a a @sddlZddlmZddlZddlmZddlZdZdZ de e j Z gdZdZd Zd Zd ZdZd Zd ZdZdZdZeeeeeeeeeeiZGdddZddZddZd;ddZdd Zde e eddfd!d"Zde e eddfd#d$Z de e eddfd%d&Z!de e eddfd'd(Z"e e fd)d*Z#d+d,Z$d-d.Z%dBsz(RootResults.__repr__..)maxmaplenjoin)rattrsrr#r__repr__>s zRootResults.__repr__N)__name__ __module__ __qualname____doc__rr+rrrrr !s r cCs0|r(|\}}}}t||||d}||fS|SdS)Nrrrrr  full_outputrxfuncallsrrresultsrrr results_cFs r8cCs,|\}}}}|r(t||||d}||fS|S)z:Select from a tuple of (root, funccalls, iterations, flag)r0r1r2rrr_results_selectRs r9r`sbO>2FTc  Cs|dkrtd|t|}|dkr.tdt|dkrRt|||||||| Sd|} d} |durt|D]2} || g|R}| d7} |dkrt| | | | tfS|| g|R}| d7} |dkrd}| r|d| d| f7}t |t |t t| | | | dt fS||}|rj|| g|R}| d7} |||d }t|dkrj|d|}| |}tj|| ||d rt| || | dtfS|} qpn|dur||krtd |}n(d }|d|}||dkr|n| 7}|| g|R}| d7} ||g|R}| d7} t|t|krJ|| ||f\} }}}t|D]} ||kr|| krd || }| r|d| d|f7}t |t |t || d}t| || | dtfSt|t|kr| ||| d||}n| || |d||}tj||||d rJt| || | dtfS||} }|}||g|R}| d7} qR| rd| d|f}t |t| || | dt fS)a} 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 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 -------- >>> 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 >>> 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,)) ... 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() rztol too small (%g <= 0)rmaxiter must be greater than 0?NzDerivative was zero.z5 Failed to converge after %d iterations, value is %s.rtolatolzx1 and x0 must be differentg-C6?zTolerance of %s reached.@z4Failed to converge after %d iterations, value is %s.) ValueErroroperatorindexnpsize _array_newtonranger9r RuntimeErrorwarningswarnRuntimeWarning _ECONVERRabsisclose)funcx0fprimeargstolmaxiterfprime2x1rAr3dispZp0r6itrfvalfdermsgZ newton_stepfder2adjpp1epsq0q1rrrr^s+                      rc Cstj|dd}tj|td} t| } |dur8t|D]} t||g|R} | sj| t} qt||g|R} | dk} | sq| | | | }|durt||g|R}|dd||| | | }tj|t||tj d}|| |8<t ||k| | <| | s8qq8nVt t j d}|d |t|dk|| }t||g|R}t||g|R}tj|td}t|D]} ||k} | s||d }q|||| ||| }tj|t|||tj d}|| ||| <| |@}|||d ||<|| M}t ||k| | <| | sfq||}}|}t||g|R}q| | @}|r"|dur||k}||@}|r tt||||d }td |tn(|rd nd}d|}t|tnF| rh| r:d nd}d||}| r\t|t|t|rtdd}||| |}|S)z 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`. T)copy)ZdtypeNrr>?gQ?rrCr?zRMS of {:g} reachedallZsomez{:s} derivatives were zeroz/{0:s} failed to converge after {1:d} iterationsresult)rrzero_der)rGarrayZ ones_likeboolrJasarrayanyZastypeZ result_typeZfloat64rPfinfofloatrcwheresqrtsumrLrMformatrNrhrKr)rRrSrTrUrVrWrXr3raZfailuresZnz_der iterationr\r]Zdpr_ZdxrbrdreactiveZactive_zero_derrjZ nonzero_dpZzero_der_nz_dpZrmsZ all_or_somer^rirrrrIns                        rIc Csjt|ts|f}t|}|dkr.td||tkrFtd|tft||||||||| } t|| S)a# 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 rxtol too small (%g <= 0)rtol too small (%g < %g)) isinstancetuplerErFrD_rtolr_bisectr8 fr"brUxtolrArWr3rZr4rrrrsJ   rc Csjt|ts|f}t|}|dkr.td||tkrFtd|tft||||||||| } t|| S)a? 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 rrwrx) ryrzrErFrDr{rZ_ridderr8r}rrrr )sV   r c Csjt|ts|f}t|}|dkr.td||tkrFtd|tft||||||||| } t|| S)a 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 rrwrx) ryrzrErFrDr{rZ_brentqr8r}rrrr sw   r c Csjt|ts|f}t|}|dkr.td||tkrFtd|tft||||||||| } t|| S)a 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-D root-finding brentq, brenth, ridder, bisect, newton : 1-D root-finding fixed_point : scalar fixed-point finder rrwrx) ryrzrErFrDr{rZ_brenthr8r}rrrr s^   r csBtosz_notclose..r)rhrGisfinitern enumerate)rrArBZ notclosefvalsrrr _notclose{s rcCs|dd\}}|dd\}}||kr.tjSt|t|krb| |||d||}n| |||d||}|S)z+Perform a secant step, taking a little careNr?r)rGnanrP)xvalsfvalsrSrYZf0f1Zx2rrr_secants rc CsR|\}}t|t|dkr$dnd}||||}}|||<|||<||fS)z?Update a bracket given (c, fc), return the discarded endpoints.rr)rGsign) abfabcfcfafbidxrxZrfxrrr_update_brackets  rc Csd|r|rt|}nt|ddd}t|}|dur<|nt||}t||g}|dd|dddf<td|D]J}t||dd|df||d|d||||d|f<qv|St|}t|}t|} |rdnd} || |d<tdt|D]T}|||t| d|dt| d} t| dd| } | | ||<q |S)aReturn a matrix of divided differences for the xvals, fvals pairs DD[i, j] = f[x_{i-j}, ..., x_i] for 0 <= j <= i If full is False, just return the main diagonal(or last row): f[a], f[a, b] and f[a, b, c]. If forward is False, return f[c], f[b, c], f[a, b, c].Nrrr)rGrmrkr(minzerosrJdiff) rrNfullforwardMZDDrddrowZidx2UseZdenomrrr_compute_divided_differencess.       ,rc Cs$t|}t|}t||g}t||g}|dd|dddf<|dd|dddf<td|D]}||d|df||d|d|df}|d|||||}||d|||||d|f<|d|||||||d|f<qht|dddf|dS)zCompute 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, fvalsNrrr)rr)rGrmr(rrJrs) rrr5rQDkalphaZdiffikrrr_interpolated_polys 0$*rcCst||||g||||gdS)zInverse cubic interpolation f-values -> x-values Given four points (fa, a), (fb, b), (fc, c), (fd, d) with fa, fb, fc, fd all distinct, find poly IP(y) through the 4 points and compute x=IP(0). r)r)r"rrdrrrfdrrr_inverse_poly_zerosrc s|\|\}t|g||gddd\}fdd}dkr\}n ttdkrxn}t|D]} |||d|} |d| kr|dksn|d|kr|dkrnn|St|d }q | }q|S) zApply Newton-Raphson like steps, using divided differences to approximate f' ab is a real interval [a, b] containing a root, fab holds the real values of f(a), f(b) d is a real number outside [ab, b] k is the number of steps to apply TFrrcs||SNr)r5ABr"rrrr_Psz_newton_quadratic.._Prr?rrC)rrGrrJrs) rrrrrr_rr4rZr1rrr_newton_quadratics$   $   rc@szeZdZdZdZdZdZddZddZdd d Z e fd d Z ddZ dddZ ddZddZdeeded fddZdS) TOMS748SolverzGSolve f(x, *args) == 0 using Algorithm748 of Alefeld, Potro & Shi. rgrrcCsnd|_d|_d|_d|_d|_tjtjg|_tjtjg|_d|_ d|_ d|_ d|_ d|_ t|_t|_t|_dS)Nrr?F)r~rUrrrrGrrrrreferZ_xtolrr{rA_iterrW)rrrrr szTOMS748Solver.__init__cCsT||_||_||_||_t||j|_|j|jkrPd|j}t |t |j|_dS)Nztoms748: Overriding k: ->%d) rZrrArWr&_K_MINr_K_MAXrLrMrN)rrrArWrZrr^rrr configures   zTOMS748Solver.configureTcCsD|j|g|jR}|jd7_t|s@|r@td||f|S)z4Call the user-supplied function, update book-keepingr$Invalid function value: f(%f) -> %s )r~rUrrGrrD)rr5errorZfxrrr_callf,s zTOMS748Solver._callfcCs||j|j|fS)z/Package the result and statistics into a tuple.)rr)rr5rrrr get_result4szTOMS748Solver.get_resultcCst|j|j||Sr)rrr)rrrrrrr8szTOMS748Solver._update_bracketrcCsHd|_d|_||_||_||g|jdd<t|rBt|dkrNtd|t|rft|dkrrtd|| |}t|rt|dkrtd||f|dkrt |fS| |}t|rt|dkrtd||f|dkrt |fSt |t |dkr$td||||f||g|j dd<t t|jdfS)zPrepare for the iterations.rNzInvalid x value: %s rz,a, b must bracket a root f(%e)=%e, f(%e)=%e rC)rrr~rUrrGrimagrDrrrr _EINPROGRESSrs)rr~r"rrUrrrrrstart;s2     zTOMS748Solver.startcCsj|jdd\}}tj|||j|jdr:tt|jdfS|j|jkrXt t|jdfSt t|jdfS)zDetermine the current status.Nr?r@rC) rrGrQrArrrsrrWrOr)rr"rrrr get_statusZs  zTOMS748Solver.get_statusc Csf|jd7_ttj}|j|j|j|jf\}}}}|j d|j d}d}t d|j dD]}t |j ||gdd|drt|j d|j d|||j d|j d||} |j d| kr|j dkrnn| }|durt|j |j |||}||} | dkrt|fS||}}||| \}}q^t|j dt|j dkrRdnd} |j | |j | } } t|j |j | dkdd\}}| d| |}t|| d |j d|j dkrt|j d }ntj|| |ddrt|j d}|| |d| d kr0d |j | |j d| d}n8| dkr>dnd }|t||j||j}| |}|j d|kr|j dksnt|j d }||} | dkrt|fS||}}||| \}}|j d|j d|j|kr6||}}t|j d }||}|dkr&t|fS|||\}}|||_|_|||_|_|\}}||fS)zkPerform one step in the algorithm. Implements Algorithm 4.1(k=1) or 4.2(k=2) in [APS1995] rrNr? r@FrrgrCr;r)rrGrorprcrrrrrrJrrrrrrrrrPrrsrQfrexprAr_MUr)rrcrrrrZab_widthrZnstepsZc0rZuixuZfurrZfrsmmr`zZfzstatusxnrrriteratecsh $    *  ("&        zTOMS748Solver.iterater?c Cs |j|||| |d|||||\} } | tkr:|| St|j|j} |jd| krh|jdksxnt|jd} || } | dkr|| S| | | \|_ |_ d\|_ |_ |jd7_|\} } | tkr|| S| tkrd}| r||jd|jf}t||| tSqdS)z3Solve f(x) = 0 given an interval containing a zero.)rrArWrZrrrrC)NNz5Failed to converge after %d iterations, bracket is %sN)rrrrrrrrsrrrrrrrrrOrK)rr~r"rrUrrArrWrZrrrrfmtr^rrrsolves, "     zTOMS748Solver.solveN)T)r)r,r-r.r/rrrrrrrrrrrrrr{rrrrrrrs    Q rc  Cs|dkrtd||tdkr0td|tft|}|dkrJtdt|s`td|t|svtd|||krtd |||dkstd |t|ts|f}t } | j ||||||||| d } | \} } }}t || | ||fS) a 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 rrwrrxrr=za is not finite %szb is not finite %sz$a and b are not an interval [{}, {}]zk too small (%s < 1))rUrrrArWrZ) rDr{rErFrGrrtryrzrrr9)r~r"rrUrrrArWr3rZZsolverrir5rrrrrrr s.j          r ) Nrr:r;NNr<FT)NTT),rL collectionsrrErZnumpyrGrrrorprcr{__all__rZ _ESIGNERRrOZ _EVALUEERRrZ CONVERGEDZSIGNERRZCONVERRZVALUEERRZ INPROGRESSrr r8r9rrIrr r r rrrrrrrrr rrrrs~   %   f U a  o   # #P