"""
Cython optimize root finding API
================================
The underlying C functions for the following root finders can be accessed
directly using Cython:

- `~scipy.optimize.bisect`
- `~scipy.optimize.ridder`
- `~scipy.optimize.brenth`
- `~scipy.optimize.brentq`

The Cython API for the root finding functions is similar except there is no
``disp`` argument. Import the root finding functions using ``cimport`` from
`scipy.optimize.cython_optimize`. ::

    from scipy.optimize.cython_optimize cimport bisect, ridder, brentq, brenth


Callback signature
------------------
The zeros functions in `~scipy.optimize.cython_optimize` expect a callback that
takes a double for the scalar independent variable as the 1st argument and a
user defined ``struct`` with any extra parameters as the 2nd argument. ::

    double (*callback_type)(double, void*)


Examples
--------
Usage of `~scipy.optimize.cython_optimize` requires Cython to write callbacks
that are compiled into C. For more information on compiling Cython, see the
`Cython Documentation <http://docs.cython.org/en/latest/index.html>`_.

These are the basic steps:

1. Create a Cython ``.pyx`` file, for example: ``myexample.pyx``.
2. Import the desired root finder from `~scipy.optimize.cython_optimize`.
3. Write the callback function, and call the selected root finding function
   passing the callback, any extra arguments, and the other solver
   parameters. ::

       from scipy.optimize.cython_optimize cimport brentq

       # import math from Cython
       from libc cimport math

       myargs = {'C0': 1.0, 'C1': 0.7}  # a dictionary of extra arguments
       XLO, XHI = 0.5, 1.0  # lower and upper search boundaries
       XTOL, RTOL, MITR = 1e-3, 1e-3, 10  # other solver parameters

       # user-defined struct for extra parameters
       ctypedef struct test_params:
           double C0
           double C1


       # user-defined callback
       cdef double f(double x, void *args):
           cdef test_params *myargs = <test_params *> args
           return myargs.C0 - math.exp(-(x - myargs.C1))


       # Cython wrapper function
       cdef double brentq_wrapper_example(dict args, double xa, double xb,
                                          double xtol, double rtol, int mitr):
           # Cython automatically casts dictionary to struct
           cdef test_params myargs = args
           return brentq(
               f, xa, xb, <test_params *> &myargs, xtol, rtol, mitr, NULL)


       # Python function
       def brentq_example(args=myargs, xa=XLO, xb=XHI, xtol=XTOL, rtol=RTOL,
                          mitr=MITR):
           '''Calls Cython wrapper from Python.'''
           return brentq_wrapper_example(args, xa, xb, xtol, rtol, mitr)

4. If you want to call your function from Python, create a Cython wrapper, and
   a Python function that calls the wrapper, or use ``cpdef``. Then, in Python,
   you can import and run the example. ::

       from myexample import brentq_example

       x = brentq_example()
       # 0.6999942848231314

5. Create a Cython ``.pxd`` file if you need to export any Cython functions.


Full output
-----------
The  functions in `~scipy.optimize.cython_optimize` can also copy the full
output from the solver to a C ``struct`` that is passed as its last argument.
If you don't want the full output, just pass ``NULL``. The full output
``struct`` must be type ``zeros_full_output``, which is defined in
`scipy.optimize.cython_optimize` with the following fields:

- ``int funcalls``: number of function calls
- ``int iterations``: number of iterations
- ``int error_num``: error number
- ``double root``: root of function

The root is copied by `~scipy.optimize.cython_optimize` to the full output
``struct``. An error number of -1 means a sign error, -2 means a convergence
error, and 0 means the solver converged. Continuing from the previous example::

    from scipy.optimize.cython_optimize cimport zeros_full_output


    # cython brentq solver with full output
    cdef zeros_full_output brentq_full_output_wrapper_example(
            dict args, double xa, double xb, double xtol, double rtol,
            int mitr):
        cdef test_params myargs = args
        cdef zeros_full_output my_full_output
        # use my_full_output instead of NULL
        brentq(f, xa, xb, &myargs, xtol, rtol, mitr, &my_full_output)
        return my_full_output


    # Python function
    def brent_full_output_example(args=myargs, xa=XLO, xb=XHI, xtol=XTOL,
                                  rtol=RTOL, mitr=MITR):
        '''Returns full output'''
        return brentq_full_output_wrapper_example(args, xa, xb, xtol, rtol,
                                                  mitr)

    result = brent_full_output_example()
    # {'error_num': 0,
    #  'funcalls': 6,
    #  'iterations': 5,
    #  'root': 0.6999942848231314}
"""
