from __future__ import division, print_function, absolute_import

import warnings
from collections import namedtuple
from . import _zeros
import numpy as np


_iter = 100
_xtol = 2e-12
_rtol = 4 * np.finfo(float).eps

__all__ = ['newton', 'bisect', 'ridder', 'brentq', 'brenth', 'toms748',
           'RootResults']

# Must agree with CONVERGED, SIGNERR, CONVERR, ...  in zeros.h
_ECONVERGED = 0
_ESIGNERR = -1
_ECONVERR = -2
_EVALUEERR = -3
_EINPROGRESS = 1

CONVERGED = 'converged'
SIGNERR = 'sign error'
CONVERR = 'convergence error'
VALUEERR = 'value error'
INPROGRESS = 'No error'


flag_map = {_ECONVERGED: CONVERGED, _ESIGNERR: SIGNERR, _ECONVERR: CONVERR,
            _EVALUEERR: VALUEERR, _EINPROGRESS: INPROGRESS}


class RootResults(object):
    """Represents the root finding result.

    Attributes
    ----------
    root : float
        Estimated root location.
    iterations : int
        Number of iterations needed to find the root.
    function_calls : int
        Number of times the function was called.
    converged : bool
        True if the routine converged.
    flag : str
        Description of the cause of termination.

    """

    def __init__(self, root, iterations, function_calls, flag):
        self.root = root
        self.iterations = iterations
        self.function_calls = function_calls
        self.converged = flag == _ECONVERGED
        self.flag = None
        try:
            self.flag = flag_map[flag]
        except KeyError:
            self.flag = 'unknown error %d' % (flag,)

    def __repr__(self):
        attrs = ['converged', 'flag', 'function_calls',
                 'iterations', 'root']
        m = max(map(len, attrs)) + 1
        return '\n'.join([a.rjust(m) + ': ' + repr(getattr(self, a))
                          for a in attrs])


def results_c(full_output, r):
    if full_output:
        x, funcalls, iterations, flag = r
        results = RootResults(root=x,
                              iterations=iterations,
                              function_calls=funcalls,
                              flag=flag)
        return x, results
    else:
        return r


def _results_select(full_output, r):
    """Select from a tuple of (root, funccalls, iterations, flag)"""
    x, funcalls, iterations, flag = r
    if full_output:
        results = RootResults(root=x,
                              iterations=iterations,
                              function_calls=funcalls,
                              flag=flag)
        return x, results
    return x


def newton(func, x0, fprime=None, args=(), tol=1.48e-8, maxiter=50,
           fprime2=None, x1=None, rtol=0.0,
           full_output=False, disp=True):
    """
    Find a zero of a real or complex function using the Newton-Raphson
    (or secant or Halley's) method.

    Find a zero of the function `func` given a nearby starting point `x0`.
    The Newton-Raphson method is used if the derivative `fprime` of `func`
    is provided, otherwise the secant method is used.  If the second order
    derivative `fprime2` of `func` is also provided, then Halley's method is
    used.

    If `x0` is a sequence with more than one item, then `newton` returns an
    array, and `func` must be vectorized and return a sequence or array of the
    same shape as its first argument. If `fprime` or `fprime2` is given then
    its return must also have the same shape.

    Parameters
    ----------
    func : callable
        The function whose zero is wanted. It must be a function of a
        single variable of the form ``f(x,a,b,c...)``, where ``a,b,c...``
        are extra arguments that can be passed in the `args` parameter.
    x0 : float, sequence, or ndarray
        An initial estimate of the zero that should be somewhere near the
        actual zero. If not scalar, then `func` must be vectorized and return
        a sequence or array of the same shape as its first argument.
    fprime : callable, optional
        The derivative of the function when available and convenient. If it
        is None (default), then the secant method is used.
    args : tuple, optional
        Extra arguments to be used in the function call.
    tol : float, optional
        The allowable error of the zero value.  If `func` is complex-valued,
        a larger `tol` is recommended as both the real and imaginary parts
        of `x` contribute to ``|x - x0|``.
    maxiter : int, optional
        Maximum number of iterations.
    fprime2 : callable, optional
        The second order derivative of the function when available and
        convenient. If it is None (default), then the normal Newton-Raphson
        or the secant method is used. If it is not None, then Halley's method
        is used.
    x1 : float, optional
        Another estimate of the zero that should be somewhere near the
        actual zero.  Used if `fprime` is not provided.
    rtol : float, optional
        Tolerance (relative) for termination.
    full_output : bool, optional
        If `full_output` is False (default), the root is returned.
        If True and `x0` is scalar, the return value is ``(x, r)``, where ``x``
        is the root and ``r`` is a `RootResults` object.
        If True and `x0` is non-scalar, the return value is ``(x, converged,
        zero_der)`` (see Returns section for details).
    disp : bool, optional
        If True, raise a RuntimeError if the algorithm didn't converge, with
        the error message containing the number of iterations and current
        function value.  Otherwise the convergence status is recorded in a
        `RootResults` return object.
        Ignored if `x0` is not scalar.
        *Note: this has little to do with displaying, however
        the `disp` keyword cannot be renamed for backwards compatibility.*

    Returns
    -------
    root : float, sequence, or ndarray
        Estimated location where function is zero.
    r : `RootResults`, optional
        Present if ``full_output=True`` and `x0` is scalar.
        Object containing information about the convergence.  In particular,
        ``r.converged`` is True if the routine converged.
    converged : ndarray of bool, optional
        Present if ``full_output=True`` and `x0` is non-scalar.
        For vector functions, indicates which elements converged successfully.
    zero_der : ndarray of bool, optional
        Present if ``full_output=True`` and `x0` is non-scalar.
        For vector functions, indicates which elements had a zero derivative.

    See Also
    --------
    brentq, brenth, ridder, bisect
    fsolve : find zeros in n dimensions.

    Notes
    -----
    The convergence rate of the Newton-Raphson method is quadratic,
    the Halley method is cubic, and the secant method is
    sub-quadratic.  This means that if the function is well behaved
    the actual error in the estimated zero after the n-th iteration
    is approximately the square (cube for Halley) of the error
    after the (n-1)-th step.  However, the stopping criterion used
    here is the step size and there is no guarantee that a zero
    has been found. Consequently the result should be verified.
    Safer algorithms are brentq, brenth, ridder, and bisect,
    but they all require that the root first be bracketed in an
    interval where the function changes sign. The brentq algorithm
    is recommended for general use in one dimensional problems
    when such an interval has been found.

    When `newton` is used with arrays, it is best suited for the following
    types of problems:

    * The initial guesses, `x0`, are all relatively the same distance from
      the roots.
    * Some or all of the extra arguments, `args`, are also arrays so that a
      class of similar problems can be solved together.
    * The size of the initial guesses, `x0`, is larger than O(100) elements.
      Otherwise, a naive loop may perform as well or better than a vector.

    Examples
    --------
    >>> from scipy import optimize
    >>> import matplotlib.pyplot as plt

    >>> def f(x):
    ...     return (x**3 - 1)  # only one real root at x = 1

    ``fprime`` is not provided, use the secant method:

    >>> root = optimize.newton(f, 1.5)
    >>> root
    1.0000000000000016
    >>> root = optimize.newton(f, 1.5, fprime2=lambda x: 6 * x)
    >>> root
    1.0000000000000016

    Only ``fprime`` is provided, use the Newton-Raphson method:

    >>> root = optimize.newton(f, 1.5, fprime=lambda x: 3 * x**2)
    >>> root
    1.0

    Both ``fprime2`` and ``fprime`` are provided, use Halley's method:

    >>> root = optimize.newton(f, 1.5, fprime=lambda x: 3 * x**2,
    ...                        fprime2=lambda x: 6 * x)
    >>> root
    1.0

    When we want to find zeros for a set of related starting values and/or
    function parameters, we can provide both of those as an array of inputs:

    >>> f = lambda x, a: x**3 - a
    >>> fder = lambda x, a: 3 * x**2
    >>> np.random.seed(4321)
    >>> x = np.random.randn(100)
    >>> a = np.arange(-50, 50)
    >>> vec_res = optimize.newton(f, x, fprime=fder, args=(a, ))

    The above is the equivalent of solving for each value in ``(x, a)``
    separately in a for-loop, just faster:

    >>> loop_res = [optimize.newton(f, x0, fprime=fder, args=(a0,))
    ...             for x0, a0 in zip(x, a)]
    >>> np.allclose(vec_res, loop_res)
    True

    Plot the results found for all values of ``a``:

    >>> analytical_result = np.sign(a) * np.abs(a)**(1/3)
    >>> fig = plt.figure()
    >>> ax = fig.add_subplot(111)
    >>> ax.plot(a, analytical_result, 'o')
    >>> ax.plot(a, vec_res, '.')
    >>> ax.set_xlabel('$a$')
    >>> ax.set_ylabel('$x$ where $f(x, a)=0$')
    >>> plt.show()

    """
    if tol <= 0:
        raise ValueError("tol too small (%g <= 0)" % tol)
    if maxiter < 1:
        raise ValueError("maxiter must be greater than 0")
    if np.size(x0) > 1:
        return _array_newton(func, x0, fprime, args, tol, maxiter, fprime2,
                             full_output)

    # Convert to float (don't use float(x0); this works also for complex x0)
    p0 = 1.0 * x0
    funcalls = 0
    if fprime is not None:
        # Newton-Raphson method
        for itr in range(maxiter):
            # first evaluate fval
            fval = func(p0, *args)
            funcalls += 1
            # If fval is 0, a root has been found, then terminate
            if fval == 0:
                return _results_select(
                    full_output, (p0, funcalls, itr, _ECONVERGED))
            fder = fprime(p0, *args)
            funcalls += 1
            if fder == 0:
                msg = "Derivative was zero."
                if disp:
                    msg += (
                        " Failed to converge after %d iterations, value is %s."
                        % (itr + 1, p0))
                    raise RuntimeError(msg)
                warnings.warn(msg, RuntimeWarning)
                return _results_select(
                    full_output, (p0, funcalls, itr + 1, _ECONVERR))
            newton_step = fval / fder
            if fprime2:
                fder2 = fprime2(p0, *args)
                funcalls += 1
                # Halley's method:
                #   newton_step /= (1.0 - 0.5 * newton_step * fder2 / fder)
                # Only do it if denominator stays close enough to 1
                # Rationale:  If 1-adj < 0, then Halley sends x in the
                # opposite direction to Newton.  Doesn't happen if x is close
                # enough to root.
                adj = newton_step * fder2 / fder / 2
                if np.abs(adj) < 1:
                    newton_step /= 1.0 - adj
            p = p0 - newton_step
            if np.isclose(p, p0, rtol=rtol, atol=tol):
                return _results_select(
                    full_output, (p, funcalls, itr + 1, _ECONVERGED))
            p0 = p
    else:
        # Secant method
        if x1 is not None:
            if x1 == x0:
                raise ValueError("x1 and x0 must be different")
            p1 = x1
        else:
            eps = 1e-4
            p1 = x0 * (1 + eps)
            p1 += (eps if p1 >= 0 else -eps)
        q0 = func(p0, *args)
        funcalls += 1
        q1 = func(p1, *args)
        funcalls += 1
        if abs(q1) < abs(q0):
            p0, p1, q0, q1 = p1, p0, q1, q0
        for itr in range(maxiter):
            if q1 == q0:
                if p1 != p0:
                    msg = "Tolerance of %s reached." % (p1 - p0)
                    if disp:
                        msg += (
                            " Failed to converge after %d iterations, value is %s."
                            % (itr + 1, p1))
                        raise RuntimeError(msg)
                    warnings.warn(msg, RuntimeWarning)
                p = (p1 + p0) / 2.0
                return _results_select(
                    full_output, (p, funcalls, itr + 1, _ECONVERGED))
            else: