a a@szgdZddlZddlZddlZddlmZmZmZmZm Z m Z m Z m Z m Z ddlmZddlmZddlmZddlmZddlmZdd lmZdd lmZdd lmZdd lmZdd lmZddlm Z m!Z!ddZ"GdddZ#ddZ$ddZ%GdddeZ&GdddZ'Gddde'Z(Gddde'Z)Gdd d Z*Gd!d"d"Z+d#d$ej,fd%d&Z-Gd'd(d(e(Z.dS)))interp1dinterp2dlagrangePPolyBPolyNdPPolyRegularGridInterpolatorinterpnN) array transpose searchsorted atleast_1d atleast_2dravelpoly1dasarrayintp)comb)prod)fitpack)dfitpack)_fitpack)_Interpolator1D)_ppoly)RectBivariateSpline)_ndim_coords_from_arrays)make_interp_splineBSplinecCsxt|}td}t|D]Z}t||}t|D]8}||kr>q0||||}|td|| g|9}q0||7}q|S)a Return a Lagrange interpolating polynomial. Given two 1-D arrays `x` and `w,` returns the Lagrange interpolating polynomial through the points ``(x, w)``. Warning: This implementation is numerically unstable. Do not expect to be able to use more than about 20 points even if they are chosen optimally. Parameters ---------- x : array_like `x` represents the x-coordinates of a set of datapoints. w : array_like `w` represents the y-coordinates of a set of datapoints, i.e., f(`x`). Returns ------- lagrange : `numpy.poly1d` instance The Lagrange interpolating polynomial. Examples -------- Interpolate :math:`f(x) = x^3` by 3 points. >>> from scipy.interpolate import lagrange >>> x = np.array([0, 1, 2]) >>> y = x**3 >>> poly = lagrange(x, y) Since there are only 3 points, Lagrange polynomial has degree 2. Explicitly, it is given by .. math:: \begin{aligned} L(x) &= 1\times \frac{x (x - 2)}{-1} + 8\times \frac{x (x-1)}{2} \\ &= x (-2 + 3x) \end{aligned} >>> from numpy.polynomial.polynomial import Polynomial >>> Polynomial(poly).coef array([ 3., -2., 0.]) ?)lenrrange)xwMpjptkZfacr*m/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/scipy/interpolate/interpolate.pyrs/    rc@s$eZdZdZd ddZd d d ZdS) ra< interp2d(x, y, z, kind='linear', copy=True, bounds_error=False, fill_value=None) Interpolate over a 2-D grid. `x`, `y` and `z` are arrays of values used to approximate some function f: ``z = f(x, y)`` which returns a scalar value `z`. This class returns a function whose call method uses spline interpolation to find the value of new points. If `x` and `y` represent a regular grid, consider using `RectBivariateSpline`. If `z` is a vector value, consider using `interpn`. Note that calling `interp2d` with NaNs present in input values results in undefined behaviour. Methods ------- __call__ Parameters ---------- x, y : array_like Arrays defining the data point coordinates. If the points lie on a regular grid, `x` can specify the column coordinates and `y` the row coordinates, for example:: >>> x = [0,1,2]; y = [0,3]; z = [[1,2,3], [4,5,6]] Otherwise, `x` and `y` must specify the full coordinates for each point, for example:: >>> x = [0,1,2,0,1,2]; y = [0,0,0,3,3,3]; z = [1,2,3,4,5,6] If `x` and `y` are multidimensional, they are flattened before use. z : array_like The values of the function to interpolate at the data points. If `z` is a multidimensional array, it is flattened before use. The length of a flattened `z` array is either len(`x`)*len(`y`) if `x` and `y` specify the column and row coordinates or ``len(z) == len(x) == len(y)`` if `x` and `y` specify coordinates for each point. kind : {'linear', 'cubic', 'quintic'}, optional The kind of spline interpolation to use. Default is 'linear'. copy : bool, optional If True, the class makes internal copies of x, y and z. If False, references may be used. The default is to copy. bounds_error : bool, optional If True, when interpolated values are requested outside of the domain of the input data (x,y), a ValueError is raised. If False, then `fill_value` is used. fill_value : number, optional If provided, the value to use for points outside of the interpolation domain. If omitted (None), values outside the domain are extrapolated via nearest-neighbor extrapolation. See Also -------- RectBivariateSpline : Much faster 2-D interpolation if your input data is on a grid bisplrep, bisplev : Spline interpolation based on FITPACK BivariateSpline : a more recent wrapper of the FITPACK routines interp1d : 1-D version of this function Notes ----- The minimum number of data points required along the interpolation axis is ``(k+1)**2``, with k=1 for linear, k=3 for cubic and k=5 for quintic interpolation. The interpolator is constructed by `bisplrep`, with a smoothing factor of 0. If more control over smoothing is needed, `bisplrep` should be used directly. Examples -------- Construct a 2-D grid and interpolate on it: >>> from scipy import interpolate >>> x = np.arange(-5.01, 5.01, 0.25) >>> y = np.arange(-5.01, 5.01, 0.25) >>> xx, yy = np.meshgrid(x, y) >>> z = np.sin(xx**2+yy**2) >>> f = interpolate.interp2d(x, y, z, kind='cubic') Now use the obtained interpolation function and plot the result: >>> import matplotlib.pyplot as plt >>> xnew = np.arange(-5.01, 5.01, 1e-2) >>> ynew = np.arange(-5.01, 5.01, 1e-2) >>> znew = f(xnew, ynew) >>> plt.plot(x, z[0, :], 'ro-', xnew, znew[0, :], 'b-') >>> plt.show() linearTFNc s~t|}t|}t|}|jt|t|k}|r|jdkrZ|jt|t|fkrZtdt|dd|ddkst |} || }|dd| f}t|dd|ddkst |} || }|| ddf}t|j }nz%interp2d.__init__..)rrsizer!ndimshape ValueErrornpallargsortTKeyErrorreprjoinmaprZbisplreptckrZ regrid_smth bounds_error fill_valuer#yzZaminZamaxx_minx_maxy_miny_max)selfr#rKrLkindr7rIrJZrectangular_gridr'Zinterpolation_typesr3r4eZnxZtxnytycfpZierr*r6r+__init__sf      2$zinterp2d.__init__r c CsDt|}t|}|jdks$|jdkr,td|sLtj|dd}tj|dd}|js\|jdur||jk||jkB}||j k||j kB}t |}t |} |jr|s| rtd|j|jf|j |j fft |||j||} t| } t| } |jdur&|r|j| dd|f<| r&|j| |ddf<t| dkr<| d} t| S)aInterpolate the function. Parameters ---------- x : 1-D array x-coordinates of the mesh on which to interpolate. y : 1-D array y-coordinates of the mesh on which to interpolate. dx : int >= 0, < kx Order of partial derivatives in x. dy : int >= 0, < ky Order of partial derivatives in y. assume_sorted : bool, optional If False, values of `x` and `y` can be in any order and they are sorted first. If True, `x` and `y` have to be arrays of monotonically increasing values. Returns ------- z : 2-D array with shape (len(y), len(x)) The interpolated values. rz!x and y should both be 1-D arrays mergesortrRNz-Values out of range; x must be in %r, y in %rr )r r=r?r@sortrIrJrMrNrOrPanyrZbisplevrHrr r!r ) rQr#rKdxZdy assume_sortedZout_of_bounds_xZout_of_bounds_yZany_out_of_bounds_xZany_out_of_bounds_yrLr*r*r+__call__s:     zinterp2d.__call__)r,TFN)r r F)__name__ __module__ __qualname____doc__rXr_r*r*r*r+rYs d 9rcCs|j}t|t|krt|ddd|dddD]\}}|dkr4||kr4qq4|jdkrx|j|krxt||j|}|Std|||fdS)z7Helper to check that arr_from broadcasts up to shape_toNr.rzE%s argument must be able to broadcast up to shape %s but had shape %s) r>r!zipr<r@Zonesdtyperr?)Zarr_fromZshape_tonameZ shape_fromtfr*r*r+_check_broadcast_up_to5s&ricCst|to|dkS)z?Helper to check if fill_value == "extrapolate" without warnings extrapolate) isinstancestr)rJr*r*r+_do_extrapolateFs rmc@seZdZdZddddejdfddZed d Zej d d Zd d Z ddZ ddZ ddZ ddZddZddZddZdS)ra Interpolate a 1-D function. `x` and `y` are arrays of values used to approximate some function f: ``y = f(x)``. This class returns a function whose call method uses interpolation to find the value of new points. Parameters ---------- x : (N,) array_like A 1-D array of real values. y : (...,N,...) array_like A N-D array of real values. The length of `y` along the interpolation axis must be equal to the length of `x`. kind : str or int, optional Specifies the kind of interpolation as a string or as an integer specifying the order of the spline interpolator to use. The string has to be one of 'linear', 'nearest', 'nearest-up', 'zero', 'slinear', 'quadratic', 'cubic', 'previous', or 'next'. 'zero', 'slinear', 'quadratic' and 'cubic' refer to a spline interpolation of zeroth, first, second or third order; 'previous' and 'next' simply return the previous or next value of the point; 'nearest-up' and 'nearest' differ when interpolating half-integers (e.g. 0.5, 1.5) in that 'nearest-up' rounds up and 'nearest' rounds down. Default is 'linear'. axis : int, optional Specifies the axis of `y` along which to interpolate. Interpolation defaults to the last axis of `y`. copy : bool, optional If True, the class makes internal copies of x and y. If False, references to `x` and `y` are used. The default is to copy. bounds_error : bool, optional If True, a ValueError is raised any time interpolation is attempted on a value outside of the range of x (where extrapolation is necessary). If False, out of bounds values are assigned `fill_value`. By default, an error is raised unless ``fill_value="extrapolate"``. fill_value : array-like or (array-like, array_like) or "extrapolate", optional - if a ndarray (or float), this value will be used to fill in for requested points outside of the data range. If not provided, then the default is NaN. The array-like must broadcast properly to the dimensions of the non-interpolation axes. - If a two-element tuple, then the first element is used as a fill value for ``x_new < x[0]`` and the second element is used for ``x_new > x[-1]``. Anything that is not a 2-element tuple (e.g., list or ndarray, regardless of shape) is taken to be a single array-like argument meant to be used for both bounds as ``below, above = fill_value, fill_value``. .. versionadded:: 0.17.0 - If "extrapolate", then points outside the data range will be extrapolated. .. versionadded:: 0.17.0 assume_sorted : bool, optional If False, values of `x` can be in any order and they are sorted first. If True, `x` has to be an array of monotonically increasing values. Attributes ---------- fill_value Methods ------- __call__ See Also -------- splrep, splev Spline interpolation/smoothing based on FITPACK. UnivariateSpline : An object-oriented wrapper of the FITPACK routines. interp2d : 2-D interpolation Notes ----- Calling `interp1d` with NaNs present in input values results in undefined behaviour. Input values `x` and `y` must be convertible to `float` values like `int` or `float`. If the values in `x` are not unique, the resulting behavior is undefined and specific to the choice of `kind`, i.e., changing `kind` will change the behavior for duplicates. Examples -------- >>> import matplotlib.pyplot as plt >>> from scipy import interpolate >>> x = np.arange(0, 10) >>> y = np.exp(-x/3.0) >>> f = interpolate.interp1d(x, y) >>> xnew = np.arange(0, 9, 0.1) >>> ynew = f(xnew) # use interpolation function returned by `interp1d` >>> plt.plot(x, y, 'o', xnew, ynew, '-') >>> plt.show() r,r.TNFc Csxtj||||d||_||_|dvr>ddddd|} d}n(t|trR|} d}n|dvrftd |t||jd }t||jd }|stj |d d } || }tj || |d}|j dkrt d |j dkrt dt |jjtjs|tj}||j |_||_||j|_||_~~||_||_|dvrd} |dkrxd|_|jd|_|jdd|jdd|_|jj|_qX|dkrd|_|jd|_|jdd|jdd|_|jj|_n|dkrd|_d|_t |jtj! |_"|jj#|_n|dkr(d|_d|_t |jtj!|_"|jj#|_n\|jjtjkoD|jjtjk} | oV|jj dk} | oft$| } | rz|jj%|_n |jj&|_n| d} d} |j|j}}| dkr*t'|j}|(r|j|}|j)dkrt dt*t+|jt,|jt-|j}d} t'|j(r*t.|j}d} t/||| dd|_0| rN|jj1|_n |jj2|_t-|j| krtt d| dS)z- Initialize a 1-D linear interpolation class.axis)ZzeroZslinearZ quadraticr1r rr-r/Zspline)r,nearest nearest-uppreviousnextz8%s is unsupported: Use fitpack routines for other types.r6rYrZz,the x array must have exactly one dimension.z-the y array must have at least one dimension.rpleftg@Nr.rqrightrrrsFz`x` array is all-nanT)r)Z check_finitez,x and y arrays must have at least %d entries)3rrXrIr7rkintNotImplementedErrorr r@rBZtaker=r? issubclassretypeinexactastypefloat_rorKZ _reshape_yi_yr#_kindrJ_sidex_bds __class__ _call_nearest_call_ind nextafterinf_x_shift_call_previousnextrm_call_linear_np _call_linearisnanr\r<ZlinspaceZnanminZnanmaxr!Z ones_liker_spline_call_nan_spline _call_spline)rQr#rKrRror7rIrJr^orderindminvalZcondZ rewrite_nanxxyymaskZsxr*r*r+rXs                          zinterp1d.__init__cCs|jS)zThe fill value.)_fill_value_origrQr*r*r+rJ4szinterp1d.fill_valuecCst|r$|jrtdd|_d|_n|jjd|j|jj|jdd}t|dkr\d}t|t rt|dkrt |dt |dg}d}t dD]}t |||||||<qnt |}t ||d gd}|\|_|_d|_|jdurd|_||_dS) Nz.Cannot extrapolate and raise at the same time.FTrr rr-)zfill_value (below)zfill_value (above)rJ)rmrIr? _extrapolaterKr>ror!rktupler@rr"ri_fill_value_below_fill_value_abover)rQrJZbroadcast_shapeZ below_abovenamesiir*r*r+rJ:s<        cCst||j|jSN)r@interpr#rKrQx_newr*r*r+r\szinterp1d._call_linear_npc Cst|j|}|dt|jdt}|d}|}|j|}|j|}|j|}|j|}||||dddf} | ||dddf|} | S)Nr)r r#clipr!r{rvr}) rQr x_new_indiceslohiZx_loZx_hiZy_loZy_hiZslopey_newr*r*r+r`s     zinterp1d._call_linearcCs<t|j||jd}|dt|jdt}|j|}|S)z5 Find nearest neighbor interpolated y_new = f(x_new).Zsider r) r rrrr!r#r{rr}rQrrrr*r*r+r}s zinterp1d._call_nearestcCsNt|j||jd}|d|jt|j|jt}|j ||jd}|S)z6Use previous/next neighbor of x_new, y_new = f(x_new).rr) r rrrrr!r#r{rr}rr*r*r+rs zinterp1d._call_previousnextcCs ||Sr)rrr*r*r+rszinterp1d._call_splinecCs||}tj|d<|S)N.)rr@nan)rQroutr*r*r+rs  zinterp1d._call_nan_splinecCsLt|}|||}|jsH||\}}t|dkrH|j||<|j||<|SNr )rrr _check_boundsr!rr)rQrr below_bounds above_boundsr*r*r+ _evaluates    zinterp1d._evaluatecCsP||jdk}||jdk}|jr2|r2td|jrH|rHtd||fS)aCheck the inputs for being in the bounds of the interpolated data. Parameters ---------- x_new : array Returns ------- out_of_bounds : bool array The mask on x_new of values that are out of the bounds. r r.z2A value in x_new is below the interpolation range.z2A value in x_new is above the interpolation range.)r#rIr\r?)rQrrrr*r*r+rszinterp1d._check_bounds)r`rarbrcr@rrXpropertyrJsetterrrrrrrrrr*r*r*r+rLs$c   !rc@sPeZdZdZdZdddZddZedd d Zd d Z dd dZ dddZ dS) _PPolyBasez%Base class for piecewise polynomials.)rVr#rjroNr cCst||_tj|tjd|_|dur,d}n|dkrdiffrA _get_dtypere)rQrVr#rjror]rer*r*r+rXs@       z_PPolyBase.__init__cCs0t|tjs t|jjtjr&tjStjSdSrr@ issubdtypecomplexfloatingrVrecomplex_r|rQrer*r*r+rs z_PPolyBase._get_dtypecCs2t|}||_||_||_|dur(d}||_|S)aH Construct the piecewise polynomial without making checks. Takes the same parameters as the constructor. Input arguments ``c`` and ``x`` must be arrays of the correct shape and type. The ``c`` array can only be of dtypes float and complex, and ``x`` array must have dtype float. NT)object__new__rVr#rorj)clsrVr#rjrorQr*r*r+construct_fast s z_PPolyBase.construct_fastcCs0|jjjs|j|_|jjjs,|j|_dS)zr c and x may be modified by the user. The Cython code expects that they are C contiguous. N)r#flags c_contiguousr7rVrr*r*r+_ensure_c_contiguous s   z_PPolyBase._ensure_c_contiguousc Cs|durtdt|}t|}|jdkr8td|jdkrJtd|jd|jdkrrtd|j|j|jdd|jjddks|j|jjkrtd |j|jj|j dkrdSt |}t |dkst |dkstd |j d |j dkr^|d |dks td |d|j d kr:d }n"|d |j dkrTd}ntdnV|d |dksxtd |d|j d krd }n"|d |j dkrd}ntd| |j}t|jd|jjd}tj||jjd|jdf|jjdd|d}|d krz|j|||jjddd|jjdf<||||jdd|jjddf<tj|j |f|_ nh|dkr||||jjddd|jdf<|j|||jdd|jddf<tj||j f|_ ||_dS)a Add additional breakpoints and coefficients to the polynomial. Parameters ---------- c : ndarray, size (k, m, ...) Additional coefficients for polynomials in intervals. Note that the first additional interval will be formed using one of the ``self.x`` end points. x : ndarray, size (m,) Additional breakpoints. Must be sorted in the same order as ``self.x`` and either to the right or to the left of the current breakpoints. right Deprecated argument. Has no effect. .. deprecated:: 0.19 Nz*`right` is deprecated and will be removed.r-zinvalid dimensions for crzinvalid dimensions for xr z(Shapes of x {} and c {} are incompatiblez-Shapes of c {} and self.c {} are incompatiblez`x` is not sorted.r.z,`x` is in the different order than `self.x`.appendprependz9`x` is neither on the left or on the right from `self.x`.r)warningswarnr@rr=r?r>formatrVr<rrAr#rremaxzerosZr_) rQrVr#rur]actionreZk2c2r*r*r+extend*sd     ,     , *& &&z_PPolyBase.extendcCs&|dur|j}t|}|j|j}}tj|tjd}|dkrr|jd||jd|jd|jd}d}tj t |t |j jddf|j j d}|||||||||j jdd}|jdkr"tt|j}||||j|d||||jd}||}|S)aX Evaluate the piecewise polynomial or its derivative. Parameters ---------- x : array_like Points to evaluate the interpolant at. nu : int, optional Order of derivative to evaluate. Must be non-negative. extrapolate : {bool, 'periodic', None}, optional If bool, determines whether to extrapolate to out-of-bounds points based on first and last intervals, or to return NaNs. If 'periodic', periodic extrapolation is used. If None (default), use `self.extrapolate`. Returns ------- y : array_like Interpolated values. Shape is determined by replacing the interpolation axis in the original array with the shape of x. Notes ----- Derivatives are evaluated piecewise for each polynomial segment, even if the polynomial is not differentiable at the breakpoints. The polynomial intervals are considered half-open, ``[a, b)``, except for the last interval which is closed ``[a, b]``. Nrrr r.Fr-)rjr@rr>r=rrr|r#emptyr!rrVrerrreshaperolistr"r )rQr#nurjx_shapeZx_ndimrlr*r*r+r_s" ,* 0 z_PPolyBase.__call__)Nr )Nr )N)r N) r`rarbrc __slots__rXr classmethodrrrr_r*r*r*r+rs .  Urc@sfeZdZdZddZdddZdddZdd d ZdddZdddZ e dddZ e dddZ d S)raL Piecewise polynomial in terms of coefficients and breakpoints The polynomial between ``x[i]`` and ``x[i + 1]`` is written in the local power basis:: S = sum(c[m, i] * (xp - x[i])**(k-m) for m in range(k+1)) where ``k`` is the degree of the polynomial. Parameters ---------- c : ndarray, shape (k, m, ...) Polynomial coefficients, order `k` and `m` intervals. x : ndarray, shape (m+1,) Polynomial breakpoints. Must be sorted in either increasing or decreasing order. extrapolate : bool or 'periodic', optional If bool, determines whether to extrapolate to out-of-bounds points based on first and last intervals, or to return NaNs. If 'periodic', periodic extrapolation is used. Default is True. axis : int, optional Interpolation axis. Default is zero. Attributes ---------- x : ndarray Breakpoints. c : ndarray Coefficients of the polynomials. They are reshaped to a 3-D array with the last dimension representing the trailing dimensions of the original coefficient array. axis : int Interpolation axis. Methods ------- __call__ derivative antiderivative integrate solve roots extend from_spline from_bernstein_basis construct_fast See also -------- BPoly : piecewise polynomials in the Bernstein basis Notes ----- High-order polynomials in the power basis can be numerically unstable. Precision problems can start to appear for orders larger than 20-30. cCs:t|j|jjd|jjdd|j||t||dSNr rr.)revaluaterVrr>r#rrQr#rrjrr*r*r+rs"zPPoly._evaluatercCs|dkr|| S|dkr(|j}n|jd| ddf}|jddkrptjd|jdd|jd}tt |jddd|}||t dfd|j d9}| ||j |j|jS)a Construct a new piecewise polynomial representing the derivative. Parameters ---------- nu : int, optional Order of derivative to evaluate. Default is 1, i.e., compute the first derivative. If negative, the antiderivative is returned. Returns ------- pp : PPoly Piecewise polynomial of order k2 = k - n representing the derivative of this polynomial. Notes ----- Derivatives are evaluated piecewise for each polynomial segment, even if the polynomial is not differentiable at the breakpoints. The polynomial intervals are considered half-open, ``[a, b)``, except for the last interval which is closed ``[a, b]``. r Nrrrr.r)antiderivativerVr7r>r@rrespecpocharangeslicer=rr#rjro)rQrrfactorr*r*r+ derivatives   zPPoly.derivativecCs|dkr|| Stj|jjd||jjdf|jjdd|jjd}|j|d| <tt|jjddd|}|d| |t dfd|j d<| t ||jd|jdd|j|d|jdkrd }n|j}|||j||jS) a= Construct a new piecewise polynomial representing the antiderivative. Antiderivative is also the indefinite integral of the function, and derivative is its inverse operation. Parameters ---------- nu : int, optional Order of antiderivative to evaluate. Default is 1, i.e., compute the first integral. If negative, the derivative is returned. Returns ------- pp : PPoly Piecewise polynomial of order k2 = k + n representing the antiderivative of this polynomial. Notes ----- The antiderivative returned by this function is continuous and continuously differentiable to order n-1, up to floating point rounding error. If antiderivative is computed and ``self.extrapolate='periodic'``, it will be set to False for the returned instance. This is done because the antiderivative is no longer periodic and its correct evaluation outside of the initially given x interval is difficult. r rr-Nrr.rrF)rr@rrVr>rerrrrr=rrfix_continuityrr#rjrro)rQrrVrrjr*r*r+r s  ..  zPPoly.antiderivativeNc Cs(|dur|j}d}||kr(||}}d}tjt|jjddf|jjd}||dkr|jd|jd}}||}||} t | |\} } | dkrt j |j |jjd|jjdd|j||d|d || 9}n | d||||}|| }t|} ||krLt j |j |jjd|jjdd|j||d| d || 7}nt j |j |jjd|jjdd|j||d| d || 7}t j |j |jjd|jjdd|j||| ||d| d || 7}n8t j |j |jjd|jjdd|j||t||d ||9}| |jjddS) a Compute a definite integral over a piecewise polynomial. Parameters ---------- a : float Lower integration bound b : float Upper integration bound extrapolate : {bool, 'periodic', None}, optional If bool, determines whether to extrapolate to out-of-bounds points based on first and last intervals, or to return NaNs. If 'periodic', periodic extrapolation is used. If None (default), use `self.extrapolate`. Returns ------- ig : array_like Definite integral of the piecewise polynomial over [a, b] Nrr.r-rrr F)r)rjr@rrrVr>rerr#divmodr integraterfill empty_liker) rQr9brjsignZ range_intxsxeperiodinterval n_periodsrtZ remainder_intr*r*r+rVsZ $          zPPoly.integraterTcCs|dur|j}|t|jjtjr0tdt|}t |j |jj d|jj dd|j |t|t|}|jjdkr|dStjt|jj ddtd}t|D]\}}|||<q| |jj ddSdS)a Find real solutions of the the equation ``pp(x) == y``. Parameters ---------- y : float, optional Right-hand side. Default is zero. discontinuity : bool, optional Whether to report sign changes across discontinuities at breakpoints as roots. extrapolate : {bool, 'periodic', None}, optional If bool, determines whether to return roots from the polynomial extrapolated based on first and last intervals, 'periodic' works the same as False. If None (default), use `self.extrapolate`. Returns ------- roots : ndarray Roots of the polynomial(s). If the PPoly object describes multiple polynomials, the return value is an object array whose each element is an ndarray containing the roots. Notes ----- This routine works only on real-valued polynomials. If the piecewise polynomial contains sections that are identically zero, the root list will contain the start point of the corresponding interval, followed by a ``nan`` value. If the polynomial is discontinuous across a breakpoint, and there is a sign change across the breakpoint, this is reported if the `discont` parameter is True. Examples -------- Finding roots of ``[x**2 - 1, (x - 1)**2]`` defined on intervals ``[-2, 1], [1, 2]``: >>> from scipy.interpolate import PPoly >>> pp = PPoly(np.array([[1, -4, 3], [1, 0, 0]]).T, [-2, 1, 2]) >>> pp.solve() array([-1., 1.]) Nz0Root finding is only for real-valued polynomialsr rr.r-r)rjrr@rrVrerr?floatrZ real_rootsrr>r#rr=rrr enumerate)rQrK discontinuityrjrZr2rrootr*r*r+solves 0"   z PPoly.solvecCs|d||S)a_ Find real roots of the the piecewise polynomial. Parameters ---------- discontinuity : bool, optional Whether to report sign changes across discontinuities at breakpoints as roots. extrapolate : {bool, 'periodic', None}, optional If bool, determines whether to return roots from the polynomial extrapolated based on first and last intervals, 'periodic' works the same as False. If None (default), use `self.extrapolate`. Returns ------- roots : ndarray Roots of the polynomial(s). If the PPoly object describes multiple polynomials, the return value is an object array whose each element is an ndarray containing the roots. See Also -------- PPoly.solve r )r)rQrrjr*r*r+rootssz PPoly.rootsc Cst|tr&|j\}}}|dur0|j}n |\}}}tj|dt|df|jd}t|ddD]>}t j |dd||d}|t |d|||ddf<q\| |||S)a Construct a piecewise polynomial from a spline Parameters ---------- tck A spline, as returned by `splrep` or a BSpline object. extrapolate : bool or 'periodic', optional If bool, determines whether to extrapolate to out-of-bounds points based on first and last intervals, or to return NaNs. If 'periodic', periodic extrapolation is used. Default is True. Nrrr.)Zder)rkrrHrjr@rr!rer"rZsplevrgammar) rrHrjrgrVr)cvalsmrKr*r*r+ from_spline s    $zPPoly.from_splinec Cst|tstdt|t|j}|jjdd}d|jj d}t |j}t |dD]|}d|t |||j|}t ||dD]L} t ||| |d| } ||| || |t df|| 7<qq^|dur|j}|||j||jS)a Construct a piecewise polynomial in the power basis from a polynomial in Bernstein basis. Parameters ---------- bp : BPoly A Bernstein basis polynomial, as created by BPoly extrapolate : bool or 'periodic', optional If bool, determines whether to extrapolate to out-of-bounds points based on first and last intervals, or to return NaNs. If 'periodic', periodic extrapolation is used. Default is True. zC.from_bernstein_basis only accepts BPoly instances. Got %s instead.r rrr-r.N)rkr TypeErrorryr@rr#rVr>r= zeros_liker"rrrjrro) rbprjr]r)restrVr9rr5valr*r*r+from_bernstein_basis)s    2zPPoly.from_bernstein_basis)r)r)N)rTN)TN)N)N) r`rarbrcrrrrrrrrrr*r*r*r+rs: , 6 R H  rc@s~eZdZdZddZdddZdddZdd d Zdd d Ze jje_e dddZ e dddZ e ddZe ddZd S)ray Piecewise polynomial in terms of coefficients and breakpoints. The polynomial between ``x[i]`` and ``x[i + 1]`` is written in the Bernstein polynomial basis:: S = sum(c[a, i] * b(a, k; x) for a in range(k+1)), where ``k`` is the degree of the polynomial, and:: b(a, k; x) = binom(k, a) * t**a * (1 - t)**(k - a), with ``t = (x - x[i]) / (x[i+1] - x[i])`` and ``binom`` is the binomial coefficient. Parameters ---------- c : ndarray, shape (k, m, ...) Polynomial coefficients, order `k` and `m` intervals x : ndarray, shape (m+1,) Polynomial breakpoints. Must be sorted in either increasing or decreasing order. extrapolate : bool, optional If bool, determines whether to extrapolate to out-of-bounds points based on first and last intervals, or to return NaNs. If 'periodic', periodic extrapolation is used. Default is True. axis : int, optional Interpolation axis. Default is zero. Attributes ---------- x : ndarray Breakpoints. c : ndarray Coefficients of the polynomials. They are reshaped to a 3-D array with the last dimension representing the trailing dimensions of the original coefficient array. axis : int Interpolation axis. Methods ------- __call__ extend derivative antiderivative integrate construct_fast from_power_basis from_derivatives See also -------- PPoly : piecewise polynomials in the power basis Notes ----- Properties of Bernstein polynomials are well documented in the literature, see for example [1]_ [2]_ [3]_. References ---------- .. [1] https://en.wikipedia.org/wiki/Bernstein_polynomial .. [2] Kenneth I. Joy, Bernstein polynomials, http://www.idav.ucdavis.edu/education/CAGDNotes/Bernstein-Polynomials.pdf .. [3] E. H. Doha, A. H. Bhrawy, and M. A. Saker, Boundary Value Problems, vol 2011, article ID 829546, :doi:`10.1155/2011/829543`. Examples -------- >>> from scipy.interpolate import BPoly >>> x = [0, 1] >>> c = [[1], [2], [3]] >>> bp = BPoly(c, x) This creates a 2nd order polynomial .. math:: B(x) = 1 \times b_{0, 2}(x) + 2 \times b_{1, 2}(x) + 3 \times b_{2, 2}(x) \\ = 1 \times (1-x)^2 + 2 \times 2 x (1 - x) + 3 \times x^2 cCs:t|j|jjd|jjdd|j||t||dSr)rZevaluate_bernsteinrVrr>r#rrr*r*r+rszBPoly._evaluatercCs|dkr|| S|dkr:|}t|D] }|}q(|S|dkrN|j}nTd|jjd}|jjdd}t|j dt df|}|tj|jdd|}|jddkrtj d|jdd|j d}| ||j |j|jS) a Construct a new piecewise polynomial representing the derivative. Parameters ---------- nu : int, optional Order of derivative to evaluate. Default is 1, i.e., compute the first derivative. If negative, the antiderivative is returned. Returns ------- bp : BPoly Piecewise polynomial of order k - nu representing the derivative of this polynomial. r rrr-Nrnrr)rr"rrVr7r=r>r@rr#rrrerrjro)rQrrr)rrr]r*r*r+rs     zBPoly.derivativec Cs4|dkr|| S|dkr:|}t|D] }|}q(|S|j|j}}|jd}tj|df|jdd|jd}tj |dd||dddf<|dd|dd}||dt dfd|j d 9}|ddddftj ||ddfdddd7<|j d krd }n|j }|j ||||jdS) a Construct a new piecewise polynomial representing the antiderivative. Parameters ---------- nu : int, optional Order of antiderivative to evaluate. Default is 1, i.e., compute the first integral. If negative, the derivative is returned. Returns ------- bp : BPoly Piecewise polynomial of order k + nu representing the antiderivative of this polynomial. Notes ----- If antiderivative is computed and ``self.extrapolate='periodic'``, it will be set to False for the returned instance. This is done because the antiderivative is no longer periodic and its correct evaluation outside of the initially given x interval is difficult. r rNrrn.r.rr-rF)rr"rrVr#r>r@rreZcumsumrr=rjrro) rQrrr)rVr#rdeltarjr*r*r+rs$    $": zBPoly.antiderivativeNc Cs|}|dur|j}|dkr$||_|dkr||kr _raise_degreerr)rQrVr#rur)r*r*r+rVsz BPoly.extendc Cst|tstdt|t|j}|jjdd}d|jj d}t |j}t |dD]l}|j|t ||||t df|||}t |||dD]"} || |t | ||7<qq^|dur|j}|||j||jS)a Construct a piecewise polynomial in Bernstein basis from a power basis polynomial. Parameters ---------- pp : PPoly A piecewise polynomial in the power basis extrapolate : bool or 'periodic', optional If bool, determines whether to extrapolate to out-of-bounds points based on first and last intervals, or to return NaNs. If 'periodic', periodic extrapolation is used. Default is True. z?.from_power_basis only accepts PPoly instances. Got %s instead.r rrr-N)rkrrryr@rr#rVr>r=rr"rrrjrro) rpprjr]r)rrVr9rr'r*r*r+from_power_basis]s   2"zBPoly.from_power_basisc sft|}t|tkr"tdt|dd|dddkrLtdt|d}ztfddt|D}Wn.ty}ztd|WYd}~n d}~00|durdg|}nBt|t tj fr|g|}t|t|}td d|Drtd g}t|D]<} | | d} } || durFt| t| } } n|| d}t |d t| } t || t| } t || t| } | | |krd || t| || dt| || f}t|| t| kr| t| kstd t || || d| d| | d| }t||kr:t ||t|}||qt|}||dd||S)a4 Construct a piecewise polynomial in the Bernstein basis, compatible with the specified values and derivatives at breakpoints. Parameters ---------- xi : array_like sorted 1-D array of x-coordinates yi : array_like or list of array_likes ``yi[i][j]`` is the ``j``th derivative known at ``xi[i]`` orders : None or int or array_like of ints. Default: None. Specifies the degree of local polynomials. If not None, some derivatives are ignored. extrapolate : bool or 'periodic', optional If bool, determines whether to extrapolate to out-of-bounds points based on first and last intervals, or to return NaNs. If 'periodic', periodic extrapolation is used. Default is True. Notes ----- If ``k`` derivatives are specified at a breakpoint ``x``, the constructed polynomial is exactly ``k`` times continuously differentiable at ``x``, unless the ``order`` is provided explicitly. In the latter case, the smoothness of the polynomial at the breakpoint is controlled by the ``order``. Deduces the number of derivatives to match at each end from ``order`` and the number of derivatives available. If possible it uses the same number of derivatives from each end; if the number is odd it tries to take the extra one from y2. In any case if not enough derivatives are available at one end or another it draws enough to make up the total from the other end. If the order is too high and not enough derivatives are available, an exception is raised. Examples -------- >>> from scipy.interpolate import BPoly >>> BPoly.from_derivatives([0, 1], [[1, 2], [3, 4]]) Creates a polynomial `f(x)` of degree 3, defined on `[0, 1]` such that `f(0) = 1, df/dx(0) = 2, f(1) = 3, df/dx(1) = 4` >>> BPoly.from_derivatives([0, 1, 2], [[0, 1], [0], [2]]) Creates a piecewise polynomial `f(x)`, such that `f(0) = f(1) = 0`, `f(2) = 2`, and `df/dx(0) = 1`. Based on the number of derivatives provided, the order of the local polynomials is 2 on `[0, 1]` and 1 on `[1, 2]`. Notice that no restriction is imposed on the derivatives at ``x = 1`` and ``x = 2``. Indeed, the explicit form of the polynomial is:: f(x) = | x * (1 - x), 0 <= x < 1 | 2 * (x - 1), 1 <= x <= 2 So that f'(1-0) = -1 and f'(1+0) = 2 z&xi and yi need to have the same lengthrNr z)x coordinates are not in increasing orderc3s*|]"}t|t|dVqdSrN)r!r8iyir*r+ r;z)BPoly.from_derivatives..z0Using a 1-D array for y? Please .reshape(-1, 1).css|]}|dkVqdS)r Nr*)r8or*r*r+r r;zOrders must be positive.r-zPPoint %g has %d derivatives, point %g has %d derivatives, but order %d requestedz0`order` input incompatible with length y1 or y2.)r@rr!r?r\rr"rrkrvintegerminr_construct_from_derivativesrrZswapaxes)rxir Zordersrjrr)rSrVry1y2Zn1Zn2nZmesgrr*rr+from_derivativess\@ "    " zBPoly.from_derivativesc Cst|t|}}|jdd|jddkrFtd|j|j|j|j}}t|tjspt|tjrxtj}ntj }t |t |}}||} tj ||f|jdd|d} t d|D]h} || t | | | ||| | | <t d| D]0} | | d| | t| | | | 8<qqt d|D]} || t | | | d| ||| | | d<t d| D]@} | | dd| dt| | d| | | 8<q|q8| S)aGCompute the coefficients of a polynomial in the Bernstein basis given the values and derivatives at the edges. Return the coefficients of a polynomial in the Bernstein basis defined on ``[xa, xb]`` and having the values and derivatives at the endpoints `xa` and `xb` as specified by `ya`` and `yb`. The polynomial constructed is of the minimal possible degree, i.e., if the lengths of `ya` and `yb` are `na` and `nb`, the degree of the polynomial is ``na + nb - 1``. Parameters ---------- xa : float Left-hand end point of the interval xb : float Right-hand end point of the interval ya : array_like Derivatives at `xa`. `ya[0]` is the value of the function, and `ya[i]` for ``i > 0`` is the value of the ``i``th derivative. yb : array_like Derivatives at `xb`. Returns ------- array coefficient array of a polynomial having specified derivatives Notes ----- This uses several facts from life of Bernstein basis functions. First of all, .. math:: b'_{a, n} = n (b_{a-1, n-1} - b_{a, n-1}) If B(x) is a linear combination of the form .. math:: B(x) = \sum_{a=0}^{n} c_a b_{a, n}, then :math: B'(x) = n \sum_{a=0}^{n-1} (c_{a+1} - c_{a}) b_{a, n-1}. Iterating the latter one, one finds for the q-th derivative .. math:: B^{q}(x) = n!/(n-q)! \sum_{a=0}^{n-q} Q_a b_{a, n-q}, with .. math:: Q_a = \sum_{j=0}^{q} (-)^{j+q} comb(q, j) c_{j+a} This way, only `a=0` contributes to :math: `B^{q}(x = xa)`, and `c_q` are found one by one by iterating `q = 0, ..., na`. At ``x = xb`` it's the same with ``a = n - q``. rNz*Shapes of ya {} and yb {} are incompatiblerr r.)r@rr>r?rrerrrr|r!rr"rrr) ZxaxbZyaZybZdtaZdtbdtnanbrrVqr'r*r*r+rs.7 "(06Bz!BPoly._construct_from_derivativesc Cs|dkr |S|jdd}tj|jd|f|jdd|jd}t|jdD]X}||t||}t|dD]4}||||t||t||||7<qtqR|S)a Raise a degree of a polynomial in the Bernstein basis. Given the coefficients of a polynomial degree `k`, return (the coefficients of) the equivalent polynomial of degree `k+d`. Parameters ---------- c : array_like coefficient array, 1-D d : integer Returns ------- array coefficient array, 1-D array of length `c.shape[0] + d` Notes ----- This uses the fact that a Bernstein polynomial `b_{a, k}` can be identically represented as a linear combination of polynomials of a higher degree `k+d`: .. math:: b_{a, k} = comb(k, a) \sum_{j=0}^{d} b_{a+j, k+d} \ comb(d, j) / comb(k+d, a+j) r rNr)r>r@rrer"r)rVdr)rr9rhr'r*r*r+rOs*4zBPoly._raise_degree)r)r)N)N)N)NN)r`rarbrcrrrrrrrrr staticmethodrrr*r*r*r+rNsU 5 8 @   " w Vrc@sveZdZdZdddZedddZddZd d Zdd d Z d dZ ddZ ddZ ddZ dddZdddZdS)raW Piecewise tensor product polynomial The value at point ``xp = (x', y', z', ...)`` is evaluated by first computing the interval indices `i` such that:: x[0][i[0]] <= x' < x[0][i[0]+1] x[1][i[1]] <= y' < x[1][i[1]+1] ... and then computing:: S = sum(c[k0-m0-1,...,kn-mn-1,i[0],...,i[n]] * (xp[0] - x[0][i[0]])**m0 * ... * (xp[n] - x[n][i[n]])**mn for m0 in range(k[0]+1) ... for mn in range(k[n]+1)) where ``k[j]`` is the degree of the polynomial in dimension j. This representation is the piecewise multivariate power basis. Parameters ---------- c : ndarray, shape (k0, ..., kn, m0, ..., mn, ...) Polynomial coefficients, with polynomial order `kj` and `mj+1` intervals for each dimension `j`. x : ndim-tuple of ndarrays, shapes (mj+1,) Polynomial breakpoints for each dimension. These must be sorted in increasing order. extrapolate : bool, optional Whether to extrapolate to out-of-bounds points based on first and last intervals, or to return NaNs. Default: True. Attributes ---------- x : tuple of ndarrays Breakpoints. c : ndarray Coefficients of the polynomials. Methods ------- __call__ derivative antiderivative integrate integrate_1d construct_fast See also -------- PPoly : piecewise polynomials in 1D Notes ----- High-order polynomials in the power basis can be numerically unstable. NcCstdd|D|_t||_|dur,d}t||_t|j}tdd|jDr\t dtdd|jDrxt d|j d|krt d td d|jDrt d td dt |j |d||jDrt d | |jj}tj|j|d|_dS)Ncss|]}tj|tjdVqdS)rN)r@rrr8vr*r*r+r r;z#NdPPoly.__init__..Tcss|]}|jdkVqdSrr=rr*r*r+r r;z"x arrays must all be 1-dimensionalcss|]}|jdkVqdS)r-Nr<rr*r*r+r r;z+x arrays must all contain at least 2 pointsr-z(c must have at least 2*len(x) dimensionscss0|](}t|dd|dddkVqdS)rNr.r )r@r\rr*r*r+r r;z)x-coordinates are not in increasing ordercss |]\}}||jdkVqdSrr)r8r9rr*r*r+r r;z/x and c do not agree on the number of intervalsr)rr#r@rrVrrjr!r\r?r=rdr>rrer)rQrVr#rjr=rer*r*r+rXs$   (zNdPPoly.__init__cCs,t|}||_||_|dur"d}||_|S)aJ Construct the piecewise polynomial without making checks. Takes the same parameters as the constructor. Input arguments ``c`` and ``x`` must be arrays of the correct shape and type. The ``c`` array can only be of dtypes float and complex, and ``x`` array must have dtype float. NT)rrrVr#rj)rrVr#rjrQr*r*r+rs zNdPPoly.construct_fastcCs0t|tjs t|jjtjr&tjStjSdSrrrr*r*r+rs zNdPPoly._get_dtypecCs2|jjjs|j|_t|jts.t|j|_dSr)rVrrr7rkr#rrr*r*r+rs   zNdPPoly._ensure_c_contiguousc Csl|dur|j}nt|}t|j}t|}|j}tj|d|jdtj d}|durjtj |ftj d}n0tj |tj d}|j dks|jd|krtdt|jjd|}t|jj|d|}t|jjd|d}tj|jjd|tj d} tj|jd|f|jjd} |t|j||||j| ||t|| | |dd|jjd|dS)a Evaluate the piecewise polynomial or its derivative Parameters ---------- x : array-like Points to evaluate the interpolant at. nu : tuple, optional Orders of derivatives to evaluate. Each must be non-negative. extrapolate : bool, optional Whether to extrapolate to out-of-bounds points based on first and last intervals, or to return NaNs. Returns ------- y : array-like Interpolated values. Shape is determined by replacing the interpolation axis in the original array with the shape of x. Notes ----- Derivatives are evaluated piecewise for each polynomial segment, even if the polynomial is not differentiable at the breakpoints. The polynomial intervals are considered half-open, ``[a, b)``, except for the last interval which is closed ``[a, b]``. Nr.rrr z&invalid number of derivative orders nur-)rjrr!r#rr>r@rrr|rZintcrr=r?rrVr rrerrZ evaluate_nd) rQr#rrjr=rZdim1Zdim2Zdim3ksrr*r*r+r_s6 zNdPPoly.__call__cCs|dkr|| |St|j}||}|dkr4dStdg|}td| d||<|jt|}|j|dkrt|j}d||<tj ||j d}t t |j|dd|}dg|j}td||<||t|9}||_dS)zz Compute 1-D derivative along a selected dimension in-place May result to non-contiguous c array. r Nrrr.)_antiderivative_inplacer!r#rrVrr>rr@rrerrrr=)rQrror=slrZshprr*r*r+_derivative_inplace.s$    zNdPPoly._derivative_inplacec Cs|dkr|| |St|j}||}tt|}|||d|d<||<|tt||jj}|j|}tj |j d|f|j dd|j d}||d| <t t|j ddd|}|d| |tdfd|jd<tt|j}||||d|d<|||<||}|}t||j d|j dd|j||d||}||}||_dS)zu Compute 1-D antiderivative along a selected dimension May result to non-contiguous c array. r rNrr.r)r"r!r#rr"rVr=r r@rr>rerrrrr7rrr) rQrror=permrVrrZperm2r*r*r+r Ps0    ."   zNdPPoly._antiderivative_inplacecCsB||j|j|j}t|D]\}}|||q ||S)a$ Construct a new piecewise polynomial representing the derivative. Parameters ---------- nu : ndim-tuple of int Order of derivatives to evaluate for each dimension. If negative, the antiderivative is returned. Returns ------- pp : NdPPoly Piecewise polynomial of orders (k[0] - nu[0], ..., k[n] - nu[n]) representing the derivative of this polynomial. Notes ----- Derivatives are evaluated piecewise for each polynomial segment, even if the polynomial is not differentiable at the breakpoints. The polynomial intervals in each dimension are considered half-open, ``[a, b)``, except for the last interval which is closed ``[a, b]``. )rrVr7r#rjrr"rrQrr&rorr*r*r+rxs zNdPPoly.derivativecCsB||j|j|j}t|D]\}}|||q ||S)a Construct a new piecewise polynomial representing the antiderivative. Antiderivative is also the indefinite integral of the function, and derivative is its inverse operation. Parameters ---------- nu : ndim-tuple of int Order of derivatives to evaluate for each dimension. If negative, the derivative is returned. Returns ------- pp : PPoly Piecewise polynomial of order k2 = k + n representing the antiderivative of this polynomial. Notes ----- The antiderivative returned by this function is continuous and continuously differentiable to order n-1, up to floating point rounding error. )rrVr7r#rjrr rr$r*r*r+rs zNdPPoly.antiderivativec Cs(|dur|j}nt|}t|j}t||}|j}tt|j}| d||||d=| d||||||d=| |}t j | |jd|jdd|j||d}|j|||d} |dkr| |jddS| |jdd}|jd||j|dd} |j || |dSdS)a Compute NdPPoly representation for one dimensional definite integral The result is a piecewise polynomial representing the integral: .. math:: p(y, z, ...) = \int_a^b dx\, p(x, y, z, ...) where the dimension integrated over is specified with the `axis` parameter. Parameters ---------- a, b : float Lower and upper bound for integration. axis : int Dimension over which to compute the 1-D integrals extrapolate : bool, optional Whether to extrapolate to out-of-bounds points based on first and last intervals, or to return NaNs. Returns ------- ig : NdPPoly or array-like Definite integral of the piecewise polynomial over [a, b]. If the polynomial was 1D, an array is returned, otherwise, an NdPPoly object. Nr rr.rjr-)rjrr!r#rvrVrr"r=insertr rrrr>r) rQr9rrorjr=rVswapr&rr#r*r*r+ integrate_1ds,     zNdPPoly.integrate_1dc Cst|j}|dur|j}nt|}t|dr8t||kr@td||j}t|D]\}\}}t t |j }| d||||||d=| |}tj||j||d} | j|||d} | |jdd}qV|S)aq Compute a definite integral over a piecewise polynomial. Parameters ---------- ranges : ndim-tuple of 2-tuples float Sequence of lower and upper bounds for each dimension, ``[(a[0], b[0]), ..., (a[ndim-1], b[ndim-1])]`` extrapolate : bool, optional Whether to extrapolate to out-of-bounds points based on first and last intervals, or to return NaNs. Returns ------- ig : array_like Definite integral of the piecewise polynomial over [a[0], b[0]] x ... x [a[ndim-1], b[ndim-1]] N__len__z&Range not a sequence of correct lengthrr%r-)r!r#rjrhasattrr?rrVrrr"r=r&r rrrrr>) rQrangesrjr=rVrr9rr'r&rr*r*r+rs"  zNdPPoly.integrate)N)N)NN)N)N)r`rarbrcrXrrrrr_r"r rrr(rr*r*r*r+rxs>   A"(!" =rc@sDeZdZdZddejfddZdddZd d Zd d Z d dZ dS)ra Interpolation on a regular grid in arbitrary dimensions The data must be defined on a regular grid; the grid spacing however may be uneven. Linear and nearest-neighbor interpolation are supported. After setting up the interpolator object, the interpolation method (*linear* or *nearest*) may be chosen at each evaluation. Parameters ---------- points : tuple of ndarray of float, with shapes (m1, ), ..., (mn, ) The points defining the regular grid in n dimensions. values : array_like, shape (m1, ..., mn, ...) The data on the regular grid in n dimensions. method : str, optional The method of interpolation to perform. Supported are "linear" and "nearest". This parameter will become the default for the object's ``__call__`` method. Default is "linear". bounds_error : bool, optional If True, when interpolated values are requested outside of the domain of the input data, a ValueError is raised. If False, then `fill_value` is used. fill_value : number, optional If provided, the value to use for points outside of the interpolation domain. If None, values outside the domain are extrapolated. Methods ------- __call__ Notes ----- Contrary to LinearNDInterpolator and NearestNDInterpolator, this class avoids expensive triangulation of the input data by taking advantage of the regular grid structure. If any of `points` have a dimension of size 1, linear interpolation will return an array of `nan` values. Nearest-neighbor interpolation will work as usual in this case. .. versionadded:: 0.14 Examples -------- Evaluate a simple example function on the points of a 3-D grid: >>> from scipy.interpolate import RegularGridInterpolator >>> def f(x, y, z): ... return 2 * x**3 + 3 * y**2 - z >>> x = np.linspace(1, 4, 11) >>> y = np.linspace(4, 7, 22) >>> z = np.linspace(7, 9, 33) >>> xg, yg ,zg = np.meshgrid(x, y, z, indexing='ij', sparse=True) >>> data = f(xg, yg, zg) ``data`` is now a 3-D array with ``data[i,j,k] = f(x[i], y[j], z[k])``. Next, define an interpolating function from this data: >>> my_interpolating_function = RegularGridInterpolator((x, y, z), data) Evaluate the interpolating function at the two points ``(x,y,z) = (2.1, 6.2, 8.3)`` and ``(3.3, 5.2, 7.1)``: >>> pts = np.array([[2.1, 6.2, 8.3], [3.3, 5.2, 7.1]]) >>> my_interpolating_function(pts) array([ 125.80469388, 146.30069388]) which is indeed a close approximation to ``[f(2.1, 6.2, 8.3), f(3.3, 5.2, 7.1)]``. See also -------- NearestNDInterpolator : Nearest neighbor interpolation on unstructured data in N dimensions LinearNDInterpolator : Piecewise linear interpolant on unstructured data in N dimensions References ---------- .. [1] Python package *regulargrid* by Johannes Buchner, see https://pypi.python.org/pypi/regulargrid/ .. [2] Wikipedia, "Trilinear interpolation", https://en.wikipedia.org/wiki/Trilinear_interpolation .. [3] Weiser, Alan, and Sergio E. Zarantonello. "A note on piecewise linear and multilinear table interpolation in many dimensions." MATH. COMPUT. 50.181 (1988): 189-196. https://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917826-0/S0025-5718-1988-0917826-0.pdf r,Tc Cs`|dvrtd|||_||_t|ds4t|}t||jkrXtdt||jft|drt|drt|j tj s| t }||_ |durt|j }t|drtj||j ddstd t|D]t\}}tt|d kstd |t|jd kstd ||j|t|kstdt||j||fqtdd|D|_||_dS)Nr,rpMethod '%s' is not definedr=7There are %d point arrays, but values has %d dimensionsrer{Z same_kind)ZcastingzDfill_value must be either 'None' or of a type compatible with valuesr5The points in dimension %d must be strictly ascendingr0The points in dimension %d must be 1-dimensional1There are %d points and %d values in dimension %dcSsg|]}t|qSr*r@rr8r&r*r*r+r: r;z4RegularGridInterpolator.__init__..)r?methodrIr*r@rr!r=rrerzr{rrJZcan_castrrArr>rgridvalues) rQpointsr6r4rIrJZfill_value_dtyperr&r*r*r+rX sJ        z RegularGridInterpolator.__init__Nc CsZ|dur|jn|}|dvr&td|t|j}t||d}|jdt|jkrftd|jd|f|j}|d|d}|jrt|j D]H\}}t t |j|d|kt ||j|dkstd |q| |j \}}} |d kr|||| } n|d kr|||| } |js8|jdur8|j| | <| |dd|jj|dS) a6 Interpolation at coordinates Parameters ---------- xi : ndarray of shape (..., ndim) The coordinates to sample the gridded data at method : str The method of interpolation to perform. 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Nr,r-rr.cThe requested sample points xi have dimension %d, but this RegularGridInterpolator has dimension %drr 8One of the requested xi is out of bounds in dimension %dr,rp)r4r?r!r5rr>rrIrrCr@ logical_andrA _find_indices_evaluate_linear_evaluate_nearestrJr6) rQrr4r=xi_shaperr&indicesnorm_distances out_of_boundsresultr*r*r+r_ sB       z RegularGridInterpolator.__call__c Cstdfd|jjt|}tjdd|D}d}|D]V}d}t|||D]$\} } } |t| | kd| | 9}qN|t |j|||7}q:|S)NrcSsg|]}||dgqSrr*rr*r*r+r: r;z.rr r) rr6r=r! itertoolsproductrdr@wherer) rQr?r@rAZvsliceedgesr6Z edge_indicesZweighteirr r*r*r+r< sz(RegularGridInterpolator._evaluate_linearcCs"ddt||D}|jt|S)NcSs&g|]\}}t|dk||dqS)g?r)r@rE)r8rr r*r*r+r: sz=RegularGridInterpolator._evaluate_nearest..)rdr6r)rQr?r@rAZidx_resr*r*r+r= sz)RegularGridInterpolator._evaluate_nearestcCsg}g}tj|jdtd}t||jD]\}}t||d}d||dk<|jd|||jdk<||||||||d|||j s(|||dk7}|||dk7}q(|||fS)Nrrr r-r.) r@rr>rrdr5r r<rrI)rQrr?r@rAr#r5rr*r*r+r; s  z%RegularGridInterpolator._find_indices)N) r`rarbrcr@rrXr_r<r=r;r*r*r*r+r) sb ) 1rr,Tc Cs|dvrtd|t|ds(t|}|j}|dkrF|dkrFtd|sb|durb|dkrbtdt||krtd t||ft||kr|dkrtd t|D]r\}}tt|d kstd |t|jd kstd||j |t|kstdt||j ||fqt dd|D} t |t| d}|j dt| krjtd|j d t| f|rt|j D]H\}}t t| |d|kt|| |dksztd|qz|dkrt||d||d} | |S|dkr t||d||d} | |S|dkr|j } |d|j d}tj| dd|dddfk|dddf| ddk| d d|ddd fk|ddd f| d dkfdd} t|dddf} t|d|d |dd} | || df|| d f| | <|| t| <| | ddSdS)a, Multidimensional interpolation on regular grids. Parameters ---------- points : tuple of ndarray of float, with shapes (m1, ), ..., (mn, ) The points defining the regular grid in n dimensions. values : array_like, shape (m1, ..., mn, ...) The data on the regular grid in n dimensions. xi : ndarray of shape (..., ndim) The coordinates to sample the gridded data at method : str, optional The method of interpolation to perform. Supported are "linear" and "nearest", and "splinef2d". "splinef2d" is only supported for 2-dimensional data. bounds_error : bool, optional If True, when interpolated values are requested outside of the domain of the input data, a ValueError is raised. If False, then `fill_value` is used. fill_value : number, optional If provided, the value to use for points outside of the interpolation domain. If None, values outside the domain are extrapolated. Extrapolation is not supported by method "splinef2d". Returns ------- values_x : ndarray, shape xi.shape[:-1] + values.shape[ndim:] Interpolated values at input coordinates. Notes ----- .. versionadded:: 0.14 Examples -------- Evaluate a simple example function on the points of a regular 3-D grid: >>> from scipy.interpolate import interpn >>> def value_func_3d(x, y, z): ... return 2 * x + 3 * y - z >>> x = np.linspace(0, 4, 5) >>> y = np.linspace(0, 5, 6) >>> z = np.linspace(0, 6, 7) >>> points = (x, y, z) >>> values = value_func_3d(*np.meshgrid(*points, indexing='ij')) Evaluate the interpolating function at a point >>> point = np.array([2.21, 3.12, 1.15]) >>> print(interpn(points, values, point)) [12.63] See also -------- NearestNDInterpolator : Nearest neighbor interpolation on unstructured data in N dimensions LinearNDInterpolator : Piecewise linear interpolant on unstructured data in N dimensions RegularGridInterpolator : Linear and nearest-neighbor Interpolation on a regular grid in arbitrary dimensions RectBivariateSpline : Bivariate spline approximation over a rectangular mesh )r,rp splinef2dz[interpn only understands the methods 'linear', 'nearest', and 'splinef2d'. 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