem(obj, constraints)

    try:
        prob.solve()
    except:
        return default_val

    if prob.status != "optimal":
        return default_val

    return vec.value[axis]


def p_norm_cont_proj(arr, axis, default_val, n_x, A_val, b_val, p):
    """Minimizes the p-Norm value with respect to value at the specified axis.

    :param arr: Array of shape (n_x - 1,) containing the independent variables of the p-norm condition.
    :type arr: numpy.ndarray
    :param axis: The axis of the dependent variable for which to solve for (i.e. z -> axis=2).
    :type axis: int
    :param default_val: The value to return if no solution for the dependent variable is found that satisfies the p-norm conditions.
    :type default_val: float
    :param n_x: The state dimension.
    :type n_x: int
    :param A_val: The matrix of shape (n_x, n_x) corresponding to the optimal p-norm ball.
    :type A_val: numpy.ndarray
    :param b_val: The vector of shape (n_x, 1) corresponding to the optimal p-norm ball.
    :type b_val: numpy.ndarray
    :param p: The order of p-norm.
    :type p: int
    :return: The value at the specified axis which corresponds the the minimum p-Norm value of the (n_x, 1)  vector.
    :rtype: float
    """
    vec = np.zeros((n_x))
    other_dims = list(range(n_x))
    other_dims.remove(axis)
    for i, j in zip(other_dims, range(n_x - 1)):
        vec[i] = arr[j]

    def f(x):
        vec[axis] = x
        return np.linalg.norm(A_val @ vec.reshape((n_x, 1)) - b_val, ord=p)

    res = minimize_scalar(f)
    vec[axis] = res.x

    if np.linalg.norm(A_val @ vec.reshape((n_x, 1)) - b_val, ord=p) <= 1:
        return res.x
    else:
        return default_val


def p_compute_contour_2D(sample, A_val, b_val, cont_axis=2, n_x=3, p=2, grid_n=200):
    """Computes the 3D contour for 2 dimensions based on sample data and the A_val, and b_val corresponding to the optimal p-norm ball.

    :param sample: Sample from dynamical system (num_samples, n_x)
    :type sample: numpy.ndarray
    :param A_val: The matrix of shape (n_x, n_x) corresponding to the optimal p-norm ball
    :type A_val: numpy.ndarray
    :param b_val: The vector of shape (n_x, 1) corresponding to the optimal p-norm ball
    :type b_val: numpy.ndarray
    :param cont_axis: The axis for which the contours are to be solved for, defaults to 2
    :type cont_axis: int, optional
    :param n_x: The state dimension, defaults to 3
    :type n_x: int, optional
    :param p: The order of p-norm, defaults to 2
    :type p: int, optional
    :param grid_n: The side length of the cube of points to be used for computing contours, defaults to 200
    :type grid_n: int, optional
    :return: The meshgrid, corresponding computed contour, and the extremum values for the chosen axis
    :rtype: tuple
    """
    x_min, x_max = sample[:, 0].min(), sample[:, 0].max()
    y_min, y_max = sample[:, 1].min(), sample[:, 1].max()
    z_min, z_max = sample[:, 2].min(), sample[:, 2].max()

    x = np.linspace(
        x_min - 0.4 * (x_max - x_min), x_max + 0.4 * (x_max - x_min), grid_n
    )
    y = np.linspace(
        y_min - 0.4 * (y_max - y_min), y_max + 0.4 * (y_max - y_min), grid_n
    )
    z = np.linspace(
        x_min - 0.4 * (z_max - z_min), z_max + 0.4 * (z_max - z_min), grid_n
    )

    if cont_axis == 2:
        d0, d1 = np.meshgrid(x, y)
        c_min, c_max = z_min, z_max
    elif cont_axis == 1:
        d0, d1 = np.meshgrid(x, z)
        c_min, c_max = y_min, y_max
    elif cont_axis == 0:
        d0, d1 = np.meshgrid(y, z)
        c_min, c_max = x_min, x_max

    d2 = np.array([d0.flatten(), d1.flatten()]).T

    solve_cont_d2 = partial(
        p_norm_cont_proj,
        axis=cont_axis,
        default_val=c_max + 1,
        n_x=n_x,
        A_val=A_val,
        b_val=b_val,
        p=p,
    )
    cont = np.fromiter(map(solve_cont_d2, d2), dtype=np.float64).reshape(grid_n, grid_n)

    return d0, d1, cont, c_min, c_max


def p_compute_contour_3D(
    sample, A_val, b_val, cont_axis=2, n_x=3, p=2, grid_n=200, stretch=0.4
):
    """Computes the 3D contour for 3 dimensions based on sample data and the A_val, and b_val corresponding to the optimal p-norm ball.

    :param sample: Sample from dynamical system (num_samples, n_x)
    :type sample: numpy.ndarray
    :param A_val: The matrix of shape (n_x, n_x) corresponding to the optimal p-norm ball
    :type A_val: numpy.ndarray
    :param b_val: The vector of shape (n_x, 1) corresponding to the optimal p-norm ball
    :type b_val: numpy.ndarray
    :param cont_axis: The axis for which the contours are to be solved for, defaults to 2
    :type cont_axis: int, optional
    :param n_x: The state dimension, defaults to 3
    :type n_x: int, optional
    :param p: The order of p-norm, defaults to 2
    :type p: int, optional
    :param grid_n: The side length of the cube of points to be used for computing contours, defaults to 200
    :type grid_n: int, optional
    :param stretch: The factor by which to stretch the grid used to compute the contour, defaults to 0.4
    :type stretch: float, optional
    :param minimum: True if optimizing for the minimal value of the dependent variable that satisfies the p-norm conditions, defaults to True
    :type minimum: bool, optional
    :return: The meshgrid, corresponding computed contour, and the extremum values for the chosen axis
    :rtype: tuple
    """
    x_min, x_max = sample[:, 0].min(), sample[:, 0].max()
    y_min, y_max = sample[:, 1].min(), sample[:, 1].max()
    z_min, z_max = sample[:, 2].min(), sample[:, 2].max()

    x = np.linspace(
        x_min - stretch * (x_max - x_min), x_max + stretch * (x_max - x_min), grid_n
    )
    y = np.linspace(
        y_min - stretch * (y_max - y_min), y_max + stretch * (y_max - y_min), grid_n
    )
    z = np.linspace(
        x_min - stretch * (z_max - z_min), z_max + stretch * (z_max - z_min), grid_n
    )

    if cont_axis == 2:
        d0, d1 = np.meshgrid(x, y)
        c_min, c_max = z_min, z_max
    elif cont_axis == 1:
        d0, d1 = np.meshgrid(x, z)
        c_min, c_max = y_min, y_max
    elif cont_axis == 0:
        d0, d1 = np.meshgrid(y, z)
        c_min, c_max = x_min, x_max

    d2 = np.array([d0.flatten(), d1.flatten()]).T

    solve_cont_d2_min = partial(
        p_norm_cont,
        axis=cont_axis,
        default_val=c_max + 1,
        n_x=n_x,
        A_val=A_val,
        b_val=b_val,
        p=p,
        minimum=True,
    )

    solve_cont_d2_max = partial(
        p_norm_cont,
        axis=cont_axis,
        default_val=c_min - 1,
        n_x=n_x,
        A_val=A_val,
        b_val=b_val,
        p=p,
        minimum=False,
    )

    cont_min = np.fromiter(map(solve_cont_d2_min, d2), dtype=np.float64).reshape(
        grid_n, grid_n
    )
    cont_max = np.fromiter(map(solve_cont_d2_max, d2), dtype=np.float64).reshape(
        grid_n, grid_n
    )

    return d0, d1, cont_min, cont_max, c_min, c_max


def p_compute_vals(sample, A_val, b_val, p=2, grid_n=200):
    """Computes the values within a p-norm ball in 1 dimension

    :param sample: The sample from a specific time step, an array of shape (num_samples,)
    :type sample: numpy.ndarray
    :param A_val: The matrix of shape (1, 1) corresponding to the optimal p-norm ball
    :type A_val: numpy.ndarray
    :param b_val: The vector of shape (1, 1) corresponding to the optimal p-norm ball
    :type b_val: numpy.ndarray
    :param p: The order of p-norm, defaults to 2
    :type p: int, optional
    :param grid_n: The number of points to test for the p-norm ball estimation at each a given time step, defaults to 200
    :type grid_n: int, optional
    :return: The values within the p-norm ball
    :rtype: list
    """
    # assuming sample is (num_samples,) shaped array
    y_min, y_max = sample.min(), sample.max()
    y = np.linspace(
        y_min - 0.4 * (y_max - y_min), y_max + 0.4 * (y_max - y_min), grid_n
    )

    vals = []
    for v in y:
        try:
            if np.linalg.norm(A_val @ np.array([[v]]) - b_val, ord=p) <= 1:
                vals.append(v)
        except ValueError:
            pass
    return vals


def p_get_dict(i, samples, solution_list, items, grid_n=50):
    """Generates a dictionary where keys are the passed in labels and values are the output of p_compute_contour_3D, the contour information for a p-norm ball reachable set estimate at a given time

    :param i: The index
    :type i: int
    :param samples: Array of shape (num_samples, time, 3)
    :type samples: numpy.ndarray
    :param solution_list: List of solutions where each solution is a dictionary with keys ("A", "b", "status"), defaults to None
    :type solution_list: list, optional
    :param items: A list of specific indices corresponding to the times at which the p-norm ball reachable set estimate will be chosen
    :type items: list
    :param grid_n: The side length of the cube of points to be used for computing contours, defaults to 50
    :type grid_n: int, optional
    :return: A dictionary where keys are the passed in labels and values are the output of p_compute_contour_3D, the contour information for a p-norm ball reachable set estimate at a given time
    :rtype: dict
    """
    labels = ["xv", "yv", "z_cont", "z_cont2", "z_min", "z_max"]
    index = items[i]
    A_i, b_i = solution_list[index]["A"], solution_list[index]["b"]
    return dict(
        zip(labels, p_compute_contour_3D(samples[:, index, :], A_i, b_i, grid_n=grid_n))
    )


def p_dict_list(
    samples, solution_list, num_parts, num_indices=20, logspace=True, grid_n=50
):
    """Generates a list of dictionaries, each containing the contour information for a p-norm ball reachable set estimate at a given time

    :param samples: Array of shape (num_samples, time, 3)
    :type samples: numpy.ndarray
    :param solution_list: List of solutions where each solution is a dictionary with keys ("A", "b", "status"), defaults to None
    :type solution_list: list, optional
    :param num_parts: The number of total timesteps for the samples
    :type num_parts: int
    :param num_indices: The number of indices corresponding to the number of reachable set estimates made at different times, defaults to 20
    :type num_indices: int, optional
    :param logspace: If True, a logarithmic scale is used for choosing times to compute reachable set estimates, defaults to True
    :type logspace: bool, optional
    :param grid_n: The side length of the cube of points to be used for computing contours, defaults to 50
    :type grid_n: int, optional
    :return: A list of dictionaries, each containing the contour information for a p-norm ball reachable set estimate at a given time
    :rtype: list
    """
    if logspace:
        log_ceil = np.log10(num_parts)
        items = list(
            OrderedDict.fromkeys(
                [int(i) - 1 for i in np.logspace(0, log_ceil, num_indices)]
            )
        )
    else:
        items = [int(i) for i in np.linspace(0, num_parts, num_indices)]

    get_dict = partial(
        p_get_dict,
        samples=samples,
        solution_list=solution_list,
        items=items,
        grid_n=grid_n,
    )

    p = Pool(cpu_count())
    dict_list = [
        d for d in tqdm(p.imap(get_dict, np.arange(len(items))), total=len(items))
    ]

    return dict_list


def p_emp_estimate(samples, A_val, b_val, n_x=3, p=2):
    """Computes the ratio of samples within the estimated reachable set for the p-norm ball reachable set estimation

    :param samples: Sample from dynamical system (num_samples, n_x)
    :type samples: numpy.ndarray
    :param A_val: The matrix of shape (n_x, n_x) corresponding to the optimal p-norm ball
    :type A_val: numpy.ndarray
    :param b_val: The vector of shape (n_x, 1) corresponding to the optimal p-norm ball
    :type b_val: numpy.ndarray
    :param n_x: The state dimension, defaults to 3
    :type n_x: int, optional
    :param p: The order of p-norm, defaults to 2
    :type p: int, optional
    :return: The ratio of samples within the estimated reachability set
    :rtype: float
    """
    num_samples = samples.shape[0]
    count = 0
    for sample in samples:
        vec = sample.reshape(n_x, 1)
        if np.linalg.norm(A_val @ vec - b_val, ord=p) <= 1:
            count += 1
    return count / num_samples


def p_get_reachable_2D(samples, A_val, b_val, p=2, grid_n=50):
    """Obtains the reachable set estimate for a set of samples at a give timestep. A utility function for plotting the reachable set estimate across all timesteps for two state variables.

    :param samples: Samples of shape (num_samples, 2)
    :type samples: numpy.ndarray
    :param A_val: Matrix corresponding to the reachable set estimate at a given timestep for two state variables, a (2, 2) array
    :type A_val: numpy.ndarray
    :param b_val: Vector corresponding to the reachable set estimate at a given timestep for two state variables, a (2, 1) array
    :type b_val: numpy.ndarray
    :param p: The order of the p-norm ball, defaults to 2
    :type p: int, optional
    :param grid_n: The side length of the square of points to be used for checking for points inside the p-norm ball, defaults to 50
    :type grid_n: int, optional
    :return: The x and y values included in the p-norm ball
    :rtype: tuple
    """
    x_min, x_max = samples[:, 0].min(), samples[:, 0].max()
    y_min, y_max = samples[:, 1].min(), samples[:, 1].max()

    x = np.linspace(
        x_min - 0.4 * (x_max - x_min), x_max + 0.4 * (x_max - x_min), grid_n
    )
    y = np.linspace(
        y_min - 0.4 * (y_max - y_min), y_max + 0.4 * (y_max - y_min), grid_n
    )

    d0, d1 = np.meshgrid(x, y)
    d2 = np.array([d0.flatten(), d1.flatten()]).T

    xs, ys = [], []
    for pair in d2:
        if np.linalg.norm(A_val @ pair.reshape((2, 1)) - b_val, ord=p) <= 1:
            xs.append(pair[0])
            ys.append(pair[1])

    return xs, ys

def p_get_reachable_3D(samples, A_val, b_val, p=2, grid_n=25):
    """Obtains the reachable set estimate for a set of samples at a give timestep. A utility function for plotting the reachable set estimate across all timesteps for three state variables.

    :param samples: Samples of shape (num_samples, 3)
    :type samples: numpy.ndarray
    :param A_val: Matrix corresponding to the reachable set estimate at a given timestep for two state variables, a (3, 3) array
    :type A_val: numpy.ndarray
    :param b_val: Vector corresponding to the reachable set estimate at a given timestep for two state variables, a (3, 1) array
    :type b_val: numpy.ndarray
    :param p: The order of the p-norm ball, defaults to 2
    :type p: int, optional
    :param grid_n: The side length of the square of points to be used for checking for points inside the p-norm ball, defaults to 50
    :type grid_n: int, optional
    :return: The x and y values included in the p-norm ball
    :rtype: tuple
    """
    x_min, x_max = samples[:, 0].min(), samples[:, 0].max()
    y_min, y_max = samples[:, 1].min(), samples[:, 1].max()
    z_min, z_max = samples[:, 2].min(), samples[:, 2].max()

    x = np.linspace(
        x_min - 0.4 * (x_max - x_min), x_max + 0.4 * (x_max - x_min), grid_n
    )
    y = np.linspace(
        y_min - 0.4 * (y_max - y_min), y_max + 0.4 * (y_max - y_min), grid_n
    )
    z = np.linspace(
        z_min - 0.4 * (z_max - z_min), z_max + 0.4 * (z_max - z_min), grid_n
    )
    
    d0, d1, d2 = np.meshgrid(x, y, z)
    d3 = np.array([d0.flatten(), d1.flatten(), d2.flatten()]).T
    
    xs, ys, zs = [], [], []
    for trio in d3:
        if np.linalg.norm(A_val @ trio.reshape((3, 1)) - b_val, ord=p) <= 1:
            xs.append(trio[0])
            ys.append(trio[1])
            zs.append(trio[2])
    
    return xs, ys, zs

Compute the number of samples needed to satisfy the specified probabilistic guarantees for the p-norm ball reachable set estimate      :param epsilon: The accuracy parameter     :type epsilon: float     :param delta: The confidence parameter     :type delta: float     :param n_x: The state dimension, defaults to 3     :type n_x: int     :param const: The constraints placed on the parameters A and b, defaults to None     :type const: string, optional     :return: The number of samples needed to satisfy the specified probabilistic guarantees     :rtype: int Solves the scenario relaxation problem for the given sample with p-Norm Balls      :param sample: Sample from dynamical system (num_samples, n_x)     :type sample: numpy.ndarray     :param n_x: The state dimension, defaults to 3     :type n_x: int, optional     :param p: The order of p-norm, defaults to 2     :type p: int, optional     :param const: The constraints placed on the parameters A and b, defaults to None     :type const: string, optional     :return: The values of matrix A and vector b corresponding to the optimal p-Norm Ball, as well as the status of the optimizer.     :rtype: tuple Computes a the p-norm ball reachable set estimates across a series of timesteps      :param samples: The samples from a dynamic system across time, an array of shape (num_samples, timesteps, state_dim)     :type samples: numpy.ndarray     :param p: The order of p-norm, defaults to 2     :type p: int, optional     :param const: The constraints placed on the parameters A and b, defaults to None     :type const: string, optional     :raises ValueError: [description]     :return: [description]     :rtype: [type] Solve for the optimal value that satisfies the p-Norm Ball conditions at the specified axis      :param arr: Array of shape (n_x - 1,) containing the independent variables of the p-norm condition     :type arr: numpy.ndarray     :param axis: The axis of the dependent variable for which to solve for (i.e. z -> axis=2).     :type axis: int     :param default_val: The value to return if no solution for the dependent variable is found that satisfies the p-norm conditions     :type default_val: float     :param n_x: The state dimension     :type n_x: int     :param A_val: The matrix of shape (n_x, n_x) corresponding to the optimal p-norm ball     :type A_val: numpy.ndarray     :param b_val: The vector of shape (n_x, 1) corresponding to the optimal p-norm ball     :type b_val: numpy.ndarray     :param p: The order of p-norm     :type p: int     :param minimum: True if optimizing for the minimal value of the dependent variable that satisfies the p-norm conditions, defaults to True     :type minimum: bool, optional     :return: The value at the specified axis which corresponds the the optimal value of the (n_x, 1) vector that satisfies the p-Norm Ball conditions at the specified axis     :rtype: float Minimizes the p-Norm value with respect to value at the specified axis.      :param arr: Array of shape (n_x - 1,) containing the independent variables of the p-norm condition.     :type arr: numpy.ndarray     :param axis: The axis of the dependent variable for which to solve for (i.e. z -> axis=2).     :type axis: int     :param default_val: The value to return if no solution for the dependent variable is found that satisfies the p-norm conditions.     :type default_val: float     :param n_x: The state dimension.     :type n_x: int     :param A_val: The matrix of shape (n_x, n_x) corresponding to the optimal p-norm ball.     :type A_val: numpy.ndarray     :param b_val: The vector of shape (n_x, 1) corresponding to the optimal p-norm ball.     :type b_val: numpy.ndarray     :param p: The order of p-norm.     :type p: int     :return: The value at the specified axis which corresponds the the minimum p-Norm value of the (n_x, 1)  vector.     :rtype: float Computes the 3D contour for 2 dimensions based on sample data and the A_val, and b_val corresponding to the optimal p-norm ball.      :param sample: Sample from dynamical system (num_samples, n_x)     :type sample: numpy.ndarray     :param A_val: The matrix of shape (n_x, n_x) corresponding to the optimal p-norm ball     :type A_val: numpy.ndarray     :param b_val: The vector of shape (n_x, 1) corresponding to the optimal p-norm ball     :type b_val: numpy.ndarray     :param cont_axis: The axis for which the contours are to be solved for, defaults to 2     :type cont_axis: int, optional     :param n_x: The state dimension, defaults to 3     :type n_x: int, optional     :param p: The order of p-norm, defaults to 2     :type p: int, optional     :param grid_n: The side length of the cube of points to be used for computing contours, defaults to 200     :type grid_n: int, optional     :return: The meshgrid, corresponding computed contour, and the extremum values for the chosen axis     :rtype: tuple Computes the 3D contour for 3 dimensions based on sample data and the A_val, and b_val corresponding to the optimal p-norm ball.      :param sample: Sample from dynamical system (num_samples, n_x)     :type sample: numpy.ndarray     :param A_val: The matrix of shape (n_x, n_x) corresponding to the optimal p-norm ball     :type A_val: numpy.ndarray     :param b_val: The vector of shape (n_x, 1) corresponding to the optimal p-norm ball     :type b_val: numpy.ndarray     :param cont_axis: The axis for which the contours are to be solved for, defaults to 2     :type cont_axis: int, optional     :param n_x: The state dimension, defaults to 3     :type n_x: int, optional     :param p: The order of p-norm, defaults to 2     :type p: int, optional     :param grid_n: The side length of the cube of points to be used for computing contours, defaults to 200     :type grid_n: int, optional     :param stretch: The factor by which to stretch the grid used to compute the contour, defaults to 0.4     :type stretch: float, optional     :param minimum: True if optimizing for the minimal value of the dependent variable that satisfies the p-norm conditions, defaults to True     :type minimum: bool, optional     :return: The meshgrid, corresponding computed contour, and the extremum values for the chosen axis     :rtype: tuple Computes the values within a p-norm ball in 1 dimension      :param sample: The sample from a specific time step, an array of shape (num_samples,)     :type sample: numpy.ndarray     :param A_val: The matrix of shape (1, 1) corresponding to the optimal p-norm ball     :type A_val: numpy.ndarray     :param b_val: The vector of shape (1, 1) corresponding to the optimal p-norm ball     :type b_val: numpy.ndarray     :param p: The order of p-norm, defaults to 2     :type p: int, optional     :param grid_n: The number of points to test for the p-norm ball estimation at each a given time step, defaults to 200     :type grid_n: int, optional     :return: The values within the p-norm ball     :rtype: list # assuming sample is (num_samples,) shaped array Generates a dictionary where keys are the passed in labels and values are the output of p_compute_contour_3D, the contour information for a p-norm ball reachable set estimate at a given time      :param i: The index     :type i: int     :param samples: Array of shape (num_samples, time, 3)     :type samples: numpy.ndarray     :param solution_list: List of solutions where each solution is a dictionary with keys ("A", "b", "status"), defaults to None     :type solution_list: list, optional     :param items: A list of specific indices corresponding to the times at which the p-norm ball reachable set estimate will be chosen     :type items: list     :param grid_n: The side length of the cube of points to be used for computing contours, defaults to 50     :type grid_n: int, optional     :return: A dictionary where keys are the passed in labels and values are the output of p_compute_contour_3D, the contour information for a p-norm ball reachable set estimate at a given time     :rtype: dict Generates a list of dictionaries, each containing the contour information for a p-norm ball reachable set estimate at a given time      :param samples: Array of shape (num_samples, time, 3)     :type samples: numpy.ndarray     :param solution_list: List of solutions where each solution is a dictionary with keys ("A", "b", "status"), defaults to None     :type solution_list: list, optional     :param num_parts: The number of total timesteps for the samples     :type num_parts: int     :param num_indices: The number of indices corresponding to the number of reachable set estimates made at different times, defaults to 20     :type num_indices: int, optional     :param logspace: If True, a logarithmic scale is used for choosing times to compute reachable set estimates, defaults to True     :type logspace: bool, optional     :param grid_n: The side length of the cube of points to be used for computing contours, defaults to 50     :type grid_n: int, optional     :return: A list of dictionaries, each containing the contour information for a p-norm ball reachable set estimate at a given time     :rtype: list Computes the ratio of samples within the estimated reachable set for the p-norm ball reachable set estimation      :param samples: Sample from dynamical system (num_samples, n_x)     :type samples: numpy.ndarray     :param A_val: The matrix of shape (n_x, n_x) corresponding to the optimal p-norm ball     :type A_val: numpy.ndarray     :param b_val: The vector of shape (n_x, 1) corresponding to the optimal p-norm ball     :type b_val: numpy.ndarray     :param n_x: The state dimension, defaults to 3     :type n_x: int, optional     :param p: The order of p-norm, defaults to 2     :type p: int, optional     :return: The ratio of samples within the estimated reachability set     :rtype: float Obtains the reachable set estimate for a set of samples at a give timestep. A utility function for plotting the reachable set estimate across all timesteps for two state variables.      :param samples: Samples of shape (num_samples, 2)     :type samples: numpy.ndarray     :param A_val: Matrix corresponding to the reachable set estimate at a given timestep for two state variables, a (2, 2) array     :type A_val: numpy.ndarray     :param b_val: Vector corresponding to the reachable set estimate at a given timestep for two state variables, a (2, 1) array     :type b_val: numpy.ndarray     :param p: The order of the p-norm ball, defaults to 2     :type p: int, optional     :param grid_n: The side length of the square of points to be used for checking for points inside the p-norm ball, defaults to 50     :type grid_n: int, optional     :return: The x and y values included in the p-norm ball     :rtype: tuple Obtains the reachable set estimate for a set of samples at a give timestep. A utility function for plotting the reachable set estimate across all timesteps for three state variables.      :param samples: Samples of shape (num_samples, 3)     :type samples: numpy.ndarray     :param A_val: Matrix corresponding to the reachable set estimate at a given timestep for two state variables, a (3, 3) array     :type A_val: numpy.ndarray     :param b_val: Vector corresponding to the reachable set estimate at a given timestep for two state variables, a (3, 1) array     :type b_val: numpy.ndarray     :param p: The order of the p-norm ball, defaults to 2     :type p: int, optional     :param grid_n: The side length of the square of points to be used for checking for points inside the p-norm ball, defaults to 50     :type grid_n: int, optional     :return: The x and y values included in the p-norm ball     :rtype: tuple