o Eb^:@sdZddlZddlmZdgZgdZGdddZ     dd dZ d d Z d dZ ddZ ddZ ddZddZddZddZdS)z Unified interfaces to root finding algorithms for real or complex scalar functions. Functions --------- - root : find a root of a scalar function. N) _zeros_py root_scalar)bisectbrentqbrenthriddertoms748newtonsecanthalleyc@s8eZdZdZddZddZddZdd Zd d Zd S) 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.cCs||_d|_d|_d|_dS)Nr)funvalsxn_calls)selfrr=/usr/lib/python3/dist-packages/scipy/optimize/_root_scalar.py__init__s zMemoizeDer.__init__cGsR|jdus ||jkr$|j|g|R}||_|jd7_|dd|_|jdS)z,Calculate f or use cached value if availableNrr)rrrr)rrargsZfgrrr__call__#s  zMemoizeDer.__call__cG.|jdus ||jkr||g|R|jdS)z/Calculate f' or use a cached value if availableNrrrrrrrrrfprime- zMemoizeDer.fprimecGr)z0Calculate f'' or use a cached value if availableNrrrrrfprime23rzMemoizeDer.fprime2cCs|jS)N)r)rrrrncalls9szMemoizeDer.ncallsN) __name__ __module__ __qualname____doc__rrrrrrrrrr s   r rc  Cst|ts|f}| duri} d} |dur+t|s+t|r)t|}d} |j}|j}nd}|durCt|sCt|rAt|}d} |j}nd}i} dD]}t|}|durW|| |<qG| r_| | | j ddd|s}|rmd}n|dur}|r{|rxd}nd}nd }|st d | }ddd }z t t |||}Wnty}zt d ||d}~ww|d vrt|tttjfst d||dd\}}||||fd|i| \}}n|dvr |durt d||durt d|d| vr| d| d<|||f|dd|d| \}}ny|dvr?|durt d||s#t d|d| vr/| d| d<|||f||dd| \}}nD|dvr}|durOt d||sXt d||sat d|d| vrm| d| d<|||f|||d| \}}nt d || r|j}||_|S)a Find a root of a scalar function. Parameters ---------- f : callable A function to find a root of. 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. 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) NFT)xtolrtolmaxiter)Z full_outputZdisprr r r zIUnable to select a solver as neither bracket nor starting point provided.)r r zUnknown solver %s)rrrrr zBracket needed for %srr)r zx0 must not be None for %szx1 must not be None for %sr$Ztol)rrrx1)r zfprime must be specified for %s)rrr)r z fprime2 must be specified for %s) isinstancetuplecallableboolr rrlocalsgetupdate ValueErrorlowergetattroptzerosAttributeErrorlistnpZndarraypoprZfunction_calls)frmethodZbracketrrZx0r'r$r%r&optionsZ is_memoizedkwargskvZmethZmap2underlyingZmethodceabrZsolrrrrr=s x                    cCdS)a Options ------- args : tuple, optional Extra arguments passed to the objective function. xtol : float, optional Tolerance (absolute) for termination. rtol : float, optional Tolerance (relative) for termination. maxiter : int, optional Maximum number of iterations. options: dict, optional Specifies any method-specific options not covered above Nrrrrr_root_scalar_brentq_doc"rBcCrAa Options ------- args : tuple, optional Extra arguments passed to the objective function. xtol : float, optional Tolerance (absolute) for termination. rtol : float, optional Tolerance (relative) for termination. maxiter : int, optional Maximum number of iterations. options: dict, optional Specifies any method-specific options not covered above. Nrrrrr_root_scalar_brenth_doc5rCrEcCrArDrrrrr_root_scalar_toms748_docGrCrFcCrA)a Options ------- args : tuple, optional Extra arguments passed to the objective function. xtol : float, optional Tolerance (absolute) for termination. rtol : float, optional Tolerance (relative) for termination. maxiter : int, optional Maximum number of iterations. x0 : float, required Initial guess. x1 : float, required A second guess. options: dict, optional Specifies any method-specific options not covered above. Nrrrrr_root_scalar_secant_docZsrGcCrA)a" Options ------- args : tuple, optional Extra arguments passed to the objective function and its derivative. xtol : float, optional Tolerance (absolute) for termination. rtol : float, optional Tolerance (relative) for termination. maxiter : int, optional Maximum number of iterations. x0 : float, required Initial guess. fprime : bool or callable, optional If `fprime` is a boolean and is True, `f` is assumed to return the value of derivative along with the objective function. `fprime` can also be a callable returning the derivative of `f`. In this case, it must accept the same arguments as `f`. options: dict, optional Specifies any method-specific options not covered above. Nrrrrr_root_scalar_newton_docqsrHcCrA)ar Options ------- args : tuple, optional Extra arguments passed to the objective function and its derivatives. xtol : float, optional Tolerance (absolute) for termination. rtol : float, optional Tolerance (relative) for termination. maxiter : int, optional Maximum number of iterations. x0 : float, required Initial guess. fprime : bool or callable, required If `fprime` is a boolean and is True, `f` is assumed to return the value of derivative along with the objective function. `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, required If `fprime2` is a boolean and is True, `f` is assumed to return the value of 1st and 2nd derivatives along with the objective function. `fprime2` can also be a callable returning the 2nd derivative of `f`. In this case, it must accept the same arguments as `f`. options: dict, optional Specifies any method-specific options not covered above. Nrrrrr_root_scalar_halley_docsrIcCrArDrrrrr_root_scalar_ridder_docrCrJcCrArDrrrrr_root_scalar_bisect_docrCrK) rNNNNNNNNNN)r#Znumpyr5rr2__all__ZROOT_SCALAR_METHODSr rrBrErFrGrHrIrJrKrrrrs* * f