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Inzynierka/Lib/site-packages/scipy/optimize/_zeros_py.py

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2023-06-02 12:51:02 +02:00
import warnings
from collections import namedtuple
import operator
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:
"""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 scalar-valued function `func` given a nearby scalar
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, `newton` returns an array:
the zeros of the function from each (scalar) starting point in `x0`.
In this case, `func` must be vectorized to return a sequence or array of
the same shape as its first argument. If `fprime` (`fprime2`) is given,
then its return must also have the same shape: each element is the first
(second) derivative of `func` with respect to its only variable evaluated
at each element of its first argument.
`newton` is for finding roots of a scalar-valued functions of a single
variable. For problems involving several variables, see `root`.
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
--------
root_scalar : interface to root solvers for scalar functions
root : interface to root solvers for multi-input, multi-output functions
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 nth 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
--------
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy import optimize
>>> 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
>>> rng = np.random.default_rng()
>>> x = rng.standard_normal(100)
>>> a = np.arange(-50, 50)
>>> vec_res = optimize.newton(f, x, fprime=fder, args=(a, ), maxiter=200)
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,),
... maxiter=200)
... 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, ax = plt.subplots()
>>> 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)
maxiter = operator.index(maxiter)
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.
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 = np.asarray(p, dtype=np.result_type(p, dp, np.float64))
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 = np.asarray(p, dtype=np.result_type(p, p1, dp, np.float64))
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,)
maxiter = operator.index(maxiter)
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 : 1-D 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,)
maxiter = operator.index(maxiter)
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-D root-finding
`fsolve`
1-D 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,)
maxiter = operator.index(maxiter)
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. Bus & Dekker (1975) guarantee convergence
for this method, claiming that the upper bound of function evaluations here
is 4 or 5 times lesser than that for bisection.
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, and implements Algorithm M of
[BusAndDekker1975]_, where further details (convergence properties,
additional remarks and such) can be found
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.
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-D root-finding
brentq, brenth, ridder, bisect, newton : 1-D root-finding
fixed_point : scalar fixed-point finder
References
----------
.. [BusAndDekker1975]
Bus, J. C. P., Dekker, T. J.,
"Two Efficient Algorithms with Guaranteed Convergence for Finding a Zero
of a Function", ACM Transactions on Mathematical Software, Vol. 1, Issue
4, Dec. 1975, pp. 330-345. Section 3: "Algorithm M".
:doi:`10.1145/355656.355659`
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
"""
if not isinstance(args, tuple):
args = (args,)
maxiter = operator.index(maxiter)
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 _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:
"""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 np.isclose(a, b, rtol=self.rtol, atol=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 kth 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))
maxiter = operator.index(maxiter)
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 [{}, {}]".format(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))
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