+ /iORt^RIt^RIHt^RIHtR.t.ROt !RR4t RRlt Rt R t R tR tR tR tRtRtR#)z Unified interfaces to root finding algorithms for real or complex scalar functions. Functions --------- - root : find a root of a scalar function. N) _zeros_pyapprox_derivative root_scalarcBa]tRt^toRtRtRtRtRtRt Rt Vt R#) MemoizeDeraDecorator that caches the value and derivative(s) of function each time it is called. This is a simplistic memoizer that calls and caches a single value of ``f(x, *args)``. It assumes that `args` does not change between invocations. It supports the use case of a root-finder where `args` is fixed, `x` changes, and only rarely, if at all, does x assume the same value more than once.c<WnRVnRVn^VnR#N)funvalsxn_calls)selfr s&&Y/var/www/html/photoedit/myenv/lib/python3.14/site-packages/scipy/optimize/_root_scalar.py__init__MemoizeDer.__init__s  c VPeWP8wdBVP!V.VO5!pWnV;P^, unVR,VnVP^,#)z,Calculate f or use cached value if available:NNN)r r r r )rr argsfgs&&* r__call__MemoizeDer.__call__$sU 99 VV !#d#BF LLA L1DIyy|rc xVPeWP8wd V!V.VO5!VP^,#)z/Calculate f' or use a cached value if availabler r rr rs&&*rfprimeMemoizeDer.fprime.- 99 VV NTNyy|rc xVPeWP8wd V!V.VO5!VP^,#)z0Calculate f'' or use a cached value if availablerrs&&*rfprime2MemoizeDer.fprime24rrcVP#r )r )rs&rncallsMemoizeDer.ncalls:s ||r)r r r r N) __name__ __module__ __qualname____firstlineno____doc__rrrrr"__static_attributes____classdictcell__) __classdict__s@rrrs(   rrc a\V\4'gV3pV f/p Rp VeK\V4'g:\V4'd'\ S4oRp SP pSP pMRpVe?\V4'g.\V4'd\ S4oRp SP pMRp/p RF&p\4PV4pVfK"WV&K( V 'dV PV 4V PRRR7V'g+VeRpM#Ve V'dV'dRpM RpM VeR pMRpV'g \R 4hVP4pRRR R/p\\VPVV44pTR9dd\T\\,\ P",4'g\R T 24hTR ,wppT!STT3RT/T BwppEM"TR9dDTf\RT 24hRT 9dT P1R4T R&T!ST3RTRRRRRT/T BwppMTR9dPTf\RT 24hT'gT3RlpRT 9dT P1R4T R&T!ST3RTRTRR/T BwppMTR9dnTf\RT 24hT'g\RT 24hT'g\RT 24hRT 9dT P1R4T R&T!ST3RTRTRT/T BwppM\R T 24hT 'dSP2pTTnT# \dp\R T 24ThRp?ii;i \ddp\%TR4'dL\P&!TP(\ P*TP,\/T4TR7pRp?LhRp?ii;i)a Find a root of a scalar function. Parameters ---------- f : callable A function to find a root of. Suppose the callable has signature ``f0(x, *my_args, **my_kwargs)``, where ``my_args`` and ``my_kwargs`` are required positional and keyword arguments. Rather than passing ``f0`` as the callable, wrap it to accept only ``x``; e.g., pass ``fun=lambda x: f0(x, *my_args, **my_kwargs)`` as the callable, where ``my_args`` (tuple) and ``my_kwargs`` (dict) have been gathered before invoking this function. args : tuple, optional Extra arguments passed to the objective function and its derivative(s). method : str, optional Type of solver. Should be one of - 'bisect' :ref:`(see here) ` - 'brentq' :ref:`(see here) ` - 'brenth' :ref:`(see here) ` - 'ridder' :ref:`(see here) ` - 'toms748' :ref:`(see here) ` - 'newton' :ref:`(see here) ` - 'secant' :ref:`(see here) ` - 'halley' :ref:`(see here) ` bracket: A sequence of 2 floats, optional An interval bracketing a root. ``f(x, *args)`` must have different signs at the two endpoints. x0 : float, optional Initial guess. x1 : float, optional A second guess. fprime : bool or callable, optional If `fprime` is a boolean and is True, `f` is assumed to return the value of the objective function and of the derivative. `fprime` can also be a callable returning the derivative of `f`. In this case, it must accept the same arguments as `f`. fprime2 : bool or callable, optional If `fprime2` is a boolean and is True, `f` is assumed to return the value of the objective function and of the first and second derivatives. `fprime2` can also be a callable returning the second derivative of `f`. In this case, it must accept the same arguments as `f`. xtol : float, optional Tolerance (absolute) for termination. rtol : float, optional Tolerance (relative) for termination. maxiter : int, optional Maximum number of iterations. options : dict, optional A dictionary of solver options. E.g., ``k``, see :obj:`show_options()` for details. Returns ------- sol : RootResults The solution represented as a ``RootResults`` object. Important attributes are: ``root`` the solution , ``converged`` a boolean flag indicating if the algorithm exited successfully and ``flag`` which describes the cause of the termination. See `RootResults` for a description of other attributes. See also -------- show_options : Additional options accepted by the solvers root : Find a root of a vector function. Notes ----- This section describes the available solvers that can be selected by the 'method' parameter. The default is to use the best method available for the situation presented. If a bracket is provided, it may use one of the bracketing methods. If a derivative and an initial value are specified, it may select one of the derivative-based methods. If no method is judged applicable, it will raise an Exception. Arguments for each method are as follows (x=required, o=optional). +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ | method | f | args | bracket | x0 | x1 | fprime | fprime2 | xtol | rtol | maxiter | options | +===============================================+===+======+=========+====+====+========+=========+======+======+=========+=========+ | :ref:`bisect ` | x | o | x | | | | | o | o | o | o | +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ | :ref:`brentq ` | x | o | x | | | | | o | o | o | o | +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ | :ref:`brenth ` | x | o | x | | | | | o | o | o | o | +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ | :ref:`ridder ` | x | o | x | | | | | o | o | o | o | +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ | :ref:`toms748 ` | x | o | x | | | | | o | o | o | o | +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ | :ref:`secant ` | x | o | | x | o | | | o | o | o | o | +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ | :ref:`newton ` | x | o | | x | | o | | o | o | o | o | +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ | :ref:`halley ` | x | o | | x | | x | x | o | o | o | o | +-----------------------------------------------+---+------+---------+----+----+--------+---------+------+------+---------+---------+ Examples -------- Find the root of a simple cubic >>> from scipy import optimize >>> def f(x): ... return (x**3 - 1) # only one real root at x = 1 >>> def fprime(x): ... return 3*x**2 The `brentq` method takes as input a bracket >>> sol = optimize.root_scalar(f, bracket=[0, 3], method='brentq') >>> sol.root, sol.iterations, sol.function_calls (1.0, 10, 11) The `newton` method takes as input a single point and uses the derivative(s). >>> sol = optimize.root_scalar(f, x0=0.2, fprime=fprime, method='newton') >>> sol.root, sol.iterations, sol.function_calls (1.0, 11, 22) The function can provide the value and derivative(s) in a single call. >>> def f_p_pp(x): ... return (x**3 - 1), 3*x**2, 6*x >>> sol = optimize.root_scalar( ... f_p_pp, x0=0.2, fprime=True, method='newton' ... ) >>> sol.root, sol.iterations, sol.function_calls (1.0, 11, 11) >>> sol = optimize.root_scalar( ... f_p_pp, x0=0.2, fprime=True, fprime2=True, method='halley' ... ) >>> sol.root, sol.iterations, sol.function_calls (1.0, 7, 8) NFTxtol) full_outputdispbrentqhalleynewtonsecantzIUnable to select a solver as neither bracket nor starting point provided.zUnknown solver zBracket needed for :NNr_x)root iterationsfunction_callsflagmethodzx0 must not be None for tolrrx1c:<V3Rlp\W RVR7^,#)c&<S!V^,.VO5!#))r rfs&*r f_wrapped.root_scalar..fprime..f_wrappedAsQqT>D>)rz2-point)r:rr)r rrBrAs&* rrroot_scalar..fprime9s*(idSTUVVrzfprime must be specified for zfprime2 must be specified for )r-rtolmaxiter)bisectridderr0brenthtoms748)r3)r2)r1) isinstancetuplecallableboolrrrlocalsgetupdate ValueErrorlowergetattroptzerosAttributeErrorlistnpndarrayhasattr RootResultsr5nan_function_callsstrpopr r8)rArr:bracketrrx0r<r-rErFoptions is_memoizedkwargskvmethmap2underlyingmethodceabrsolr sf&&&&&&&&&&& rrr>sr dE " "wK8G#4#4 ==1 AKiiGXXFG (6"2"2 <<1 AKXXFFF ( HLLO =1I)  g MMdM/   F ^%F%F!! 89 9 <<>D(H=N:(N$6$6tT$BC BB'4%<"**#<==26(;< <r{1 Q1:4:6:FAs   :7x@A A V "JJv.F5MB*T*$***"(*3   :7x@A A W V "JJv.F5MB#T#&#$#!#3   :7x@A Ars^#' /5''T^B  , * , 0 4 > , r