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from __future__ import division, print_function, absolute_import

import warnings
from . import _minpack

import numpy as np
from numpy import (atleast_1d, dot, take, triu, shape, eye,
                   transpose, zeros, product, greater, array,
                   all, where, isscalar, asarray, inf, abs,
                   finfo, inexact, issubdtype, dtype)
from .optimize import OptimizeResult, _check_unknown_options

error = _minpack.error

__all__ = ['fsolve', 'leastsq', 'fixed_point', 'curve_fit']


def _check_func(checker, argname, thefunc, x0, args, numinputs,
                output_shape=None):
    res = atleast_1d(thefunc(*((x0[:numinputs],) + args)))
    if (output_shape is not None) and (shape(res) != output_shape):
        if (output_shape[0] != 1):
            if len(output_shape) > 1:
                if output_shape[1] == 1:
                    return shape(res)
            msg = "%s: there is a mismatch between the input and output " \
                  "shape of the '%s' argument" % (checker, argname)
            func_name = getattr(thefunc, '__name__', None)
            if func_name:
                msg += " '%s'." % func_name
            else:
                msg += "."
            raise TypeError(msg)
    if issubdtype(res.dtype, inexact):
        dt = res.dtype
    else:
        dt = dtype(float)
    return shape(res), dt


def fsolve(func, x0, args=(), fprime=None, full_output=0,
           col_deriv=0, xtol=1.49012e-8, maxfev=0, band=None,
           epsfcn=None, factor=100, diag=None):
    """
Find the roots of a function.

Return the roots of the (non-linear) equations defined by
``func(x) = 0`` given a starting estimate.

Parameters
----------
func : callable ``f(x, *args)``
A function that takes at least one (possibly vector) argument.
x0 : ndarray
The starting estimate for the roots of ``func(x) = 0``.
args : tuple, optional
Any extra arguments to `func`.
fprime : callable(x), optional
A function to compute the Jacobian of `func` with derivatives
across the rows. By default, the Jacobian will be estimated.
full_output : bool, optional
If True, return optional outputs.
col_deriv : bool, optional
Specify whether the Jacobian function computes derivatives down
the columns (faster, because there is no transpose operation).
xtol : float
The calculation will terminate if the relative error between two
consecutive iterates is at most `xtol`.
maxfev : int, optional
The maximum number of calls to the function. If zero, then
``100*(N+1)`` is the maximum where N is the number of elements
in `x0`.
band : tuple, optional
If set to a two-sequence containing the number of sub- and
super-diagonals within the band of the Jacobi matrix, the
Jacobi matrix is considered banded (only for ``fprime=None``).
epsfcn : float, optional
A suitable step length for the forward-difference
approximation of the Jacobian (for ``fprime=None``). If
`epsfcn` is less than the machine precision, it is assumed
that the relative errors in the functions are of the order of
the machine precision.
factor : float, optional
A parameter determining the initial step bound
(``factor * || diag * x||``). Should be in the interval
``(0.1, 100)``.
diag : sequence, optional
N positive entries that serve as a scale factors for the
variables.

Returns
-------
x : ndarray
The solution (or the result of the last iteration for
an unsuccessful call).
infodict : dict
A dictionary of optional outputs with the keys:

``nfev``
number of function calls
``njev``
number of Jacobian calls
``fvec``
function evaluated at the output
``fjac``
the orthogonal matrix, q, produced by the QR
factorization of the final approximate Jacobian
matrix, stored column wise
``r``
upper triangular matrix produced by QR factorization
of the same matrix
``qtf``
the vector ``(transpose(q) * fvec)``

ier : int
An integer flag. Set to 1 if a solution was found, otherwise refer
to `mesg` for more information.
mesg : str
If no solution is found, `mesg` details the cause of failure.

See Also
--------
root : Interface to root finding algorithms for multivariate
functions. See the 'hybr' `method` in particular.

Notes
-----
``fsolve`` is a wrapper around MINPACK's hybrd and hybrj algorithms.

"""
    options = {'col_deriv': col_deriv,
               'xtol': xtol,
               'maxfev': maxfev,
               'band': band,
               'eps': epsfcn,
               'factor': factor,
               'diag': diag,
               'full_output': full_output}

    res = _root_hybr(func, x0, args, jac=fprime, **options)
    if full_output:
        x = res['x']
        info = dict((k, res.get(k))
                    for k in ('nfev', 'njev', 'fjac', 'r', 'qtf') if k in res)
        info['fvec'] = res['fun']
        return x, info, res['status'], res['message']
    else:
        return res['x']


def _root_hybr(func, x0, args=(), jac=None,
               col_deriv=0, xtol=1.49012e-08, maxfev=0, band=None, eps=None,
               factor=100, diag=None, full_output=0, **unknown_options):
    """
Find the roots of a multivariate function using MINPACK's hybrd and
hybrj routines (modified Powell method).

Options for the hybrd algorithm are:
col_deriv : bool
Specify whether the Jacobian function computes derivatives down
the columns (faster, because there is no transpose operation).
xtol : float
The calculation will terminate if the relative error between two
consecutive iterates is at most `xtol`.
maxfev : int
The maximum number of calls to the function. If zero, then
``100*(N+1)`` is the maximum where N is the number of elements
in `x0`.
band : tuple
If set to a two-sequence containing the number of sub- and
super-diagonals within the band of the Jacobi matrix, the
Jacobi matrix is considered banded (only for ``fprime=None``).
eps : float
A suitable step length for the forward-difference
approximation of the Jacobian (for ``fprime=None``). If
`eps` is less than the machine precision, it is assumed
that the relative errors in the functions are of the order of
the machine precision.
factor : float
A parameter determining the initial step bound
(``factor * || diag * x||``). Should be in the interval
``(0.1, 100)``.
diag : sequence
N positive entries that serve as a scale factors for the
variables.

This function is called by the `root` function with `method=hybr`. It
is not supposed to be called directly.
"""
    _check_unknown_options(unknown_options)
    epsfcn = eps

    x0 = asarray(x0).flatten()
    n = len(x0)
    if not isinstance(args, tuple):
        args = (args,)
    shape, dtype = _check_func('fsolve', 'func', func, x0, args, n, (n,))
    if epsfcn is None:
        epsfcn = finfo(dtype).eps
    Dfun = jac
    if Dfun is None:
        if band is None:
            ml, mu = -10, -10
        else:
            ml, mu = band[:2]
        if maxfev == 0:
            maxfev = 200 * (n + 1)
        retval = _minpack._hybrd(func, x0, args, 1, xtol, maxfev,
                                 ml, mu, epsfcn, factor, diag)
    else:
        _check_func('fsolve', 'fprime', Dfun, x0, args, n, (n, n))
        if (maxfev == 0):
            maxfev = 100 * (n + 1)
        retval = _minpack._hybrj(func, Dfun, x0, args, 1,
                                 col_deriv, xtol, maxfev, factor, diag)

    x, status = retval[0], retval[-1]

    errors = {0: ["Improper input parameters were entered.", TypeError],
              1: ["The solution converged.", None],
              2: ["The number of calls to function has "
                  "reached maxfev = %d." % maxfev, ValueError],
              3: ["xtol=%f is too small, no further improvement "
                  "in the approximate\n solution "
                  "is possible." % xtol, ValueError],
              4: ["The iteration is not making good progress, as measured "
                  "by the \n improvement from the last five "
                  "Jacobian evaluations.", ValueError],
              5: ["The iteration is not making good progress, "
                  "as measured by the \n improvement from the last "
                  "ten iterations.", ValueError],
              'unknown': ["An error occurred.", TypeError]}

    if status != 1 and not full_output:
        if status in [2, 3, 4, 5]:
            msg = errors[status][0]
            warnings.warn(msg, RuntimeWarning)
        else:
            try:
                raise errors[status][1](errors[status][0])
            except KeyError:
                raise errors['unknown'][1](errors['unknown'][0])

    info = retval[1]
    info['fun'] = info.pop('fvec')
    sol = OptimizeResult(x=x, success=(status == 1), status=status)
    sol.update(info)
    try:
        sol['message'] = errors[status][0]
    except KeyError:
        info['message'] = errors['unknown'][0]

    return sol


def leastsq(func, x0, args=(), Dfun=None, full_output=0,
            col_deriv=0, ftol=1.49012e-8, xtol=1.49012e-8,
            gtol=0.0, maxfev=0, epsfcn=None, factor=100, diag=None):
    """
Minimize the sum of squares of a set of equations.

::

x = arg min(sum(func(y)**2,axis=0))
y

Parameters
----------
func : callable
should take at least one (possibly length N vector) argument and
returns M floating point numbers.
x0 : ndarray
The starting estimate for the minimization.
args : tuple
Any extra arguments to func are placed in this tuple.
Dfun : callable
A function or method to compute the Jacobian of func with derivatives
across the rows. If this is None, the Jacobian will be estimated.
full_output : bool
non-zero to return all optional outputs.
col_deriv : bool
non-zero to specify that the Jacobian function computes derivatives
down the columns (faster, because there is no transpose operation).
ftol : float
Relative error desired in the sum of squares.
xtol : float
Relative error desired in the approximate solution.
gtol : float
Orthogonality desired between the function vector and the columns of
the Jacobian.
maxfev : int
The maximum number of calls to the function. If zero, then 100*(N+1) is
the maximum where N is the number of elements in x0.
epsfcn : float
A variable used in determining a suitable step length for the forward-
difference approximation of the Jacobian (for Dfun=None).
Normally the actual step length will be sqrt(epsfcn)*x
If epsfcn is less than the machine precision, it is assumed that the
relative errors are of the order of the machine precision.
factor : float
A parameter determining the initial step bound
(``factor * || diag * x||``). Should be in interval ``(0.1, 100)``.
diag : sequence
N positive entries that serve as a scale factors for the variables.

Returns
-------
x : ndarray
The solution (or the result of the last iteration for an unsuccessful
call).
cov_x : ndarray
Uses the fjac and ipvt optional outputs to construct an
estimate of the jacobian around the solution. None if a
singular matrix encountered (indicates very flat curvature in
some direction). This matrix must be multiplied by the
residual variance to get the covariance of the
parameter estimates -- see curve_fit.
infodict : dict
a dictionary of optional outputs with the key s:

``nfev``
The number of function calls
``fvec``
The function evaluated at the output
``fjac``
A permutation of the R matrix of a QR
factorization of the final approximate
Jacobian matrix, stored column wise.
Together with ipvt, the covariance of the
estimate can be approximated.
``ipvt``
An integer array of length N which defines
a permutation matrix, p, such that
fjac*p = q*r, where r is upper triangular
with diagonal elements of nonincreasing
magnitude. Column j of p is column ipvt(j)
of the identity matrix.
``qtf``
The vector (transpose(q) * fvec).

mesg : str
A string message giving information about the cause of failure.
ier : int
An integer flag. If it is equal to 1, 2, 3 or 4, the solution was
found. Otherwise, the solution was not found. In either case, the
optional output variable 'mesg' gives more information.

Notes
-----
"leastsq" is a wrapper around MINPACK's lmdif and lmder algorithms.

cov_x is a Jacobian approximation to the Hessian of the least squares
objective function.
This approximation assumes that the objective function is based on the
difference between some observed target data (ydata) and a (non-linear)
function of the parameters `f(xdata, params)` ::

func(params) = ydata - f(xdata, params)

so that the objective function is ::

min sum((ydata - f(xdata, params))**2, axis=0)
params

"""
    x0 = asarray(x0).flatten()
    n = len(x0)
    if not isinstance(args, tuple):
        args = (args,)
    shape, dtype = _check_func('leastsq', 'func', func, x0, args, n)
    m = shape[0]
    if n > m:
        raise TypeError('Improper input: N=%s must not exceed M=%s' % (n, m))
    if epsfcn is None:
        epsfcn = finfo(dtype).eps
    if Dfun is None:
        if maxfev == 0:
            maxfev = 200*(n + 1)
        retval = _minpack._lmdif(func, x0, args, full_output, ftol, xtol,
                                 gtol, maxfev, epsfcn, factor, diag)
    else:
        if col_deriv:
            _check_func('leastsq', 'Dfun', Dfun, x0, args, n, (n, m))
        else:
            _check_func('leastsq', 'Dfun', Dfun, x0, args, n, (m, n))
        if maxfev == 0:
            maxfev = 100 * (n + 1)
        retval = _minpack._lmder(func, Dfun, x0, args, full_output, col_deriv,
                                 ftol, xtol, gtol, maxfev, factor, diag)

    errors = {0: ["Improper input parameters.", TypeError],
              1: ["Both actual and predicted relative reductions "
                  "in the sum of squares\n are at most %f" % ftol, None],
              2: ["The relative error between two consecutive "
                  "iterates is at most %f" % xtol, None],
              3: ["Both actual and predicted relative reductions in "
                  "the sum of squares\n are at most %f and the "
                  "relative error between two consecutive "
                  "iterates is at \n most %f" % (ftol, xtol), None],
              4: ["The cosine of the angle between func(x) and any "
                  "column of the\n Jacobian is at most %f in "
                  "absolute value" % gtol, None],
              5: ["Number of calls to function has reached "
                  "maxfev = %d." % maxfev, ValueError],
              6: ["ftol=%f is too small, no further reduction "
                  "in the sum of squares\n is possible.""" % ftol,
                  ValueError],
              7: ["xtol=%f is too small, no further improvement in "
                  "the approximate\n solution is possible." % xtol,
                  ValueError],
              8: ["gtol=%f is too small, func(x) is orthogonal to the "
                  "columns of\n the Jacobian to machine "
                  "precision." % gtol, ValueError],
              'unknown': ["Unknown error.", TypeError]}

    info = retval[-1] # The FORTRAN return value

    if info not in [1, 2, 3, 4] and not full_output:
        if info in [5, 6, 7, 8]:
            warnings.warn(errors[info][0], RuntimeWarning)
        else:
            try:
                raise errors[info][1](errors[info][0])
            except KeyError:
                raise errors['unknown'][1](errors['unknown'][0])

    mesg = errors[info][0]
    if full_output:
        cov_x = None
        if info in [1, 2, 3, 4]:
            from numpy.dual import inv
            from numpy.linalg import LinAlgError
            perm = take(eye(n), retval[1]['ipvt'] - 1, 0)
            r = triu(transpose(retval[1]['fjac'])[:n, :])
            R = dot(r, perm)
            try:
                cov_x = inv(dot(transpose(R), R))
            except (LinAlgError, ValueError):
                pass
        return (retval[0], cov_x) + retval[1:-1] + (mesg, info)
    else:
        return (retval[0], info)


def _general_function(params, xdata, ydata, function):
    return function(xdata, *params) - ydata


def _weighted_general_function(params, xdata, ydata, function, weights):
    return weights * (function(xdata, *params) - ydata)


def curve_fit(f, xdata, ydata, p0=None, sigma=None, absolute_sigma=False, **kw):
    """
Use non-linear least squares to fit a function, f, to data.

Assumes ``ydata = f(xdata, *params) + eps``

Parameters
----------
f : callable
The model function, f(x, ...). It must take the independent
variable as the first argument and the parameters to fit as
separate remaining arguments.
xdata : An M-length sequence or an (k,M)-shaped array
for functions with k predictors.
The independent variable where the data is measured.
ydata : M-length sequence
The dependent data --- nominally f(xdata, ...)
p0 : None, scalar, or N-length sequence
Initial guess for the parameters. If None, then the initial
values will all be 1 (if the number of parameters for the function
can be determined using introspection, otherwise a ValueError
is raised).
sigma : None or M-length sequence, optional
If not None, these values are used as weights in the
least-squares problem.
absolute_sigma : bool, optional
If False, `sigma` denotes relative weights of the data points.
The returned covariance matrix `pcov` is based on *estimated*
errors in the data, and is not affected by the overall
magnitude of the values in `sigma`. Only the relative
magnitudes of the `sigma` values matter.

If True, `sigma` describes one standard deviation errors of
the input data points. The estimated covariance in `pcov` is
based on these values.

Returns
-------
popt : array
Optimal values for the parameters so that the sum of the squared error
of ``f(xdata, *popt) - ydata`` is minimized
pcov : 2d array
The estimated covariance of popt. The diagonals provide the variance
of the parameter estimate. To compute one standard deviation errors
on the parameters use ``perr = np.sqrt(np.diag(pcov))``.

How the `sigma` parameter affects the estimated covariance
depends on `absolute_sigma` argument, as described above.

See Also
--------
leastsq

Notes
-----
The algorithm uses the Levenberg-Marquardt algorithm through `leastsq`.
Additional keyword arguments are passed directly to that algorithm.

Examples
--------
>>> import numpy as np
>>> from scipy.optimize import curve_fit
>>> def func(x, a, b, c):
... return a * np.exp(-b * x) + c

>>> xdata = np.linspace(0, 4, 50)
>>> y = func(xdata, 2.5, 1.3, 0.5)
>>> ydata = y + 0.2 * np.random.normal(size=len(xdata))

>>> popt, pcov = curve_fit(func, xdata, ydata)

"""
    if p0 is None:
        # determine number of parameters by inspecting the function
        import inspect
        args, varargs, varkw, defaults = inspect.getargspec(f)
        if len(args) < 2:
            msg = "Unable to determine number of fit parameters."
            raise ValueError(msg)
        if 'self' in args:
            p0 = [1.0] * (len(args)-2)
        else:
            p0 = [1.0] * (len(args)-1)

    # Check input arguments
    if isscalar(p0):
        p0 = array([p0])

    ydata = np.asanyarray(ydata)
    if isinstance(xdata, (list, tuple)):
        # `xdata` is passed straight to the user-defined `f`, so allow
        # non-array_like `xdata`.
        xdata = np.asarray(xdata)

    args = (xdata, ydata, f)
    if sigma is None:
        func = _general_function
    else:
        func = _weighted_general_function
        args += (1.0 / asarray(sigma),)

    # Remove full_output from kw, otherwise we're passing it in twice.
    return_full = kw.pop('full_output', False)
    res = leastsq(func, p0, args=args, full_output=1, **kw)
    (popt, pcov, infodict, errmsg, ier) = res

    if ier not in [1, 2, 3, 4]:
        msg = "Optimal parameters not found: " + errmsg
        raise RuntimeError(msg)

    if pcov is None:
        # indeterminate covariance
        pcov = zeros((len(popt), len(popt)), dtype=float)
        pcov.fill(inf)
    elif not absolute_sigma:
        if len(ydata) > len(p0):
            s_sq = (asarray(func(popt, *args))**2).sum() / (len(ydata) - len(p0))
            pcov = pcov * s_sq
        else:
            pcov.fill(inf)

    if return_full:
        return popt, pcov, infodict, errmsg, ier
    else:
        return popt, pcov


def check_gradient(fcn, Dfcn, x0, args=(), col_deriv=0):
    """Perform a simple check on the gradient for correctness.

"""

    x = atleast_1d(x0)
    n = len(x)
    x = x.reshape((n,))
    fvec = atleast_1d(fcn(x, *args))
    m = len(fvec)
    fvec = fvec.reshape((m,))
    ldfjac = m
    fjac = atleast_1d(Dfcn(x, *args))
    fjac = fjac.reshape((m, n))
    if col_deriv == 0:
        fjac = transpose(fjac)

    xp = zeros((n,), float)
    err = zeros((m,), float)
    fvecp = None
    _minpack._chkder(m, n, x, fvec, fjac, ldfjac, xp, fvecp, 1, err)

    fvecp = atleast_1d(fcn(xp, *args))
    fvecp = fvecp.reshape((m,))
    _minpack._chkder(m, n, x, fvec, fjac, ldfjac, xp, fvecp, 2, err)

    good = (product(greater(err, 0.5), axis=0))

    return (good, err)


def fixed_point(func, x0, args=(), xtol=1e-8, maxiter=500):
    """
Find a fixed point of the function.

Given a function of one or more variables and a starting point, find a
fixed-point of the function: i.e. where ``func(x0) == x0``.

Parameters
----------
func : function
Function to evaluate.
x0 : array_like
Fixed point of function.
args : tuple, optional
Extra arguments to `func`.
xtol : float, optional
Convergence tolerance, defaults to 1e-08.
maxiter : int, optional
Maximum number of iterations, defaults to 500.

Notes
-----
Uses Steffensen's Method using Aitken's ``Del^2`` convergence acceleration.
See Burden, Faires, "Numerical Analysis", 5th edition, pg. 80

Examples
--------
>>> from scipy import optimize
>>> def func(x, c1, c2):
.... return np.sqrt(c1/(x+c2))
>>> c1 = np.array([10,12.])
>>> c2 = np.array([3, 5.])
>>> optimize.fixed_point(func, [1.2, 1.3], args=(c1,c2))
array([ 1.4920333 , 1.37228132])

"""
    if not isscalar(x0):
        x0 = asarray(x0)
        p0 = x0
        for iter in range(maxiter):
            p1 = func(p0, *args)
            p2 = func(p1, *args)
            d = p2 - 2.0 * p1 + p0
            p = where(d == 0, p2, p0 - (p1 - p0)*(p1 - p0) / d)
            relerr = where(p0 == 0, p, (p-p0)/p0)
            if all(abs(relerr) < xtol):
                return p
            p0 = p
    else:
        p0 = x0
        for iter in range(maxiter):
            p1 = func(p0, *args)
            p2 = func(p1, *args)
            d = p2 - 2.0 * p1 + p0
            if d == 0.0:
                return p2
            else:
                p = p0 - (p1 - p0)*(p1 - p0) / d
            if p0 == 0:
                relerr = p
            else:
                relerr = (p - p0)/p0
            if abs(relerr) < xtol:
                return p
            p0 = p
    msg = "Failed to converge after %d iterations, value is %s" % (maxiter, p)
    raise RuntimeError(msg)
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