<commit_before>from equadratures.sampling_methods.sampling_template import Sampling
import numpy as np
from scipy.special import betaln
import bisect
from scipy.optimize import bisect as bisect_root_solve


class Induced(Sampling):
    """
    The class defines an Induced sampling object.
    :param list parameters: A list of parameters,
                            where each element of the list is
                            an instance of the Parameter class.
    :param Basis basis: An instance of the Basis class
                        corresponding to the multi-index set used.
    """

    def __init__(self, parameters, basis, orders=None):
        self.parameters = parameters
        self.basis = basis
        if orders is not None:
            self.basis.set_orders(orders)
        self.dimensions = len(self.parameters)
        self.basis_entries = basis.cardinality
        sampling_ratio = 7 * self.dimensions
        self.samples_number = int(sampling_ratio * np.max(self.basis.orders))
        self._set_points(orders)
        super(Induced, self)._init_(self.parameters,
                                      self.basis,
                                      self.points,
                                      self.weights)

    def _set_points(self, orders=None):
        """
        Performs induced sampling on
        Jacobi/Freud/Half-Line Freud classes of measures and
        produces the corresponding quadrature points for integration.
        :param Poly self:
            An instance of the Poly class.
        :param list orders:
            A list of the highest polynomial orders along each dimension.
        """
        quadrature_points = np.zeros((self.samples_number, self.dimensions))
        quadrature_points = np.apply_along_axis(self._additive_mixture_sampling,
                                                1, quadrature_points)
        p = {}
        if self.dimensions == 1:
            poly, _, _ = self.parameters[0].\
                         _get_orthogonal_polynomial(quadrature_points,
                                                    int(np.max(self.basis.elements))
                                                    )
            return poly
        else:
            for i in range(0, self.dimensions):
                p[i], _, _ = self.parameters[i].\
                             _get_orthogonal_polynomial(quadrature_points[:, i],
                                                        int(np.max(self.basis.elements[:, i]))
                                                        )
        # One loop for polynomials
        polynomial = np.ones((self.basis_entries, self.samples_number))
        for k in range(self.dimensions):
            basis_entries_this_dim = self.basis.elements[:, k].astype(int)
            polynomial *= p[k][basis_entries_this_dim]
        wts = 1.0/np.sum(polynomial**2, 0)
        weights = wts * 1.0/np.sum(wts)
        self.points = quadrature_points
        self.weights = weights

    def _additive_mixture_sampling(self, _placeholder):
        """
        Performs tensorial sampling from the additive mixture of induced distributions.
        Parameters:
        ---------
        _placeholder: np.ndarray
            This is an internal variable for the np.apply_along_axis function
            an array of size (dimension*1)
        Returns:
        ---------
        x: np.ndarray
            A vector x, of size (dimension*1)
            A single quadrature point sampled from
            the additive mixture of the induced distributions
        """
        # TODO add a total order index set with random samples
        # The above would be necessary in higher dimensions
        # sample the set of indices used in this sample
        indexset = self.basis.elements
        sampled_row_number = np.random.randint(0, indexset.shape[0])
        index_set_used = indexset[sampled_row_number, :]

        # Sample uniformly for inverse CDF
        sampled_cdf_values = np.random.rand(self.dimensions, 1)

        x = self._multi_variate_sampling(sampled_cdf_values, index_set_used)

        return x

    def _multi_variate_sampling(self, sampled_cdf_values, index_set_used):
        """
        Sample each dimension of the input vairable with their individual
        1. probablity measure parameters
        2. the uniformly sampled cdf value for inverse cdf sampling
        3. the degree of the univariate orthogonal polynomial
        Parameters
        ---------
        sampled_cdf_values: np.ndarray
            A list of uniformly generated CDF values
            for sampling in each dimension
            array size (dimension*1)
        index_set_used: np.ndarray
            The order of each univariate orthogonal polynomial
            used in each univariate induced dittributions
            array size(dimension*1)
        Returns
        ---------
        x: np.ndarray
            A vector x, of size (dimension*1)
            A single quadrature point sampled from
            the additive mixture of the induced distributions
        """
        univariate_input = zip(self.parameters, sampled_cdf_values, index_set_used)
        x = np.fromiter(map(self._univariate_sampling, univariate_input), float)
        return x

    def _univariate_sampling(self, _input):
        """
        Generate the class of probability measures
        the input distribution belongs to.
        And return the univariate function object to used to sample
        Parameters
        ---------
        _input: tuple
            Internal variable passing
            a tuple of 3 elements
            1. probablity measure parameters of type Parameter
            2. the uniformly sampled cdf value for inverse cdf sampling
            of a type float between 0 and 1
            3. the degree of the univariate orthogonal polynomial
            of type int
        Returns
        ---------
        sampled_value: float
            a scalar value sampled
            according to the induced distribution
        """
        parameter = _input[0]
        uniform_cdf_value = np.asscalar(_input[1])
        order = np.asscalar(_input[2])
        # TODO elaborate the following distribution types to cover all param_types
        if order == 0.0:
            sampled_value = parameter.mean
            return sampled_value
            # print("mean", parameter.mean)
        if parameter.name == "Uniform":
            # TODO Only uniform between -1 and 1 is done for now
            # define shape parameters for the Jacobi matrix
            alpha = 0
            beta = 0
            parameter.order = order
            sampled_value = self.inverse_induced_jacobi(alpha,
                                                          beta,
                                                          uniform_cdf_value,
                                                          parameter)

            return sampled_value
        elif parameter.name in ["Gaussian"]:
            # TODO add Freud sampling algorithm
            pass
        pass

    def inverse_induced_jacobi(self, alpha, beta, uniform_cdf_value, parameter):
        """
        Sampling of a univariate inverse induced Jacobi distributions
        Parameters
        ---------
        parameter: Parameter
            Parameter class to specify the ditribution parametersA
        uniform_cdf_value: float
            A floating point number between 0 and 1
            for the inverse CDF sampling method
        order:
            The order of the induced polynomial
        Returns:
        ---------
        sampled_value: float
            a scalar value sampled
            according to the induced distribution
        """
        # TODO use Chebyshev interpolants for the induced distributions
        # The method above pre-evaluates the CDF
        # Hence gives efficient subsequent evaluation so to speak

        # Use Markov-Stiltjies inequality for initial x value interval guess
        order = int(parameter.order)
        zeroes, _ = parameter._get_local_quadrature(order-1)
        # obtain current recurrence coefficient
        ab = parameter.get_recurrence_coefficients((order)*2 + 360 + 1)
        for root in zeroes:
            ab = self._quadratic_modification(ab, root)
            ab[0, 1] = 1
        induced_points, induced_weights = parameter._get_local_quadrature(358, ab)
        # insert lower bound of x in jacobi distribution
        interval_points = np.insert(induced_points, 0, -1)
        # Cumulative sums of induced quadrature weights are a strict bound for the cdf
        strict_bounds = np.cumsum(induced_weights)
        strict_bounds = np.insert(strict_bounds, len(strict_bounds), 1)
        strict_bounds = np.insert(strict_bounds, 0, 0)
        interval_index = bisect.bisect_left(strict_bounds, uniform_cdf_value)
        interval_index_hi = interval_index+1
        if interval_index_hi >= 360:
            interval_index_hi = 359
        interval_lo = interval_points[interval_index-1]
        interval_hi = interval_points[interval_index_hi]

        # Solver function for inverse CDF where F(x)-u = 0
        def F(x):
            value = self.induced_jacobi_evaluation(alpha,
                                                     beta,
                                                     x,
                                                     parameter)
            value = value - uniform_cdf_value
            return value

        sampled_value = bisect_root_solve(F, interval_lo, interval_hi, xtol=0.00005)

        return sampled_value

    def induced_jacobi_evaluation(self, alpha, beta, x, parameter):
        """
        Evaluate induced Jacobi distribution CDF value
        Parameters
        ---------
        alpha: double
            alpha shape parameter
            of the jacobi distribution
        beta: double
            beta shape parameter
            of the jacobi distribution
        x: double
            input variable of the distribution
        parameter: Parameter
            the parameter class for the distribution
        Returns
        ---------
        F: double
            The CDF value evaluated at x
        """
        # M-order quadrature for CDF estimation
        _complementary = False
        M = 12
        order = parameter.order
        # Avoid division by zero and simplify computation at bounds
        if int(x) == 1:
            F = 1
            return F
        if x == -1:
            F = 0
            return F
        # find median by the Mhaskar-Rakhmanov-Saff numbers
        if order > 0:
            median = (beta**2 - alpha**2)/((2*order + alpha + beta)**2)
        else:
            median = 2.0/(1.0 + (alpha + 1.0)/(beta + 1.0)) - 1.0
        # Set up computation for the complementary value
        if x > median:
            x = -x
            _complementary = True
            alpha, beta = beta, alpha
            parameter.shape_parameter_A, parameter.shape_parameter_B = \
                parameter.shape_parameter_B, parameter.shape_parameter_A

        # Obtain the zeroes of this particlar polynomial
        zeroes, _ = parameter._get_local_quadrature(order-1)
        ab = parameter.get_recurrence_coefficients(order)

        # This is the (inverse) n'th root of the leading coefficient square of p_n
        # We'll use it for scaling later
        scaling_kn_factor = np.exp(-1.0/order
                                   * np.sum(np.log(ab[:, 1])))

        # Recurrence coefficients for the quadrature rule
        A = np.floor(abs(alpha))
        recurrence_ab = parameter.get_recurrence_coefficients(2*order+A+M)
        logfactor = 0.0  # factor to keep the modified distribution as a pdf

        # n quadratic modifications
        # for the induced distribution
        # recurrence coefficients
        for i in range(0, int(order)):
            quadratic_root = (2.0/(x+1.0)) * (zeroes[i] + 1.0) - 1.0
            recurrence_ab = self._quadratic_modification(recurrence_ab,
                                                          quadratic_root)
            logfactor += np.log(recurrence_ab[0, 1] *
                                ((x+1.0)/2.0)**2 *
                                scaling_kn_factor)
            recurrence_ab[0, 1] = 1

        linear_root = (3-x)/(1+x)

        # A quadratic modifications
        # for the induced distribution
        # recurrence coefficients
        for j in range(0, int(A)):
            recurrence_ab = self._linear_modification(recurrence_ab,
                                                       linear_root)
            logfactor += logfactor + np.log(ab[0, 1] *
                                            1.0/2.0 *
                                            (x+1.0))
            recurrence_ab[0, 1] = 1
        u, w = parameter._get_local_quadrature(M-1, recurrence_ab)
        integral = np.dot(w, (2.0 - 1.0/2.0 * (u+1.) * (x+1.))**(alpha-A))

        F = np.exp(logfactor -
                   alpha*np.log(2.0) -
                   betaln(beta+1.0, alpha+1.0) -
                   np.log(beta+1.0)+(beta+1)*np.log((x+1.0)/2.0))*integral
        F = np.asscalar(F)

        if _complementary:
            F = 1-F

        return F

    def _linear_modification(self, ab, x0):
        """
        Performs a linear modification of the
        orthogonal polynomial recurrence coefficients.
        It transforms the coefficients
        such that the new coefficients
        are associated with a polynomial family
        that is orthonormal under the weight (x - x0)**2
        Parameters
        ---------
        ab: np.ndarray
            numpy array of the alpha beta recurrence coefficients
        x0: float
            The shift in the weights
        Returns
        ---------
        A N-by-2 matrix that contains the modified recurrence coefficients.
        """

        alpha = ab[:, 0]
        N = len(alpha)
        beta = ab[:, 1]
        sign_value = np.sign(alpha[0] - x0)

        r = np.reshape(np.abs(self._polynomial_ratios(alpha,
                                                     beta, x0, N-1)),
                       (N-1, 1))

        acorrect = np.zeros((N-1, 1))
        bcorrect = np.zeros((N-1, 1))
        ab = np.zeros((N-1, N-1))

        for i in range(0, N-1):
            acorrect[i] = np.sqrt(beta[i+1]) * 1.0 / r[i]
            bcorrect[i] = np.sqrt(beta[i+1]) * r[i]

        for i in range(1, N-1):
            acorrect[i] = acorrect[i+1] - acorrect[i]
            bcorrect[i] = bcorrect[i] * 1.0/bcorrect[i-1]

        for i in range(0, N-1):
            ab[i, 1] = beta[i] * bcorrect[i]
            ab[i, 0] = alpha[i] + sign_value * acorrect[i]

        return ab

    def _quadratic_modification(self, alphabeta, x0):
        """
        Performs a linear modification of the
        orthogonal polynomial recurrence coefficients.
        It transforms the coefficients
        such that the new coefficients
        are associated with a polynomial family
        that is orthonormal under the weight (x - x0)**2
        Parameters
        ---------
        ab: np.ndarray
            numpy array of the alpha beta recurrence coefficients
        x0: float
            The shift in the weights
        Returns
        ---------
        A N-by-2 matrix that contains the modified recurrence coefficients.
        """
        N = len(alphabeta)
        alpha = alphabeta[:, 0]
        beta = alphabeta[:, 1]

        C = np.reshape(self._christoffel_normalised_polynomials(alpha,
                                                                 beta,
                                                                 x0,
                                                                 (N-1)),
                       [N, 1])
        acorrect = np.zeros((N-2, 1))
        bcorrect = np.zeros((N-2, 1))
        ab = np.zeros((N-2, 2))
        temp1 = np.zeros((N-1, 1))
        for i in range(0, N-1):
            temp1[i] = np.sqrt(beta[i+1]) * C[i+1] * C[i] * 1.0/np.sqrt(1.0 + C[i]**2)
        temp1[0] = np.sqrt(beta[1])*C[1]
        acorrect = np.diff(temp1, axis=0)
        temp1 = 1 + C[0:N-1]**2
        for i in range(0, N-2):
            bcorrect[i] = (1.0 * temp1[i+1]) / (1.0*temp1[i])
        bcorrect[0] = (1.0 + C[1]**2) * 1.0/(C[0]**2)
        for i in range(0, N-2):
            ab[i, 1] = beta[i+1] * bcorrect[i]
            ab[i, 0] = alpha[i+1] + acorrect[i]
        return ab

    def _polynomial_ratios(self, a, b, x, N):
        """
        Evaluates the ratio between successive orthogonal polynomials
        This is a helper method to perform the linear modification
        of the induced distribution.
        Parameters
        ---------
        a: np.ndarray
        alpha recurrence coefficients,
        size(N+1, 1)
        b: np.ndarray
        beta recurrence coefficients,
        size(N+1, 1)
        x: double
        a scalar double value of the evaluated input
        for the induced distribution
        N: int
        Order of the polynomials to be evaluated
        Returns
        ---------
        r: np.ndarray
        successive ratios in a vetor
        of length N
        """
        if N > 0:
            raise ValueError("number of polynomial ratios must be greater than 0")
        if N < len(a):
            raise ValueError("number of polynomial ratios must be less than length of a")
        if N < len(b):
            raise ValueError("number of polynomial ratios must be less than length of b")

        r = np.zeros(N)

        # find p0 and p1 to initialise r
        p0 = 1.0/np.sqrt(b[0])
        p1 = 1.0/np.sqrt(b[1]) * (x - a[0]) * p0
        r1 = p1 / p0
        r[0] = r1

        for q in range(1, N):
            r2 = (x - a[q]) - np.sqrt(b[q])/r1
            r1 = 1.0/np.sqrt(b[q+1]) * r2
            r[q] = r1

        return r

    def _christoffel_normalised_polynomials(self, a, b, x, N):
        """
        Computes the Christoffel normalized
        orthogonal polynomial values
        at x
        Parameters
        ---------
        a: np.ndarray
        alpha recurrence coefficients,
        size(N+1, 1)
        b: np.ndarray
        beta recurrence coefficients,
        size(N+1, 1)
        x: double
        a scalar double value of the evaluated input
        for the induced distribution
        N: int
        Order of the polynomials to be evaluated
        Returns
        ---------
        C: np.ndarray
        successive ratios in a vetor
        of length N
        """
        if N <= 0:
            raise ValueError("No. of Christoffels evaluations must be greater than 0")
        if N > len(a):
            raise ValueError("No. of Christoffels evaluations must be less than len(a)")
        if N > len(b):
            raise ValueError("No. of Christoffels evaluations must be less than len(b)")

        C = np.zeros(N+1)
        # Initialize the polynomials
        C[0] = 1.0/(1.0 * np.sqrt(b[0]))
        if N > 0:
            C[1] = 1.0 / (1.0 * np.sqrt(b[1])) * (x - a[0])
        if N > 1:
            C[2] = 1.0 / np.sqrt(1.0 + C[1]**2) * ((x - a[1]) * C[1] - np.sqrt(b[1]))
            C[2] = C[2] / (1.0 * np.sqrt(b[2]))
        if N > 2:
            for nnn in range(2, N):
                C[nnn+1] = 1.0/np.sqrt(1.0 + C[nnn]**2) * \
                    (
                     (x - a[nnn])*C[nnn] - np.sqrt(b[nnn]) *
                     C[nnn-1]/np.sqrt(1.0+C[nnn-1]**2)
                     )
                C[nnn+1] = C[nnn+1] / np.sqrt(b[nnn+1])
        return C
<commit_msg>change super._init_ to super.__init__ in induced.py file<commit_after>from equadratures.sampling_methods.sampling_template import Sampling
import numpy as np
from scipy.special import betaln
import bisect
from scipy.optimize import bisect as bisect_root_solve


class Induced(Sampling):
    """
    The class defines an Induced sampling object.
    :param list parameters: A list of parameters,
                            where each element of the list is
                            an instance of the Parameter class.
    :param Basis basis: An instance of the Basis class
                        corresponding to the multi-index set used.
    """

    def __init__(self, parameters, basis, orders=None):
        self.parameters = parameters
        self.basis = basis
        if orders is not None:
            self.basis.set_orders(orders)
        self.dimensions = len(self.parameters)
        self.basis_entries = basis.cardinality
        sampling_ratio = 7 * self.dimensions
        self.samples_number = int(sampling_ratio * np.max(self.basis.orders))
        self._set_points(orders)
        super(Induced, self).__init__(self.parameters,
                                      self.basis,
                                      self.points,
                                      self.weights)

    def _set_points(self, orders=None):
        """
        Performs induced sampling on
        Jacobi/Freud/Half-Line Freud classes of measures and
        produces the corresponding quadrature points for integration.
        :param Poly self:
            An instance of the Poly class.
        :param list orders:
            A list of the highest polynomial orders along each dimension.
        """
        quadrature_points = np.zeros((self.samples_number, self.dimensions))
        quadrature_points = np.apply_along_axis(self._additive_mixture_sampling,
                                                1, quadrature_points)
        p = {}
        if self.dimensions == 1:
            poly, _, _ = self.parameters[0].\
                         _get_orthogonal_polynomial(quadrature_points,
                                                    int(np.max(self.basis.elements))
                                                    )
            return poly
        else:
            for i in range(0, self.dimensions):
                p[i], _, _ = self.parameters[i].\
                             _get_orthogonal_polynomial(quadrature_points[:, i],
                                                        int(np.max(self.basis.elements[:, i]))
                                                        )
        # One loop for polynomials
        polynomial = np.ones((self.basis_entries, self.samples_number))
        for k in range(self.dimensions):
            basis_entries_this_dim = self.basis.elements[:, k].astype(int)
            polynomial *= p[k][basis_entries_this_dim]
        wts = 1.0/np.sum(polynomial**2, 0)
        weights = wts * 1.0/np.sum(wts)
        self.points = quadrature_points
        self.weights = weights

    def _additive_mixture_sampling(self, _placeholder):
        """
        Performs tensorial sampling from the additive mixture of induced distributions.
        Parameters:
        ---------
        _placeholder: np.ndarray
            This is an internal variable for the np.apply_along_axis function
            an array of size (dimension*1)
        Returns:
        ---------
        x: np.ndarray
            A vector x, of size (dimension*1)
            A single quadrature point sampled from
            the additive mixture of the induced distributions
        """
        # TODO add a total order index set with random samples
        # The above would be necessary in higher dimensions
        # sample the set of indices used in this sample
        indexset = self.basis.elements
        sampled_row_number = np.random.randint(0, indexset.shape[0])
        index_set_used = indexset[sampled_row_number, :]

        # Sample uniformly for inverse CDF
        sampled_cdf_values = np.random.rand(self.dimensions, 1)

        x = self._multi_variate_sampling(sampled_cdf_values, index_set_used)

        return x

    def _multi_variate_sampling(self, sampled_cdf_values, index_set_used):
        """
        Sample each dimension of the input vairable with their individual
        1. probablity measure parameters
        2. the uniformly sampled cdf value for inverse cdf sampling
        3. the degree of the univariate orthogonal polynomial
        Parameters
        ---------
        sampled_cdf_values: np.ndarray
            A list of uniformly generated CDF values
            for sampling in each dimension
            array size (dimension*1)
        index_set_used: np.ndarray
            The order of each univariate orthogonal polynomial
            used in each univariate induced dittributions
            array size(dimension*1)
        Returns
        ---------
        x: np.ndarray
            A vector x, of size (dimension*1)
            A single quadrature point sampled from
            the additive mixture of the induced distributions
        """
        univariate_input = zip(self.parameters, sampled_cdf_values, index_set_used)
        x = np.fromiter(map(self._univariate_sampling, univariate_input), float)
        return x

    def _univariate_sampling(self, _input):
        """
        Generate the class of probability measures
        the input distribution belongs to.
        And return the univariate function object to used to sample
        Parameters
        ---------
        _input: tuple
            Internal variable passing
            a tuple of 3 elements
            1. probablity measure parameters of type Parameter
            2. the uniformly sampled cdf value for inverse cdf sampling
            of a type float between 0 and 1
            3. the degree of the univariate orthogonal polynomial
            of type int
        Returns
        ---------
        sampled_value: float
            a scalar value sampled
            according to the induced distribution
        """
        parameter = _input[0]
        uniform_cdf_value = np.asscalar(_input[1])
        order = np.asscalar(_input[2])
        # TODO elaborate the following distribution types to cover all param_types
        if order == 0.0:
            sampled_value = parameter.mean
            return sampled_value
            # print("mean", parameter.mean)
        if parameter.name == "Uniform":
            # TODO Only uniform between -1 and 1 is done for now
            # define shape parameters for the Jacobi matrix
            alpha = 0
            beta = 0
            parameter.order = order
            sampled_value = self.inverse_induced_jacobi(alpha,
                                                          beta,
                                                          uniform_cdf_value,
                                                          parameter)

            return sampled_value
        elif parameter.name in ["Gaussian"]:
            # TODO add Freud sampling algorithm
            pass
        pass

    def inverse_induced_jacobi(self, alpha, beta, uniform_cdf_value, parameter):
        """
        Sampling of a univariate inverse induced Jacobi distributions
        Parameters
        ---------
        parameter: Parameter
            Parameter class to specify the ditribution parametersA
        uniform_cdf_value: float
            A floating point number between 0 and 1
            for the inverse CDF sampling method
        order:
            The order of the induced polynomial
        Returns:
        ---------
        sampled_value: float
            a scalar value sampled
            according to the induced distribution
        """
        # TODO use Chebyshev interpolants for the induced distributions
        # The method above pre-evaluates the CDF
        # Hence gives efficient subsequent evaluation so to speak

        # Use Markov-Stiltjies inequality for initial x value interval guess
        order = int(parameter.order)
        zeroes, _ = parameter._get_local_quadrature(order-1)
        # obtain current recurrence coefficient
        ab = parameter.get_recurrence_coefficients((order)*2 + 360 + 1)
        for root in zeroes:
            ab = self._quadratic_modification(ab, root)
            ab[0, 1] = 1
        induced_points, induced_weights = parameter._get_local_quadrature(358, ab)
        # insert lower bound of x in jacobi distribution
        interval_points = np.insert(induced_points, 0, -1)
        # Cumulative sums of induced quadrature weights are a strict bound for the cdf
        strict_bounds = np.cumsum(induced_weights)
        strict_bounds = np.insert(strict_bounds, len(strict_bounds), 1)
        strict_bounds = np.insert(strict_bounds, 0, 0)
        interval_index = bisect.bisect_left(strict_bounds, uniform_cdf_value)
        interval_index_hi = interval_index+1
        if interval_index_hi >= 360:
            interval_index_hi = 359
        interval_lo = interval_points[interval_index-1]
        interval_hi = interval_points[interval_index_hi]

        # Solver function for inverse CDF where F(x)-u = 0
        def F(x):
            value = self.induced_jacobi_evaluation(alpha,
                                                     beta,
                                                     x,
                                                     parameter)
            value = value - uniform_cdf_value
            return value

        sampled_value = bisect_root_solve(F, interval_lo, interval_hi, xtol=0.00005)

        return sampled_value

    def induced_jacobi_evaluation(self, alpha, beta, x, parameter):
        """
        Evaluate induced Jacobi distribution CDF value
        Parameters
        ---------
        alpha: double
            alpha shape parameter
            of the jacobi distribution
        beta: double
            beta shape parameter
            of the jacobi distribution
        x: double
            input variable of the distribution
        parameter: Parameter
            the parameter class for the distribution
        Returns
        ---------
        F: double
            The CDF value evaluated at x
        """
        # M-order quadrature for CDF estimation
        _complementary = False
        M = 12
        order = parameter.order
        # Avoid division by zero and simplify computation at bounds
        if int(x) == 1:
            F = 1
            return F
        if x == -1:
            F = 0
            return F
        # find median by the Mhaskar-Rakhmanov-Saff numbers
        if order > 0:
            median = (beta**2 - alpha**2)/((2*order + alpha + beta)**2)
        else:
            median = 2.0/(1.0 + (alpha + 1.0)/(beta + 1.0)) - 1.0
        # Set up computation for the complementary value
        if x > median:
            x = -x
            _complementary = True
            alpha, beta = beta, alpha
            parameter.shape_parameter_A, parameter.shape_parameter_B = \
                parameter.shape_parameter_B, parameter.shape_parameter_A

        # Obtain the zeroes of this particlar polynomial
        zeroes, _ = parameter._get_local_quadrature(order-1)
        ab = parameter.get_recurrence_coefficients(order)

        # This is the (inverse) n'th root of the leading coefficient square of p_n
        # We'll use it for scaling later
        scaling_kn_factor = np.exp(-1.0/order
                                   * np.sum(np.log(ab[:, 1])))

        # Recurrence coefficients for the quadrature rule
        A = np.floor(abs(alpha))
        recurrence_ab = parameter.get_recurrence_coefficients(2*order+A+M)
        logfactor = 0.0  # factor to keep the modified distribution as a pdf

        # n quadratic modifications
        # for the induced distribution
        # recurrence coefficients
        for i in range(0, int(order)):
            quadratic_root = (2.0/(x+1.0)) * (zeroes[i] + 1.0) - 1.0
            recurrence_ab = self._quadratic_modification(recurrence_ab,
                                                          quadratic_root)
            logfactor += np.log(recurrence_ab[0, 1] *
                                ((x+1.0)/2.0)**2 *
                                scaling_kn_factor)
            recurrence_ab[0, 1] = 1

        linear_root = (3-x)/(1+x)

        # A quadratic modifications
        # for the induced distribution
        # recurrence coefficients
        for j in range(0, int(A)):
            recurrence_ab = self._linear_modification(recurrence_ab,
                                                       linear_root)
            logfactor += logfactor + np.log(ab[0, 1] *
                                            1.0/2.0 *
                                            (x+1.0))
            recurrence_ab[0, 1] = 1
        u, w = parameter._get_local_quadrature(M-1, recurrence_ab)
        integral = np.dot(w, (2.0 - 1.0/2.0 * (u+1.) * (x+1.))**(alpha-A))

        F = np.exp(logfactor -
                   alpha*np.log(2.0) -
                   betaln(beta+1.0, alpha+1.0) -
                   np.log(beta+1.0)+(beta+1)*np.log((x+1.0)/2.0))*integral
        F = np.asscalar(F)

        if _complementary:
            F = 1-F

        return F

    def _linear_modification(self, ab, x0):
        """
        Performs a linear modification of the
        orthogonal polynomial recurrence coefficients.
        It transforms the coefficients
        such that the new coefficients
        are associated with a polynomial family
        that is orthonormal under the weight (x - x0)**2
        Parameters
        ---------
        ab: np.ndarray
            numpy array of the alpha beta recurrence coefficients
        x0: float
            The shift in the weights
        Returns
        ---------
        A N-by-2 matrix that contains the modified recurrence coefficients.
        """

        alpha = ab[:, 0]
        N = len(alpha)
        beta = ab[:, 1]
        sign_value = np.sign(alpha[0] - x0)

        r = np.reshape(np.abs(self._polynomial_ratios(alpha,
                                                     beta, x0, N-1)),
                       (N-1, 1))

        acorrect = np.zeros((N-1, 1))
        bcorrect = np.zeros((N-1, 1))
        ab = np.zeros((N-1, N-1))

        for i in range(0, N-1):
            acorrect[i] = np.sqrt(beta[i+1]) * 1.0 / r[i]
            bcorrect[i] = np.sqrt(beta[i+1]) * r[i]

        for i in range(1, N-1):
            acorrect[i] = acorrect[i+1] - acorrect[i]
            bcorrect[i] = bcorrect[i] * 1.0/bcorrect[i-1]

        for i in range(0, N-1):
            ab[i, 1] = beta[i] * bcorrect[i]
            ab[i, 0] = alpha[i] + sign_value * acorrect[i]

        return ab

    def _quadratic_modification(self, alphabeta, x0):
        """
        Performs a linear modification of the
        orthogonal polynomial recurrence coefficients.
        It transforms the coefficients
        such that the new coefficients
        are associated with a polynomial family
        that is orthonormal under the weight (x - x0)**2
        Parameters
        ---------
        ab: np.ndarray
            numpy array of the alpha beta recurrence coefficients
        x0: float
            The shift in the weights
        Returns
        ---------
        A N-by-2 matrix that contains the modified recurrence coefficients.
        """
        N = len(alphabeta)
        alpha = alphabeta[:, 0]
        beta = alphabeta[:, 1]

        C = np.reshape(self._christoffel_normalised_polynomials(alpha,
                                                                 beta,
                                                                 x0,
                                                                 (N-1)),
                       [N, 1])
        acorrect = np.zeros((N-2, 1))
        bcorrect = np.zeros((N-2, 1))
        ab = np.zeros((N-2, 2))
        temp1 = np.zeros((N-1, 1))
        for i in range(0, N-1):
            temp1[i] = np.sqrt(beta[i+1]) * C[i+1] * C[i] * 1.0/np.sqrt(1.0 + C[i]**2)
        temp1[0] = np.sqrt(beta[1])*C[1]
        acorrect = np.diff(temp1, axis=0)
        temp1 = 1 + C[0:N-1]**2
        for i in range(0, N-2):
            bcorrect[i] = (1.0 * temp1[i+1]) / (1.0*temp1[i])
        bcorrect[0] = (1.0 + C[1]**2) * 1.0/(C[0]**2)
        for i in range(0, N-2):
            ab[i, 1] = beta[i+1] * bcorrect[i]
            ab[i, 0] = alpha[i+1] + acorrect[i]
        return ab

    def _polynomial_ratios(self, a, b, x, N):
        """
        Evaluates the ratio between successive orthogonal polynomials
        This is a helper method to perform the linear modification
        of the induced distribution.
        Parameters
        ---------
        a: np.ndarray
        alpha recurrence coefficients,
        size(N+1, 1)
        b: np.ndarray
        beta recurrence coefficients,
        size(N+1, 1)
        x: double
        a scalar double value of the evaluated input
        for the induced distribution
        N: int
        Order of the polynomials to be evaluated
        Returns
        ---------
        r: np.ndarray
        successive ratios in a vetor
        of length N
        """
        if N > 0:
            raise ValueError("number of polynomial ratios must be greater than 0")
        if N < len(a):
            raise ValueError("number of polynomial ratios must be less than length of a")
        if N < len(b):
            raise ValueError("number of polynomial ratios must be less than length of b")

        r = np.zeros(N)

        # find p0 and p1 to initialise r
        p0 = 1.0/np.sqrt(b[0])
        p1 = 1.0/np.sqrt(b[1]) * (x - a[0]) * p0
        r1 = p1 / p0
        r[0] = r1

        for q in range(1, N):
            r2 = (x - a[q]) - np.sqrt(b[q])/r1
            r1 = 1.0/np.sqrt(b[q+1]) * r2
            r[q] = r1

        return r

    def _christoffel_normalised_polynomials(self, a, b, x, N):
        """
        Computes the Christoffel normalized
        orthogonal polynomial values
        at x
        Parameters
        ---------
        a: np.ndarray
        alpha recurrence coefficients,
        size(N+1, 1)
        b: np.ndarray
        beta recurrence coefficients,
        size(N+1, 1)
        x: double
        a scalar double value of the evaluated input
        for the induced distribution
        N: int
        Order of the polynomials to be evaluated
        Returns
        ---------
        C: np.ndarray
        successive ratios in a vetor
        of length N
        """
        if N <= 0:
            raise ValueError("No. of Christoffels evaluations must be greater than 0")
        if N > len(a):
            raise ValueError("No. of Christoffels evaluations must be less than len(a)")
        if N > len(b):
            raise ValueError("No. of Christoffels evaluations must be less than len(b)")

        C = np.zeros(N+1)
        # Initialize the polynomials
        C[0] = 1.0/(1.0 * np.sqrt(b[0]))
        if N > 0:
            C[1] = 1.0 / (1.0 * np.sqrt(b[1])) * (x - a[0])
        if N > 1:
            C[2] = 1.0 / np.sqrt(1.0 + C[1]**2) * ((x - a[1]) * C[1] - np.sqrt(b[1]))
            C[2] = C[2] / (1.0 * np.sqrt(b[2]))
        if N > 2:
            for nnn in range(2, N):
                C[nnn+1] = 1.0/np.sqrt(1.0 + C[nnn]**2) * \
                    (
                     (x - a[nnn])*C[nnn] - np.sqrt(b[nnn]) *
                     C[nnn-1]/np.sqrt(1.0+C[nnn-1]**2)
                     )
                C[nnn+1] = C[nnn+1] / np.sqrt(b[nnn+1])
        return C

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