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""" 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. """ self.root = root self.iterations = iterations self.function_calls = function_calls self.converged = flag == 0 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])
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
# Newton-Raphson method fprime2=None): """ Find a zero using the Newton-Raphson or secant 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 provided, then Halley's method is used.
Parameters ---------- func : function 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 An initial estimate of the zero that should be somewhere near the actual zero. fprime : function, 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. maxiter : int, optional Maximum number of iterations. fprime2 : function, 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.
Returns ------- zero : float Estimated location where function is zero.
See Also -------- brentq, brenth, ridder, bisect fsolve : find zeroes 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 is approximately the square (cube for Halley) of the requested tolerance up to roundoff error. 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.
Examples --------
>>> def f(x): ... return (x**3 - 1) # only one real root at x = 1
>>> from scipy import optimize
``fprime`` not provided, use 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`` provided, use Newton Raphson method
>>> root = optimize.newton(f, 1.5, fprime=lambda x: 3 * x**2) >>> root 1.0
Both ``fprime2`` and ``fprime`` 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
""" if tol <= 0: raise ValueError("tol too small (%g <= 0)" % tol) if maxiter < 1: raise ValueError("maxiter must be greater than 0") # Multiply by 1.0 to convert to floating point. We don't use float(x0) # so it still works if x0 is complex. p0 = 1.0 * x0 if fprime is not None: # Newton-Rapheson method for iter in range(maxiter): fder = fprime(p0, *args) if fder == 0: msg = "derivative was zero." warnings.warn(msg, RuntimeWarning) return p0 fval = func(p0, *args) newton_step = fval / fder if fprime2 is None: # Newton step p = p0 - newton_step else: fder2 = fprime2(p0, *args) # Halley's method p = p0 - newton_step / (1.0 - 0.5 * newton_step * fder2 / fder) if abs(p - p0) < tol: return p p0 = p else: # Secant method if x0 >= 0: p1 = x0*(1 + 1e-4) + 1e-4 else: p1 = x0*(1 + 1e-4) - 1e-4 q0 = func(p0, *args) q1 = func(p1, *args) for iter in range(maxiter): if q1 == q0: if p1 != p0: msg = "Tolerance of %s reached" % (p1 - p0) warnings.warn(msg, RuntimeWarning) return (p1 + p0)/2.0 else: p = p1 - q1*(p1 - p0)/(q1 - q0) if abs(p - p1) < tol: return p p0 = p1 q0 = q1 p1 = p q1 = func(p1, *args) msg = "Failed to converge after %d iterations, value is %s" % (maxiter, p) raise RuntimeError(msg)
xtol=_xtol, rtol=_rtol, maxiter=_iter, full_output=False, disp=True): """ Find root of a function within an interval.
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 : number One end of the bracketing interval [a,b]. b : number 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 : number, 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.
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)
xtol=_xtol, rtol=_rtol, maxiter=_iter, full_output=False, disp=True): """ Find a root of a function in an interval.
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 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 : number, 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.
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.
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
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.
""" 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)
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]_. 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. The function :math:`f` must be continuous, and :math:`f(a)` and :math:`f(b)` must have opposite signs. a : number One end of the bracketing interval :math:`[a, b]`. b : number 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 : number, 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.
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 `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`
Notes ----- `f` must be continuous. f(a) and f(b) must have opposite signs.
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
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 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)
xtol=_xtol, rtol=_rtol, maxiter=_iter, full_output=False, disp=True): """Find root of f in [a,b].
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 : number One end of the bracketing interval [a,b]. b : number 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 : number, 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.
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) |