Comparison of scalar scipy.optimze.zeros.newton from v1.0.0 with proposed vectorized version from pr #8357

bench_scalar_array_newton.py

from __future__ import division, print_function, absolute_import
from math import sqrt, exp, sin, cos
import warnings
import numpy as np


# Newton-Raphson method
def scalar_newton(func, x0, fprime=None, args=(), tol=1.48e-8, maxiter=50,
           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
    derivate `fprime2` of `func` is provided, parabolic 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 given, parabolic 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`` and ``fprime2`` not provided, use secant method
    
    >>> root = optimize.newton(f, 1.5)
    >>> 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
    
    ``fprime2`` provided, ``fprime`` provided/not provided use parabolic
    Halley's method

    >>> root = optimize.newton(f, 1.5, fprime2=lambda x: 6 * x)
    >>> root
    1.0000000000000016
    >>> 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")
    if fprime is not None:
        # Newton-Rapheson method
        # 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
        fder2 = 0
        for iter in range(maxiter):
            myargs = (p0,) + args
            fder = fprime(*myargs)
            if fder == 0:
                msg = "derivative was zero."
                warnings.warn(msg, RuntimeWarning)
                return p0
            fval = func(*myargs)
            if fprime2 is not None:
                fder2 = fprime2(*myargs)
            if fder2 == 0:
                # Newton step
                p = p0 - fval / fder
            else:
                # Parabolic Halley's method
                discr = fder ** 2 - 2 * fval * fder2
                if discr < 0:
                    p = p0 - fder / fder2
                else:
                    p = p0 - 2*fval / (fder + np.sign(fder) * sqrt(discr))
            if abs(p - p0) < tol:
                return p
            p0 = p
    else:
        # Secant method
        p0 = x0
        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)


# Newton-Raphson method
def array_newton(func, x0, fprime=None, args=(), tol=1.48e-8, maxiter=50,
           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
    derivate `fprime2` of `func` is provided, parabolic 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 given, parabolic 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`` and ``fprime2`` not provided, use secant method
    
    >>> root = optimize.newton(f, 1.5)
    >>> 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
    
    ``fprime2`` provided, ``fprime`` provided/not provided use parabolic
    Halley's method

    >>> root = optimize.newton(f, 1.5, fprime2=lambda x: 6 * x)
    >>> root
    1.0000000000000016
    >>> 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")
    if fprime is not None:
        # Newton-Rapheson method
        # 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 * np.asarray(x0)  # convert to ndarray
        for iter in range(maxiter):
            myargs = (p0,) + args
            fder = np.asarray(fprime(*myargs))  # convert to ndarray
            if (fder == 0).any():
                msg = "derivative was zero."
                warnings.warn(msg, RuntimeWarning)
                return p0
            fval = np.asarray(func(*myargs))  # convert to ndarray
            if fprime2 is None:
                # Newton step
                p = p0 - fval / fder
            else:
                fder2 = np.asarray(fprime2(*myargs))  # convert to ndarray
                # Halley's method
                # https://en.wikipedia.org/wiki/Halley%27s_method
                p = p0 - 2 * fval * fder / (2 * fder ** 2 - fval * fder2)
            if np.abs(p - p0).max() < tol:
                return p
            p0 = p
    else:
        # Secant method
        p0 = np.asarray(x0)
        dx = np.finfo(float).eps**0.33
        dp = np.where(p0 >= 0, dx, -dx)
        p1 = p0 * (1 + dx) + dp
        q0 = np.asarray(func(*((p0,) + args)))
        q1 = np.asarray(func(*((p1,) + args)))
        for iter in range(maxiter):
            divide_by_zero = (q1 == q0)
            if divide_by_zero.any():
                tolerance_reached = (p1 != p0)
                if (divide_by_zero & tolerance_reached).any():
                    msg = "Tolerance of %s reached" % np.sqrt(sum((p1 - p0)**2))
                    warnings.warn(msg, RuntimeWarning)
                return (p1 + p0)/2.0
            else:
                p = p1 - q1*(p1 - p0)/(q1 - q0)
            if np.abs(p - p1).max() < tol:
                return p
            p0 = p1
            q0 = q1
            p1 = p
            q1 = np.asarray(func(*((p1,) + args)))
    msg = "Failed to converge after %d iterations, value is %s" % (maxiter, p)
    raise RuntimeError(msg)


def test_newton(method):
    f1 = lambda x: x**2 - 2*x - 1
    f1_1 = lambda x: 2*x - 2
    f1_2 = lambda x: 2.0 + 0*x

    f2 = lambda x: exp(x) - cos(x)
    f2_1 = lambda x: exp(x) + sin(x)
    f2_2 = lambda x: exp(x) + cos(x)

    for f, f_1, f_2 in [(f1, f1_1, f1_2), (f2, f2_1, f2_2)]:
        x = method(f, 3, tol=1e-6)
        x = method(f, 3, fprime=f_1, tol=1e-6)
        x = method(f, 3, fprime=f_1, fprime2=f_2, tol=1e-6)


def test_newton_array(method):
    """test newton with array"""

    def f1(x, *a):
        b = a[0] + x * a[3]
        return a[1] - a[2] * (np.exp(b / a[5]) - 1.0) - b / a[4] - x

    def f1_1(x, *a):
        b = a[3] / a[5]
        return -a[2] * np.exp(a[0] / a[5] + x * b) * b - a[3] / a[4] - 1

    def f1_2(x, *a):
        b = a[3] / a[5]
        return -a[2] * np.exp(a[0] / a[5] + x * b) * b ** 2

    args = (5.32725221, 7.0, 1e-09, 0.004, 10, 0.27456)
    x0 = 7.0
    x = method(f1, x0, f1_1, args)

    # test halley's
    x = method(f1, x0, f1_1, args, fprime2=f1_2)

    # test secant
    x = method(lambda y, z: z - y ** 2, 4.0, args=(15.0,))