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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: 

if abs(q1) > abs(q0): 

p = (-q0 / q1 * p1 + p0) / (1 - q0 / q1) 

else: 

p = (-q1 / q0 * p0 + p1) / (1 - q1 / q0) 

if np.isclose(p, p1, rtol=rtol, atol=tol): 

return _results_select( 

full_output, (p, funcalls, itr + 1, _ECONVERGED)) 

p0, q0 = p1, q1 

p1 = p 

q1 = func(p1, *args) 

funcalls += 1 

 

if disp: 

msg = ("Failed to converge after %d iterations, value is %s." 

% (itr + 1, p)) 

raise RuntimeError(msg) 

 

return _results_select(full_output, (p, funcalls, itr + 1, _ECONVERR)) 

 

 

def _array_newton(func, x0, fprime, args, tol, maxiter, fprime2, full_output): 

""" 

A vectorized version of Newton, Halley, and secant methods for arrays. 

 

Do not use this method directly. This method is called from `newton` 

when ``np.size(x0) > 1`` is ``True``. For docstring, see `newton`. 

""" 

# Explicitly copy `x0` as `p` will be modified inplace, but, the 

# user's array should not be altered. 

try: 

p = np.array(x0, copy=True, dtype=float) 

except TypeError: 

# can't convert complex to float 

p = np.array(x0, copy=True) 

 

failures = np.ones_like(p, dtype=bool) 

nz_der = np.ones_like(failures) 

if fprime is not None: 

# Newton-Raphson method 

for iteration in range(maxiter): 

# first evaluate fval 

fval = np.asarray(func(p, *args)) 

# If all fval are 0, all roots have been found, then terminate 

if not fval.any(): 

failures = fval.astype(bool) 

break 

fder = np.asarray(fprime(p, *args)) 

nz_der = (fder != 0) 

# stop iterating if all derivatives are zero 

if not nz_der.any(): 

break 

# Newton step 

dp = fval[nz_der] / fder[nz_der] 

if fprime2 is not None: 

fder2 = np.asarray(fprime2(p, *args)) 

dp = dp / (1.0 - 0.5 * dp * fder2[nz_der] / fder[nz_der]) 

# only update nonzero derivatives 

p[nz_der] -= dp 

failures[nz_der] = np.abs(dp) >= tol # items not yet converged 

# stop iterating if there aren't any failures, not incl zero der 

if not failures[nz_der].any(): 

break 

else: 

# Secant method 

dx = np.finfo(float).eps**0.33 

p1 = p * (1 + dx) + np.where(p >= 0, dx, -dx) 

q0 = np.asarray(func(p, *args)) 

q1 = np.asarray(func(p1, *args)) 

active = np.ones_like(p, dtype=bool) 

for iteration in range(maxiter): 

nz_der = (q1 != q0) 

# stop iterating if all derivatives are zero 

if not nz_der.any(): 

p = (p1 + p) / 2.0 

break 

# Secant Step 

dp = (q1 * (p1 - p))[nz_der] / (q1 - q0)[nz_der] 

# only update nonzero derivatives 

p[nz_der] = p1[nz_der] - dp 

active_zero_der = ~nz_der & active 

p[active_zero_der] = (p1 + p)[active_zero_der] / 2.0 

active &= nz_der # don't assign zero derivatives again 

failures[nz_der] = np.abs(dp) >= tol # not yet converged 

# stop iterating if there aren't any failures, not incl zero der 

if not failures[nz_der].any(): 

break 

p1, p = p, p1 

q0 = q1 

q1 = np.asarray(func(p1, *args)) 

 

zero_der = ~nz_der & failures # don't include converged with zero-ders 

if zero_der.any(): 

# Secant warnings 

if fprime is None: 

nonzero_dp = (p1 != p) 

# non-zero dp, but infinite newton step 

zero_der_nz_dp = (zero_der & nonzero_dp) 

if zero_der_nz_dp.any(): 

rms = np.sqrt( 

sum((p1[zero_der_nz_dp] - p[zero_der_nz_dp]) ** 2) 

) 

warnings.warn( 

'RMS of {:g} reached'.format(rms), RuntimeWarning) 

# Newton or Halley warnings 

else: 

all_or_some = 'all' if zero_der.all() else 'some' 

msg = '{:s} derivatives were zero'.format(all_or_some) 

warnings.warn(msg, RuntimeWarning) 

elif failures.any(): 

all_or_some = 'all' if failures.all() else 'some' 

msg = '{0:s} failed to converge after {1:d} iterations'.format( 

all_or_some, maxiter 

) 

if failures.all(): 

raise RuntimeError(msg) 

warnings.warn(msg, RuntimeWarning) 

 

if full_output: 

result = namedtuple('result', ('root', 'converged', 'zero_der')) 

p = result(p, ~failures, zero_der) 

 

return p 

 

 

def bisect(f, a, b, args=(), 

xtol=_xtol, rtol=_rtol, maxiter=_iter, 

full_output=False, disp=True): 

""" 

Find root of a function within an interval using bisection. 

 

Basic bisection routine to find a zero of the function `f` between the 

arguments `a` and `b`. `f(a)` and `f(b)` cannot have the same signs. 

Slow but sure. 

 

Parameters 

---------- 

f : function 

Python function returning a number. `f` must be continuous, and 

f(a) and f(b) must have opposite signs. 

a : scalar 

One end of the bracketing interval [a,b]. 

b : scalar 

The other end of the bracketing interval [a,b]. 

xtol : number, optional 

The computed root ``x0`` will satisfy ``np.allclose(x, x0, 

atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The 

parameter must be nonnegative. 

rtol : number, optional 

The computed root ``x0`` will satisfy ``np.allclose(x, x0, 

atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The 

parameter cannot be smaller than its default value of 

``4*np.finfo(float).eps``. 

maxiter : int, optional 

if convergence is not achieved in `maxiter` iterations, an error is 

raised. Must be >= 0. 

args : tuple, optional 

containing extra arguments for the function `f`. 

`f` is called by ``apply(f, (x)+args)``. 

full_output : bool, optional 

If `full_output` is False, the root is returned. If `full_output` is 

True, the return value is ``(x, r)``, where x is the root, and r is 

a `RootResults` object. 

disp : bool, optional 

If True, raise RuntimeError if the algorithm didn't converge. 

Otherwise the convergence status is recorded in a `RootResults` 

return object. 

 

Returns 

------- 

x0 : float 

Zero of `f` between `a` and `b`. 

r : `RootResults` (present if ``full_output = True``) 

Object containing information about the convergence. In particular, 

``r.converged`` is True if the routine converged. 

 

Examples 

-------- 

 

>>> def f(x): 

... return (x**2 - 1) 

 

>>> from scipy import optimize 

 

>>> root = optimize.bisect(f, 0, 2) 

>>> root 

1.0 

 

>>> root = optimize.bisect(f, -2, 0) 

>>> root 

-1.0 

 

See Also 

-------- 

brentq, brenth, bisect, newton 

fixed_point : scalar fixed-point finder 

fsolve : n-dimensional root-finding 

 

""" 

if not isinstance(args, tuple): 

args = (args,) 

if xtol <= 0: 

raise ValueError("xtol too small (%g <= 0)" % xtol) 

if rtol < _rtol: 

raise ValueError("rtol too small (%g < %g)" % (rtol, _rtol)) 

r = _zeros._bisect(f, a, b, xtol, rtol, maxiter, args, full_output, disp) 

return results_c(full_output, r) 

 

 

def ridder(f, a, b, args=(), 

xtol=_xtol, rtol=_rtol, maxiter=_iter, 

full_output=False, disp=True): 

""" 

Find a root of a function in an interval using Ridder's method. 

 

Parameters 

---------- 

f : function 

Python function returning a number. f must be continuous, and f(a) and 

f(b) must have opposite signs. 

a : scalar 

One end of the bracketing interval [a,b]. 

b : scalar 

The other end of the bracketing interval [a,b]. 

xtol : number, optional 

The computed root ``x0`` will satisfy ``np.allclose(x, x0, 

atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The 

parameter must be nonnegative. 

rtol : number, optional 

The computed root ``x0`` will satisfy ``np.allclose(x, x0, 

atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The 

parameter cannot be smaller than its default value of 

``4*np.finfo(float).eps``. 

maxiter : int, optional 

if convergence is not achieved in `maxiter` iterations, an error is 

raised. Must be >= 0. 

args : tuple, optional 

containing extra arguments for the function `f`. 

`f` is called by ``apply(f, (x)+args)``. 

full_output : bool, optional 

If `full_output` is False, the root is returned. If `full_output` is 

True, the return value is ``(x, r)``, where `x` is the root, and `r` is 

a `RootResults` object. 

disp : bool, optional 

If True, raise RuntimeError if the algorithm didn't converge. 

Otherwise the convergence status is recorded in any `RootResults` 

return object. 

 

Returns 

------- 

x0 : float 

Zero of `f` between `a` and `b`. 

r : `RootResults` (present if ``full_output = True``) 

Object containing information about the convergence. 

In particular, ``r.converged`` is True if the routine converged. 

 

See Also 

-------- 

brentq, brenth, bisect, newton : one-dimensional root-finding 

fixed_point : scalar fixed-point finder 

 

Notes 

----- 

Uses [Ridders1979]_ method to find a zero of the function `f` between the 

arguments `a` and `b`. Ridders' method is faster than bisection, but not 

generally as fast as the Brent routines. [Ridders1979]_ provides the 

classic description and source of the algorithm. A description can also be 

found in any recent edition of Numerical Recipes. 

 

The routine used here diverges slightly from standard presentations in 

order to be a bit more careful of tolerance. 

 

References 

---------- 

.. [Ridders1979] 

Ridders, C. F. J. "A New Algorithm for Computing a 

Single Root of a Real Continuous Function." 

IEEE Trans. Circuits Systems 26, 979-980, 1979. 

 

Examples 

-------- 

 

>>> def f(x): 

... return (x**2 - 1) 

 

>>> from scipy import optimize 

 

>>> root = optimize.ridder(f, 0, 2) 

>>> root 

1.0 

 

>>> root = optimize.ridder(f, -2, 0) 

>>> root 

-1.0 

""" 

if not isinstance(args, tuple): 

args = (args,) 

if xtol <= 0: 

raise ValueError("xtol too small (%g <= 0)" % xtol) 

if rtol < _rtol: 

raise ValueError("rtol too small (%g < %g)" % (rtol, _rtol)) 

r = _zeros._ridder(f, a, b, xtol, rtol, maxiter, args, full_output, disp) 

return results_c(full_output, r) 

 

 

def brentq(f, a, b, args=(), 

xtol=_xtol, rtol=_rtol, maxiter=_iter, 

full_output=False, disp=True): 

""" 

Find a root of a function in a bracketing interval using Brent's method. 

 

Uses the classic Brent's method to find a zero of the function `f` on 

the sign changing interval [a , b]. Generally considered the best of the 

rootfinding routines here. It is a safe version of the secant method that 

uses inverse quadratic extrapolation. Brent's method combines root 

bracketing, interval bisection, and inverse quadratic interpolation. It is 

sometimes known as the van Wijngaarden-Dekker-Brent method. Brent (1973) 

claims convergence is guaranteed for functions computable within [a,b]. 

 

[Brent1973]_ provides the classic description of the algorithm. Another 

description can be found in a recent edition of Numerical Recipes, including 

[PressEtal1992]_. A third description is at 

http://mathworld.wolfram.com/BrentsMethod.html. It should be easy to 

understand the algorithm just by reading our code. Our code diverges a bit 

from standard presentations: we choose a different formula for the 

extrapolation step. 

 

Parameters 

---------- 

f : function 

Python function returning a number. The function :math:`f` 

must be continuous, and :math:`f(a)` and :math:`f(b)` must 

have opposite signs. 

a : scalar 

One end of the bracketing interval :math:`[a, b]`. 

b : scalar 

The other end of the bracketing interval :math:`[a, b]`. 

xtol : number, optional 

The computed root ``x0`` will satisfy ``np.allclose(x, x0, 

atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The 

parameter must be nonnegative. For nice functions, Brent's 

method will often satisfy the above condition with ``xtol/2`` 

and ``rtol/2``. [Brent1973]_ 

rtol : number, optional 

The computed root ``x0`` will satisfy ``np.allclose(x, x0, 

atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The 

parameter cannot be smaller than its default value of 

``4*np.finfo(float).eps``. For nice functions, Brent's 

method will often satisfy the above condition with ``xtol/2`` 

and ``rtol/2``. [Brent1973]_ 

maxiter : int, optional 

if convergence is not achieved in `maxiter` iterations, an error is 

raised. Must be >= 0. 

args : tuple, optional 

containing extra arguments for the function `f`. 

`f` is called by ``apply(f, (x)+args)``. 

full_output : bool, optional 

If `full_output` is False, the root is returned. If `full_output` is 

True, the return value is ``(x, r)``, where `x` is the root, and `r` is 

a `RootResults` object. 

disp : bool, optional 

If True, raise RuntimeError if the algorithm didn't converge. 

Otherwise the convergence status is recorded in any `RootResults` 

return object. 

 

Returns 

------- 

x0 : float 

Zero of `f` between `a` and `b`. 

r : `RootResults` (present if ``full_output = True``) 

Object containing information about the convergence. In particular, 

``r.converged`` is True if the routine converged. 

 

Notes 

----- 

`f` must be continuous. f(a) and f(b) must have opposite signs. 

 

Related functions fall into several classes: 

 

multivariate local optimizers 

`fmin`, `fmin_powell`, `fmin_cg`, `fmin_bfgs`, `fmin_ncg` 

nonlinear least squares minimizer 

`leastsq` 

constrained multivariate optimizers 

`fmin_l_bfgs_b`, `fmin_tnc`, `fmin_cobyla` 

global optimizers 

`basinhopping`, `brute`, `differential_evolution` 

local scalar minimizers 

`fminbound`, `brent`, `golden`, `bracket` 

n-dimensional root-finding 

`fsolve` 

one-dimensional root-finding 

`brenth`, `ridder`, `bisect`, `newton` 

scalar fixed-point finder 

`fixed_point` 

 

References 

---------- 

.. [Brent1973] 

Brent, R. P., 

*Algorithms for Minimization Without Derivatives*. 

Englewood Cliffs, NJ: Prentice-Hall, 1973. Ch. 3-4. 

 

.. [PressEtal1992] 

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. 

*Numerical Recipes in FORTRAN: The Art of Scientific Computing*, 2nd ed. 

Cambridge, England: Cambridge University Press, pp. 352-355, 1992. 

Section 9.3: "Van Wijngaarden-Dekker-Brent Method." 

 

Examples 

-------- 

>>> def f(x): 

... return (x**2 - 1) 

 

>>> from scipy import optimize 

 

>>> root = optimize.brentq(f, -2, 0) 

>>> root 

-1.0 

 

>>> root = optimize.brentq(f, 0, 2) 

>>> root 

1.0 

""" 

if not isinstance(args, tuple): 

args = (args,) 

if xtol <= 0: 

raise ValueError("xtol too small (%g <= 0)" % xtol) 

if rtol < _rtol: 

raise ValueError("rtol too small (%g < %g)" % (rtol, _rtol)) 

r = _zeros._brentq(f, a, b, xtol, rtol, maxiter, args, full_output, disp) 

return results_c(full_output, r) 

 

 

def brenth(f, a, b, args=(), 

xtol=_xtol, rtol=_rtol, maxiter=_iter, 

full_output=False, disp=True): 

"""Find a root of a function in a bracketing interval using Brent's 

method with hyperbolic extrapolation. 

 

A variation on the classic Brent routine to find a zero of the function f 

between the arguments a and b that uses hyperbolic extrapolation instead of 

inverse quadratic extrapolation. There was a paper back in the 1980's ... 

f(a) and f(b) cannot have the same signs. Generally on a par with the 

brent routine, but not as heavily tested. It is a safe version of the 

secant method that uses hyperbolic extrapolation. The version here is by 

Chuck Harris. 

 

Parameters 

---------- 

f : function 

Python function returning a number. f must be continuous, and f(a) and 

f(b) must have opposite signs. 

a : scalar 

One end of the bracketing interval [a,b]. 

b : scalar 

The other end of the bracketing interval [a,b]. 

xtol : number, optional 

The computed root ``x0`` will satisfy ``np.allclose(x, x0, 

atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The 

parameter must be nonnegative. As with `brentq`, for nice 

functions the method will often satisfy the above condition 

with ``xtol/2`` and ``rtol/2``. 

rtol : number, optional 

The computed root ``x0`` will satisfy ``np.allclose(x, x0, 

atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The 

parameter cannot be smaller than its default value of 

``4*np.finfo(float).eps``. As with `brentq`, for nice functions 

the method will often satisfy the above condition with 

``xtol/2`` and ``rtol/2``. 

maxiter : int, optional 

if convergence is not achieved in `maxiter` iterations, an error is 

raised. Must be >= 0. 

args : tuple, optional 

containing extra arguments for the function `f`. 

`f` is called by ``apply(f, (x)+args)``. 

full_output : bool, optional 

If `full_output` is False, the root is returned. If `full_output` is 

True, the return value is ``(x, r)``, where `x` is the root, and `r` is 

a `RootResults` object. 

disp : bool, optional 

If True, raise RuntimeError if the algorithm didn't converge. 

Otherwise the convergence status is recorded in any `RootResults` 

return object. 

 

Returns 

------- 

x0 : float 

Zero of `f` between `a` and `b`. 

r : `RootResults` (present if ``full_output = True``) 

Object containing information about the convergence. In particular, 

``r.converged`` is True if the routine converged. 

 

Examples 

-------- 

>>> def f(x): 

... return (x**2 - 1) 

 

>>> from scipy import optimize 

 

>>> root = optimize.brenth(f, -2, 0) 

>>> root 

-1.0 

 

>>> root = optimize.brenth(f, 0, 2) 

>>> root 

1.0 

 

See Also 

-------- 

fmin, fmin_powell, fmin_cg, 

fmin_bfgs, fmin_ncg : multivariate local optimizers 

 

leastsq : nonlinear least squares minimizer 

 

fmin_l_bfgs_b, fmin_tnc, fmin_cobyla : constrained multivariate optimizers 

 

basinhopping, differential_evolution, brute : global optimizers 

 

fminbound, brent, golden, bracket : local scalar minimizers 

 

fsolve : n-dimensional root-finding 

 

brentq, brenth, ridder, bisect, newton : one-dimensional root-finding 

 

fixed_point : scalar fixed-point finder 

 

""" 

if not isinstance(args, tuple): 

args = (args,) 

if xtol <= 0: 

raise ValueError("xtol too small (%g <= 0)" % xtol) 

if rtol < _rtol: 

raise ValueError("rtol too small (%g < %g)" % (rtol, _rtol)) 

r = _zeros._brenth(f, a, b, xtol, rtol, maxiter, args, full_output, disp) 

return results_c(full_output, r) 

 

 

################################ 

# TOMS "Algorithm 748: Enclosing Zeros of Continuous Functions", by 

# Alefeld, G. E. and Potra, F. A. and Shi, Yixun, 

# See [1] 

 

 

def _within_tolerance(x, y, rtol, atol): 

diff = np.abs(x - y) 

z = np.abs(y) 

result = (diff <= (atol + rtol * z)) 

return result 

 

 

def _notclose(fs, rtol=_rtol, atol=_xtol): 

# Ensure not None, not 0, all finite, and not very close to each other 

notclosefvals = ( 

all(fs) and all(np.isfinite(fs)) and 

not any(any(np.isclose(_f, fs[i + 1:], rtol=rtol, atol=atol)) 

for i, _f in enumerate(fs[:-1]))) 

return notclosefvals 

 

 

def _secant(xvals, fvals): 

"""Perform a secant step, taking a little care""" 

# Secant has many "mathematically" equivalent formulations 

# x2 = x0 - (x1 - x0)/(f1 - f0) * f0 

# = x1 - (x1 - x0)/(f1 - f0) * f1 

# = (-x1 * f0 + x0 * f1) / (f1 - f0) 

# = (-f0 / f1 * x1 + x0) / (1 - f0 / f1) 

# = (-f1 / f0 * x0 + x1) / (1 - f1 / f0) 

x0, x1 = xvals[:2] 

f0, f1 = fvals[:2] 

if f0 == f1: 

return np.nan 

if np.abs(f1) > np.abs(f0): 

x2 = (-f0 / f1 * x1 + x0) / (1 - f0 / f1) 

else: 

x2 = (-f1 / f0 * x0 + x1) / (1 - f1 / f0) 

return x2 

 

 

def _update_bracket(ab, fab, c, fc): 

"""Update a bracket given (c, fc), return the discarded endpoints.""" 

fa, fb = fab 

idx = (0 if np.sign(fa) * np.sign(fc) > 0 else 1) 

rx, rfx = ab[idx], fab[idx] 

fab[idx] = fc 

ab[idx] = c 

return rx, rfx 

 

 

def _compute_divided_differences(xvals, fvals, N=None, full=True, 

forward=True): 

"""Return a matrix of divided differences for the xvals, fvals pairs 

 

DD[i, j] = f[x_{i-j}, ..., x_i] for 0 <= j <= i 

 

If full is False, just return the main diagonal(or last row): 

f[a], f[a, b] and f[a, b, c]. 

If forward is False, return f[c], f[b, c], f[a, b, c].""" 

if full: 

if forward: 

xvals = np.asarray(xvals) 

else: 

xvals = np.array(xvals)[::-1] 

M = len(xvals) 

N = M if N is None else min(N, M) 

DD = np.zeros([M, N]) 

DD[:, 0] = fvals[:] 

for i in range(1, N): 

DD[i:, i] = (np.diff(DD[i - 1:, i - 1]) / 

(xvals[i:] - xvals[:M - i])) 

return DD 

 

xvals = np.asarray(xvals) 

dd = np.array(fvals) 

row = np.array(fvals) 

idx2Use = (0 if forward else -1) 

dd[0] = fvals[idx2Use] 

for i in range(1, len(xvals)): 

denom = xvals[i:i + len(row) - 1] - xvals[:len(row) - 1] 

row = np.diff(row)[:] / denom 

dd[i] = row[idx2Use] 

return dd 

 

 

def _interpolated_poly(xvals, fvals, x): 

"""Compute p(x) for the polynomial passing through the specified locations. 

 

Use Neville's algorithm to compute p(x) where p is the minimal degree 

polynomial passing through the points xvals, fvals""" 

xvals = np.asarray(xvals) 

N = len(xvals) 

Q = np.zeros([N, N]) 

D = np.zeros([N, N]) 

Q[:, 0] = fvals[:] 

D[:, 0] = fvals[:] 

for k in range(1, N): 

alpha = D[k:, k - 1] - Q[k - 1:N - 1, k - 1] 

diffik = xvals[0:N - k] - xvals[k:N] 

Q[k:, k] = (xvals[k:] - x) / diffik * alpha 

D[k:, k] = (xvals[:N - k] - x) / diffik * alpha 

# Expect Q[-1, 1:] to be small relative to Q[-1, 0] as x approaches a root 

return np.sum(Q[-1, 1:]) + Q[-1, 0] 

 

 

def _inverse_poly_zero(a, b, c, d, fa, fb, fc, fd): 

"""Inverse cubic interpolation f-values -> x-values 

 

Given four points (fa, a), (fb, b), (fc, c), (fd, d) with 

fa, fb, fc, fd all distinct, find poly IP(y) through the 4 points 

and compute x=IP(0). 

""" 

return _interpolated_poly([fa, fb, fc, fd], [a, b, c, d], 0) 

 

 

def _newton_quadratic(ab, fab, d, fd, k): 

"""Apply Newton-Raphson like steps, using divided differences to approximate f' 

 

ab is a real interval [a, b] containing a root, 

fab holds the real values of f(a), f(b) 

d is a real number outside [ab, b] 

k is the number of steps to apply 

""" 

a, b = ab 

fa, fb = fab 

_, B, A = _compute_divided_differences([a, b, d], [fa, fb, fd], 

forward=True, full=False) 

 

# _P is the quadratic polynomial through the 3 points 

def _P(x): 

# Horner evaluation of fa + B * (x - a) + A * (x - a) * (x - b) 

return (A * (x - b) + B) * (x - a) + fa 

 

if A == 0: 

r = a - fa / B 

else: 

r = (a if np.sign(A) * np.sign(fa) > 0 else b) 

# Apply k Newton-Raphson steps to _P(x), starting from x=r 

for i in range(k): 

r1 = r - _P(r) / (B + A * (2 * r - a - b)) 

if not (ab[0] < r1 < ab[1]): 

if (ab[0] < r < ab[1]): 

return r 

r = sum(ab) / 2.0 

break 

r = r1 

 

return r 

 

 

class TOMS748Solver(object): 

"""Solve f(x, *args) == 0 using Algorithm748 of Alefeld, Potro & Shi. 

""" 

_MU = 0.5 

_K_MIN = 1 

_K_MAX = 100 # A very high value for real usage. Expect 1, 2, maybe 3. 

 

def __init__(self): 

self.f = None 

self.args = None 

self.function_calls = 0 

self.iterations = 0 

self.k = 2 

# ab=[a,b] is a global interval containing a root 

self.ab = [np.nan, np.nan] 

# fab is function values at a, b 

self.fab = [np.nan, np.nan] 

self.d = None 

self.fd = None 

self.e = None 

self.fe = None 

self.disp = False 

self.xtol = _xtol 

self.rtol = _rtol 

self.maxiter = _iter 

 

def configure(self, xtol, rtol, maxiter, disp, k): 

self.disp = disp 

self.xtol = xtol 

self.rtol = rtol 

self.maxiter = maxiter 

# Silently replace a low value of k with 1 

self.k = max(k, self._K_MIN) 

# Noisily replace a high value of k with self._K_MAX 

if self.k > self._K_MAX: 

msg = "toms748: Overriding k: ->%d" % self._K_MAX 

warnings.warn(msg, RuntimeWarning) 

self.k = self._K_MAX 

 

def _callf(self, x, error=True): 

"""Call the user-supplied function, update book-keeping""" 

fx = self.f(x, *self.args) 

self.function_calls += 1 

if not np.isfinite(fx) and error: 

raise ValueError("Invalid function value: f(%f) -> %s " % (x, fx)) 

return fx 

 

def get_result(self, x, flag=_ECONVERGED): 

r"""Package the result and statistics into a tuple.""" 

return (x, self.function_calls, self.iterations, flag) 

 

def _update_bracket(self, c, fc): 

return _update_bracket(self.ab, self.fab, c, fc) 

 

def start(self, f, a, b, args=()): 

r"""Prepare for the iterations.""" 

self.function_calls = 0 

self.iterations = 0 

 

self.f = f 

self.args = args 

self.ab[:] = [a, b] 

if not np.isfinite(a) or np.imag(a) != 0: 

raise ValueError("Invalid x value: %s " % (a)) 

if not np.isfinite(b) or np.imag(b) != 0: 

raise ValueError("Invalid x value: %s " % (b)) 

 

fa = self._callf(a) 

if not np.isfinite(fa) or np.imag(fa) != 0: 

raise ValueError("Invalid function value: f(%f) -> %s " % (a, fa)) 

if fa == 0: 

return _ECONVERGED, a 

fb = self._callf(b) 

if not np.isfinite(fb) or np.imag(fb) != 0: 

raise ValueError("Invalid function value: f(%f) -> %s " % (b, fb)) 

if fb == 0: 

return _ECONVERGED, b 

 

if np.sign(fb) * np.sign(fa) > 0: 

raise ValueError("a, b must bracket a root f(%e)=%e, f(%e)=%e " % 

(a, fa, b, fb)) 

self.fab[:] = [fa, fb] 

 

return _EINPROGRESS, sum(self.ab) / 2.0 

 

def get_status(self): 

"""Determine the current status.""" 

a, b = self.ab[:2] 

if _within_tolerance(a, b, self.rtol, self.xtol): 

return _ECONVERGED, sum(self.ab) / 2.0 

if self.iterations >= self.maxiter: 

return _ECONVERR, sum(self.ab) / 2.0 

return _EINPROGRESS, sum(self.ab) / 2.0 

 

def iterate(self): 

"""Perform one step in the algorithm. 

 

Implements Algorithm 4.1(k=1) or 4.2(k=2) in [APS1995] 

""" 

self.iterations += 1 

eps = np.finfo(float).eps 

d, fd, e, fe = self.d, self.fd, self.e, self.fe 

ab_width = self.ab[1] - self.ab[0] # Need the start width below 

c = None 

 

for nsteps in range(2, self.k+2): 

# If the f-values are sufficiently separated, perform an inverse 

# polynomial interpolation step. Otherwise nsteps repeats of 

# an approximate Newton-Raphson step. 

if _notclose(self.fab + [fd, fe], rtol=0, atol=32*eps): 

c0 = _inverse_poly_zero(self.ab[0], self.ab[1], d, e, 

self.fab[0], self.fab[1], fd, fe) 

if self.ab[0] < c0 < self.ab[1]: 

c = c0 

if c is None: 

c = _newton_quadratic(self.ab, self.fab, d, fd, nsteps) 

 

fc = self._callf(c) 

if fc == 0: 

return _ECONVERGED, c 

 

# re-bracket 

e, fe = d, fd 

d, fd = self._update_bracket(c, fc) 

 

# u is the endpoint with the smallest f-value 

uix = (0 if np.abs(self.fab[0]) < np.abs(self.fab[1]) else 1) 

u, fu = self.ab[uix], self.fab[uix] 

 

_, A = _compute_divided_differences(self.ab, self.fab, 

forward=(uix == 0), full=False) 

c = u - 2 * fu / A 

if np.abs(c - u) > 0.5 * (self.ab[1] - self.ab[0]): 

c = sum(self.ab) / 2.0 

else: 

if np.isclose(c, u, rtol=eps, atol=0): 

# c didn't change (much). 

# Either because the f-values at the endpoints have vastly 

# differing magnitudes, or because the root is very close to 

# that endpoint 

frs = np.frexp(self.fab)[1] 

if frs[uix] < frs[1 - uix] - 50: # Differ by more than 2**50 

c = (31 * self.ab[uix] + self.ab[1 - uix]) / 32 

else: 

# Make a bigger adjustment, about the 

# size of the requested tolerance. 

mm = (1 if uix == 0 else -1) 

adj = mm * np.abs(c) * self.rtol + mm * self.xtol 

c = u + adj 

if not self.ab[0] < c < self.ab[1]: 

c = sum(self.ab) / 2.0 

 

fc = self._callf(c) 

if fc == 0: 

return _ECONVERGED, c 

 

e, fe = d, fd 

d, fd = self._update_bracket(c, fc) 

 

# If the width of the new interval did not decrease enough, bisect 

if self.ab[1] - self.ab[0] > self._MU * ab_width: 

e, fe = d, fd 

z = sum(self.ab) / 2.0 

fz = self._callf(z) 

if fz == 0: 

return _ECONVERGED, z 

d, fd = self._update_bracket(z, fz) 

 

# Record d and e for next iteration 

self.d, self.fd = d, fd 

self.e, self.fe = e, fe 

 

status, xn = self.get_status() 

return status, xn 

 

def solve(self, f, a, b, args=(), 

xtol=_xtol, rtol=_rtol, k=2, maxiter=_iter, disp=True): 

r"""Solve f(x) = 0 given an interval containing a zero.""" 

self.configure(xtol=xtol, rtol=rtol, maxiter=maxiter, disp=disp, k=k) 

status, xn = self.start(f, a, b, args) 

if status == _ECONVERGED: 

return self.get_result(xn) 

 

# The first step only has two x-values. 

c = _secant(self.ab, self.fab) 

if not self.ab[0] < c < self.ab[1]: 

c = sum(self.ab) / 2.0 

fc = self._callf(c) 

if fc == 0: 

return self.get_result(c) 

 

self.d, self.fd = self._update_bracket(c, fc) 

self.e, self.fe = None, None 

self.iterations += 1 

 

while True: 

status, xn = self.iterate() 

if status == _ECONVERGED: 

return self.get_result(xn) 

if status == _ECONVERR: 

fmt = "Failed to converge after %d iterations, bracket is %s" 

if disp: 

msg = fmt % (self.iterations + 1, self.ab) 

raise RuntimeError(msg) 

return self.get_result(xn, _ECONVERR) 

 

 

def toms748(f, a, b, args=(), k=1, 

xtol=_xtol, rtol=_rtol, maxiter=_iter, 

full_output=False, disp=True): 

""" 

Find a zero using TOMS Algorithm 748 method. 

 

Implements the Algorithm 748 method of Alefeld, Potro and Shi to find a 

zero of the function `f` on the interval `[a , b]`, where `f(a)` and 

`f(b)` must have opposite signs. 

 

It uses a mixture of inverse cubic interpolation and 

"Newton-quadratic" steps. [APS1995]. 

 

Parameters 

---------- 

f : function 

Python function returning a scalar. The function :math:`f` 

must be continuous, and :math:`f(a)` and :math:`f(b)` 

have opposite signs. 

a : scalar, 

lower boundary of the search interval 

b : scalar, 

upper boundary of the search interval 

args : tuple, optional 

containing extra arguments for the function `f`. 

`f` is called by ``f(x, *args)``. 

k : int, optional 

The number of Newton quadratic steps to perform each 

iteration. ``k>=1``. 

xtol : scalar, optional 

The computed root ``x0`` will satisfy ``np.allclose(x, x0, 

atol=xtol, rtol=rtol)``, where ``x`` is the exact root. The 

parameter must be nonnegative. 

rtol : scalar, optional 

The computed root ``x0`` will satisfy ``np.allclose(x, x0, 

atol=xtol, rtol=rtol)``, where ``x`` is the exact root. 

maxiter : int, optional 

if convergence is not achieved in `maxiter` iterations, an error is 

raised. Must be >= 0. 

full_output : bool, optional 

If `full_output` is False, the root is returned. If `full_output` is 

True, the return value is ``(x, r)``, where `x` is the root, and `r` is 

a `RootResults` object. 

disp : bool, optional 

If True, raise RuntimeError if the algorithm didn't converge. 

Otherwise the convergence status is recorded in the `RootResults` 

return object. 

 

Returns 

------- 

x0 : float 

Approximate Zero of `f` 

r : `RootResults` (present if ``full_output = True``) 

Object containing information about the convergence. In particular, 

``r.converged`` is True if the routine converged. 

 

See Also 

-------- 

brentq, brenth, ridder, bisect, newton 

fsolve : find zeroes in n dimensions. 

 

Notes 

----- 

`f` must be continuous. 

Algorithm 748 with ``k=2`` is asymptotically the most efficient 

algorithm known for finding roots of a four times continuously 

differentiable function. 

In contrast with Brent's algorithm, which may only decrease the length of 

the enclosing bracket on the last step, Algorithm 748 decreases it each 

iteration with the same asymptotic efficiency as it finds the root. 

 

For easy statement of efficiency indices, assume that `f` has 4 

continuouous deriviatives. 

For ``k=1``, the convergence order is at least 2.7, and with about 

asymptotically 2 function evaluations per iteration, the efficiency 

index is approximately 1.65. 

For ``k=2``, the order is about 4.6 with asymptotically 3 function 

evaluations per iteration, and the efficiency index 1.66. 

For higher values of `k`, the efficiency index approaches 

the `k`-th root of ``(3k-2)``, hence ``k=1`` or ``k=2`` are 

usually appropriate. 

 

References 

---------- 

.. [APS1995] 

Alefeld, G. E. and Potra, F. A. and Shi, Yixun, 

*Algorithm 748: Enclosing Zeros of Continuous Functions*, 

ACM Trans. Math. Softw. Volume 221(1995) 

doi = {10.1145/210089.210111} 

 

Examples 

-------- 

>>> def f(x): 

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

 

>>> from scipy import optimize 

>>> root, results = optimize.toms748(f, 0, 2, full_output=True) 

>>> root 

1.0 

>>> results 

converged: True 

flag: 'converged' 

function_calls: 11 

iterations: 5 

root: 1.0 

""" 

if xtol <= 0: 

raise ValueError("xtol too small (%g <= 0)" % xtol) 

if rtol < _rtol / 4: 

raise ValueError("rtol too small (%g < %g)" % (rtol, _rtol)) 

if maxiter < 1: 

raise ValueError("maxiter must be greater than 0") 

if not np.isfinite(a): 

raise ValueError("a is not finite %s" % a) 

if not np.isfinite(b): 

raise ValueError("b is not finite %s" % b) 

if a >= b: 

raise ValueError("a and b are not an interval [%d, %d]" % (a, b)) 

if not k >= 1: 

raise ValueError("k too small (%s < 1)" % k) 

 

if not isinstance(args, tuple): 

args = (args,) 

solver = TOMS748Solver() 

result = solver.solve(f, a, b, args=args, k=k, xtol=xtol, rtol=rtol, 

maxiter=maxiter, disp=disp) 

x, function_calls, iterations, flag = result 

return _results_select(full_output, (x, function_calls, iterations, flag))