def brentq(f, a, b, args=(),
xtol=_xtol, rtol=_rtol, maxiter=_iter,
full_output=False, disp=True):
"""
Find a root of a function in given interval.
Return float, a zero of `f` between `a` and `b`. `f` must be a continuous
function, and [a,b] must be a sign changing interval.
Description:
Uses the classic Brent (1973) 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-Deker-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]_. Another 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. f must be continuous, and f(a) and
f(b) must have opposite signs.
a : number
One end of the bracketing interval [a,b].
b : number
The other end of the bracketing interval [a,b].
xtol : number, optional
The routine converges when a root is known to lie within xtol of the
value return. Should be >= 0. The routine modifies this to take into
account the relative precision of doubles.
maxiter : number, optional
if convergence is not achieved in maxiter iterations, and 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.
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
--------
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
`anneal`, `brute`
local scalar minimizers
`fminbound`, `brent`, `golden`, `bracket`
n-dimensional root-finding
`fsolve`
one-dimensional root-finding
`brentq`, `brenth`, `ridder`, `bisect`, `newton`
scalar fixed-point finder
`fixed_point`
Notes
-----
`f` must be continuous. f(a) and f(b) must have opposite signs.
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."
"""
if type(args) != type(()) :
args = (args,)
r = _zeros._brentq(f,a,b,xtol,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=False):
"""Find root of f in [a,b]
The classic Brent routine to find a zero of the function
f between the arguments a and b. f(a) and f(b) can not
have the same signs. Generally the best of the routines here.
It is a safe version of the secant method that uses inverse
quadratic extrapolation. The version here is a slight
modification that uses a different formula in the extrapolation
step. A description may be found in Numerical Recipes, but the
code here is probably easier to understand.
f : Python function returning a number.
a : Number, one end of the bracketing interval.
b : Number, the other end of the bracketing interval.
xtol : Number, the routine converges when a root is known
to lie within xtol of the value return. Should be >= 0.
The routine modifies this to take into account the relative
precision of doubles.
maxiter : Number, if convergence is not achieved in
maxiter iterations, and error is raised. Must be
>= 0.
args : tuple containing extra arguments for the function f.
f is called by apply(f,(x)+args).
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 containing information
about the convergence. In particular, r.converged is True if the
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
anneal, brute -- global optimizers
fminbound, brent, golden, bracket -- local scalar minimizers
fsolve -- n-dimenstional root-finding
brentq, brenth, ridder, bisect, newton -- one-dimensional root-finding
fixed_point -- scalar fixed-point finder
"""
if type(args) != type(()) :
args = (args,)
r = _zeros._brentq(f,a,b,xtol,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 given interval.
Return float, a zero of `f` between `a` and `b`. `f` must be a continuous
function, and [a,b] must be a sign changing interval.
Description:
Uses the classic Brent (1973) 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-Deker-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]_. Another 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. f must be continuous, and f(a) and
f(b) must have opposite signs.
a : number
One end of the bracketing interval [a,b].
b : number
The other end of the bracketing interval [a,b].
xtol : number, optional
The routine converges when a root is known to lie within xtol of the
value return. Should be >= 0. The routine modifies this to take into
account the relative precision of doubles.
maxiter : number, optional
if convergence is not achieved in maxiter iterations, and 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.
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
--------
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
`anneal`, `brute`
local scalar minimizers
`fminbound`, `brent`, `golden`, `bracket`
n-dimensional root-finding
`fsolve`
one-dimensional root-finding
`brentq`, `brenth`, `ridder`, `bisect`, `newton`
scalar fixed-point finder
`fixed_point`
Notes
-----
`f` must be continuous. f(a) and f(b) must have opposite signs.
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."
"""
if type(args) != type(()):
args = (args, )
r = _zeros._brentq(f, a, b, xtol, 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, )
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)
