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""" Classes for interpolating values. """
'lagrange', 'PPoly', 'BPoly', 'NdPPoly', 'RegularGridInterpolator', 'interpn']
dot, ravel, poly1d, asarray, intp)
"""Product of a list of numbers; ~40x faster vs np.prod for Python tuples""" if len(x) == 0: return 1 return functools.reduce(operator.mul, x)
r""" 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.])
"""
M = len(x) p = poly1d(0.0) for j in xrange(M): pt = poly1d(w[j]) for k in xrange(M): if k == j: continue fac = x[j]-x[k] pt *= poly1d([1.0, -x[k]])/fac p += pt return p
# !! Need to find argument for keeping initialize. If it isn't # !! found, get rid of it!
""" interp2d(x, y, z, kind='linear', copy=True, bounds_error=False, fill_value=nan)
Interpolate over a 2-D grid.
`x`, `y` and `z` are arrays of values used to approximate some function f: ``z = f(x, y)``. 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.
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 multi-dimensional, they are flattened before use. z : array_like The values of the function to interpolate at the data points. If `z` is a multi-dimensional 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.
See Also -------- RectBivariateSpline : Much faster 2D 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 : one dimension 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() """
fill_value=None): x = ravel(x) y = ravel(y) z = asarray(z)
rectangular_grid = (z.size == len(x) * len(y)) if rectangular_grid: if z.ndim == 2: if z.shape != (len(y), len(x)): raise ValueError("When on a regular grid with x.size = m " "and y.size = n, if z.ndim == 2, then z " "must have shape (n, m)") if not np.all(x[1:] >= x[:-1]): j = np.argsort(x) x = x[j] z = z[:, j] if not np.all(y[1:] >= y[:-1]): j = np.argsort(y) y = y[j] z = z[j, :] z = ravel(z.T) else: z = ravel(z) if len(x) != len(y): raise ValueError( "x and y must have equal lengths for non rectangular grid") if len(z) != len(x): raise ValueError( "Invalid length for input z for non rectangular grid")
try: kx = ky = {'linear': 1, 'cubic': 3, 'quintic': 5}[kind] except KeyError: raise ValueError("Unsupported interpolation type.")
if not rectangular_grid: # TODO: surfit is really not meant for interpolation! self.tck = fitpack.bisplrep(x, y, z, kx=kx, ky=ky, s=0.0) else: nx, tx, ny, ty, c, fp, ier = dfitpack.regrid_smth( x, y, z, None, None, None, None, kx=kx, ky=ky, s=0.0) self.tck = (tx[:nx], ty[:ny], c[:(nx - kx - 1) * (ny - ky - 1)], kx, ky)
self.bounds_error = bounds_error self.fill_value = fill_value self.x, self.y, self.z = [array(a, copy=copy) for a in (x, y, z)]
self.x_min, self.x_max = np.amin(x), np.amax(x) self.y_min, self.y_max = np.amin(y), np.amax(y)
"""Interpolate the function.
Parameters ---------- x : 1D array x-coordinates of the mesh on which to interpolate. y : 1D 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 : 2D array with shape (len(y), len(x)) The interpolated values. """
x = atleast_1d(x) y = atleast_1d(y)
if x.ndim != 1 or y.ndim != 1: raise ValueError("x and y should both be 1-D arrays")
if not assume_sorted: x = np.sort(x) y = np.sort(y)
if self.bounds_error or self.fill_value is not None: out_of_bounds_x = (x < self.x_min) | (x > self.x_max) out_of_bounds_y = (y < self.y_min) | (y > self.y_max)
any_out_of_bounds_x = np.any(out_of_bounds_x) any_out_of_bounds_y = np.any(out_of_bounds_y)
if self.bounds_error and (any_out_of_bounds_x or any_out_of_bounds_y): raise ValueError("Values out of range; x must be in %r, y in %r" % ((self.x_min, self.x_max), (self.y_min, self.y_max)))
z = fitpack.bisplev(x, y, self.tck, dx, dy) z = atleast_2d(z) z = transpose(z)
if self.fill_value is not None: if any_out_of_bounds_x: z[:, out_of_bounds_x] = self.fill_value if any_out_of_bounds_y: z[out_of_bounds_y, :] = self.fill_value
if len(z) == 1: z = z[0] return array(z)
"""Helper to check that arr_from broadcasts up to shape_to""" if f != 1 and f != t: break else: # all checks pass, do the upcasting that we need later arr_from = np.ones(shape_to, arr_from.dtype) * arr_from # at least one check failed raise ValueError('%s argument must be able to broadcast up ' 'to shape %s but had shape %s' % (name, shape_to, shape_from))
"""Helper to check if fill_value == "extrapolate" without warnings""" fill_value == 'extrapolate')
""" 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.
Note that calling `interp1d` with NaNs present in input values results in undefined behaviour.
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 ('linear', 'nearest', 'zero', 'slinear', 'quadratic', 'cubic', 'previous', 'next', where '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) or as an integer specifying the order of the spline interpolator to use. 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.
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
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() """
copy=True, bounds_error=None, fill_value=np.nan, assume_sorted=False): """ Initialize a 1D linear interpolation class."""
order = {'zero': 0, 'slinear': 1, 'quadratic': 2, 'cubic': 3}[kind] kind = 'spline' order = kind kind = 'spline' raise NotImplementedError("%s is unsupported: Use fitpack " "routines for other types." % kind)
raise ValueError("the x array must have exactly one dimension.") raise ValueError("the y array must have at least one dimension.")
# Force-cast y to a floating-point type, if it's not yet one y = y.astype(np.float_)
# Backward compatibility
# Interpolation goes internally along the first axis
# Adjust to interpolation kind; store reference to *unbound* # interpolation methods, in order to avoid circular references to self # stored in the bound instance methods, and therefore delayed garbage # collection. See: http://docs.python.org/2/reference/datamodel.html # Make a "view" of the y array that is rotated to the interpolation # axis. # Do division before addition to prevent possible integer # overflow self.x_bds = self.x / 2.0 self.x_bds = self.x_bds[1:] + self.x_bds[:-1]
self._call = self.__class__._call_nearest # Side for np.searchsorted and index for clipping self._side = 'left' self._ind = 0 # Move x by one floating point value to the left self._x_shift = np.nextafter(self.x, -np.inf) self._call = self.__class__._call_previousnext self._side = 'right' self._ind = 1 # Move x by one floating point value to the right self._x_shift = np.nextafter(self.x, np.inf) self._call = self.__class__._call_previousnext else: # Check if we can delegate to numpy.interp (2x-10x faster).
else: self._call = self.__class__._call_linear else: minval = order + 1
rewrite_nan = False xx, yy = self.x, self._y if order > 1: # Quadratic or cubic spline. If input contains even a single # nan, then the output is all nans. We cannot just feed data # with nans to make_interp_spline because it calls LAPACK. # So, we make up a bogus x and y with no nans and use it # to get the correct shape of the output, which we then fill # with nans. # For slinear or zero order spline, we just pass nans through. if np.isnan(self.x).any(): xx = np.linspace(min(self.x), max(self.x), len(self.x)) rewrite_nan = True if np.isnan(self._y).any(): yy = np.ones_like(self._y) rewrite_nan = True
self._spline = make_interp_spline(xx, yy, k=order, check_finite=False) if rewrite_nan: self._call = self.__class__._call_nan_spline else: self._call = self.__class__._call_spline
raise ValueError("x and y arrays must have at " "least %d entries" % minval)
def fill_value(self): # backwards compat: mimic a public attribute return self._fill_value_orig
def fill_value(self, fill_value): # extrapolation only works for nearest neighbor and linear methods if self.bounds_error: raise ValueError("Cannot extrapolate and raise " "at the same time.") self.bounds_error = False self._extrapolate = True else: self.y.shape[self.axis + 1:]) # it's either a pair (_below_range, _above_range) or a single value # for both above and below range below_above = [np.asarray(fill_value[0]), np.asarray(fill_value[1])] names = ('fill_value (below)', 'fill_value (above)') for ii in range(2): below_above[ii] = _check_broadcast_up_to( below_above[ii], broadcast_shape, names[ii]) else: fill_value, broadcast_shape, 'fill_value')] * 2 # backwards compat: fill_value was a public attr; make it writeable
# Note that out-of-bounds values are taken care of in self._evaluate
# 2. Find where in the original data, the values to interpolate # would be inserted. # Note: If x_new[n] == x[m], then m is returned by searchsorted. x_new_indices = searchsorted(self.x, x_new)
# 3. Clip x_new_indices so that they are within the range of # self.x indices and at least 1. Removes mis-interpolation # of x_new[n] = x[0] x_new_indices = x_new_indices.clip(1, len(self.x)-1).astype(int)
# 4. Calculate the slope of regions that each x_new value falls in. lo = x_new_indices - 1 hi = x_new_indices
x_lo = self.x[lo] x_hi = self.x[hi] y_lo = self._y[lo] y_hi = self._y[hi]
# Note that the following two expressions rely on the specifics of the # broadcasting semantics. slope = (y_hi - y_lo) / (x_hi - x_lo)[:, None]
# 5. Calculate the actual value for each entry in x_new. y_new = slope*(x_new - x_lo)[:, None] + y_lo
return y_new
""" Find nearest neighbour interpolated y_new = f(x_new)."""
# 2. Find where in the averaged data the values to interpolate # would be inserted. # Note: use side='left' (right) to searchsorted() to define the # halfway point to be nearest to the left (right) neighbour x_new_indices = searchsorted(self.x_bds, x_new, side='left')
# 3. Clip x_new_indices so that they are within the range of x indices. x_new_indices = x_new_indices.clip(0, len(self.x)-1).astype(intp)
# 4. Calculate the actual value for each entry in x_new. y_new = self._y[x_new_indices]
return y_new
"""Use previous/next neighbour of x_new, y_new = f(x_new)."""
# 1. Get index of left/right value x_new_indices = searchsorted(self._x_shift, x_new, side=self._side)
# 2. Clip x_new_indices so that they are within the range of x indices. x_new_indices = x_new_indices.clip(1-self._ind, len(self.x)-self._ind).astype(intp)
# 3. Calculate the actual value for each entry in x_new. y_new = self._y[x_new_indices+self._ind-1]
return y_new
return self._spline(x_new)
out = self._spline(x_new) out[...] = np.nan return out
# 1. Handle values in x_new that are outside of x. Throw error, # or return a list of mask array indicating the outofbounds values. # The behavior is set by the bounds_error variable. # Note fill_value must be broadcast up to the proper size # and flattened to work here
"""Check 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. """
# If self.bounds_error is True, we raise an error if any x_new values # fall outside the range of x. Otherwise, we return an array indicating # which values are outside the boundary region.
# !! Could provide more information about which values are out of bounds raise ValueError("A value in x_new is below the interpolation " "range.") raise ValueError("A value in x_new is above the interpolation " "range.")
# !! Should we emit a warning if some values are out of bounds? # !! matlab does not.
"""Base class for piecewise polynomials."""
self.c = np.asarray(c) self.x = np.ascontiguousarray(x, dtype=np.float64)
if extrapolate is None: extrapolate = True elif extrapolate != 'periodic': extrapolate = bool(extrapolate) self.extrapolate = extrapolate
if self.c.ndim < 2: raise ValueError("Coefficients array must be at least " "2-dimensional.")
if not (0 <= axis < self.c.ndim - 1): raise ValueError("axis=%s must be between 0 and %s" % (axis, self.c.ndim-1))
self.axis = axis if axis != 0: # roll the interpolation axis to be the first one in self.c # More specifically, the target shape for self.c is (k, m, ...), # and axis !=0 means that we have c.shape (..., k, m, ...) # ^ # axis # So we roll two of them. self.c = np.rollaxis(self.c, axis+1) self.c = np.rollaxis(self.c, axis+1)
if self.x.ndim != 1: raise ValueError("x must be 1-dimensional") if self.x.size < 2: raise ValueError("at least 2 breakpoints are needed") if self.c.ndim < 2: raise ValueError("c must have at least 2 dimensions") if self.c.shape[0] == 0: raise ValueError("polynomial must be at least of order 0") if self.c.shape[1] != self.x.size-1: raise ValueError("number of coefficients != len(x)-1") dx = np.diff(self.x) if not (np.all(dx >= 0) or np.all(dx <= 0)): raise ValueError("`x` must be strictly increasing or decreasing.")
dtype = self._get_dtype(self.c.dtype) self.c = np.ascontiguousarray(self.c, dtype=dtype)
if np.issubdtype(dtype, np.complexfloating) \ or np.issubdtype(self.c.dtype, np.complexfloating): return np.complex_ else: return np.float_
""" 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. """ self = object.__new__(cls) self.c = c self.x = x self.axis = axis if extrapolate is None: extrapolate = True self.extrapolate = extrapolate return self
""" c and x may be modified by the user. The Cython code expects that they are C contiguous. """ if not self.x.flags.c_contiguous: self.x = self.x.copy() if not self.c.flags.c_contiguous: self.c = self.c.copy()
""" 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 """ if right is not None: warnings.warn("`right` is deprecated and will be removed.")
c = np.asarray(c) x = np.asarray(x)
if c.ndim < 2: raise ValueError("invalid dimensions for c") if x.ndim != 1: raise ValueError("invalid dimensions for x") if x.shape[0] != c.shape[1]: raise ValueError("x and c have incompatible sizes") if c.shape[2:] != self.c.shape[2:] or c.ndim != self.c.ndim: raise ValueError("c and self.c have incompatible shapes")
if c.size == 0: return
dx = np.diff(x) if not (np.all(dx >= 0) or np.all(dx <= 0)): raise ValueError("`x` is not sorted.")
if self.x[-1] >= self.x[0]: if not x[-1] >= x[0]: raise ValueError("`x` is in the different order " "than `self.x`.")
if x[0] >= self.x[-1]: action = 'append' elif x[-1] <= self.x[0]: action = 'prepend' else: raise ValueError("`x` is neither on the left or on the right " "from `self.x`.") else: if not x[-1] <= x[0]: raise ValueError("`x` is in the different order " "than `self.x`.")
if x[0] <= self.x[-1]: action = 'append' elif x[-1] >= self.x[0]: action = 'prepend' else: raise ValueError("`x` is neither on the left or on the right " "from `self.x`.")
dtype = self._get_dtype(c.dtype)
k2 = max(c.shape[0], self.c.shape[0]) c2 = np.zeros((k2, self.c.shape[1] + c.shape[1]) + self.c.shape[2:], dtype=dtype)
if action == 'append': c2[k2-self.c.shape[0]:, :self.c.shape[1]] = self.c c2[k2-c.shape[0]:, self.c.shape[1]:] = c self.x = np.r_[self.x, x] elif action == 'prepend': c2[k2-self.c.shape[0]:, :c.shape[1]] = c c2[k2-c.shape[0]:, c.shape[1]:] = self.c self.x = np.r_[x, self.x]
self.c = c2
""" 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]``. """ if extrapolate is None: extrapolate = self.extrapolate x = np.asarray(x) x_shape, x_ndim = x.shape, x.ndim x = np.ascontiguousarray(x.ravel(), dtype=np.float_)
# With periodic extrapolation we map x to the segment # [self.x[0], self.x[-1]]. if extrapolate == 'periodic': x = self.x[0] + (x - self.x[0]) % (self.x[-1] - self.x[0]) extrapolate = False
out = np.empty((len(x), prod(self.c.shape[2:])), dtype=self.c.dtype) self._ensure_c_contiguous() self._evaluate(x, nu, extrapolate, out) out = out.reshape(x_shape + self.c.shape[2:]) if self.axis != 0: # transpose to move the calculated values to the interpolation axis l = list(range(out.ndim)) l = l[x_ndim:x_ndim+self.axis] + l[:x_ndim] + l[x_ndim+self.axis:] out = out.transpose(l) return out
""" 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-dimensional 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. """ _ppoly.evaluate(self.c.reshape(self.c.shape[0], self.c.shape[1], -1), self.x, x, nu, bool(extrapolate), out)
""" 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]``. """ if nu < 0: return self.antiderivative(-nu)
# reduce order if nu == 0: c2 = self.c.copy() else: c2 = self.c[:-nu, :].copy()
if c2.shape[0] == 0: # derivative of order 0 is zero c2 = np.zeros((1,) + c2.shape[1:], dtype=c2.dtype)
# multiply by the correct rising factorials factor = spec.poch(np.arange(c2.shape[0], 0, -1), nu) c2 *= factor[(slice(None),) + (None,)*(c2.ndim-1)]
# construct a compatible polynomial return self.construct_fast(c2, self.x, self.extrapolate, self.axis)
""" 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. """ if nu <= 0: return self.derivative(-nu)
c = np.zeros((self.c.shape[0] + nu, self.c.shape[1]) + self.c.shape[2:], dtype=self.c.dtype) c[:-nu] = self.c
# divide by the correct rising factorials factor = spec.poch(np.arange(self.c.shape[0], 0, -1), nu) c[:-nu] /= factor[(slice(None),) + (None,)*(c.ndim-1)]
# fix continuity of added degrees of freedom self._ensure_c_contiguous() _ppoly.fix_continuity(c.reshape(c.shape[0], c.shape[1], -1), self.x, nu - 1)
if self.extrapolate == 'periodic': extrapolate = False else: extrapolate = self.extrapolate
# construct a compatible polynomial return self.construct_fast(c, self.x, extrapolate, self.axis)
""" 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] """ if extrapolate is None: extrapolate = self.extrapolate
# Swap integration bounds if needed sign = 1 if b < a: a, b = b, a sign = -1
range_int = np.empty((prod(self.c.shape[2:]),), dtype=self.c.dtype) self._ensure_c_contiguous()
# Compute the integral. if extrapolate == 'periodic': # Split the integral into the part over period (can be several # of them) and the remaining part.
xs, xe = self.x[0], self.x[-1] period = xe - xs interval = b - a n_periods, left = divmod(interval, period)
if n_periods > 0: _ppoly.integrate( self.c.reshape(self.c.shape[0], self.c.shape[1], -1), self.x, xs, xe, False, out=range_int) range_int *= n_periods else: range_int.fill(0)
# Map a to [xs, xe], b is always a + left. a = xs + (a - xs) % period b = a + left
# If b <= xe then we need to integrate over [a, b], otherwise # over [a, xe] and from xs to what is remained. remainder_int = np.empty_like(range_int) if b <= xe: _ppoly.integrate( self.c.reshape(self.c.shape[0], self.c.shape[1], -1), self.x, a, b, False, out=remainder_int) range_int += remainder_int else: _ppoly.integrate( self.c.reshape(self.c.shape[0], self.c.shape[1], -1), self.x, a, xe, False, out=remainder_int) range_int += remainder_int
_ppoly.integrate( self.c.reshape(self.c.shape[0], self.c.shape[1], -1), self.x, xs, xs + left + a - xe, False, out=remainder_int) range_int += remainder_int else: _ppoly.integrate( self.c.reshape(self.c.shape[0], self.c.shape[1], -1), self.x, a, b, bool(extrapolate), out=range_int)
# Return range_int *= sign return range_int.reshape(self.c.shape[2:])
""" 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.roots() array([-1., 1.]) """ if extrapolate is None: extrapolate = self.extrapolate
self._ensure_c_contiguous()
if np.issubdtype(self.c.dtype, np.complexfloating): raise ValueError("Root finding is only for " "real-valued polynomials")
y = float(y) r = _ppoly.real_roots(self.c.reshape(self.c.shape[0], self.c.shape[1], -1), self.x, y, bool(discontinuity), bool(extrapolate)) if self.c.ndim == 2: return r[0] else: r2 = np.empty(prod(self.c.shape[2:]), dtype=object) # this for-loop is equivalent to ``r2[...] = r``, but that's broken # in numpy 1.6.0 for ii, root in enumerate(r): r2[ii] = root
return r2.reshape(self.c.shape[2:])
""" 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 """ return self.solve(0, discontinuity, extrapolate)
""" 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. """ if isinstance(tck, BSpline): t, c, k = tck.tck if extrapolate is None: extrapolate = tck.extrapolate else: t, c, k = tck
cvals = np.empty((k + 1, len(t)-1), dtype=c.dtype) for m in xrange(k, -1, -1): y = fitpack.splev(t[:-1], tck, der=m) cvals[k - m, :] = y/spec.gamma(m+1)
return cls.construct_fast(cvals, t, extrapolate)
""" 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. """ dx = np.diff(bp.x) k = bp.c.shape[0] - 1 # polynomial order
rest = (None,)*(bp.c.ndim-2)
c = np.zeros_like(bp.c) for a in range(k+1): factor = (-1)**a * comb(k, a) * bp.c[a] for s in range(a, k+1): val = comb(k-a, s-a) * (-1)**s c[k-s] += factor * val / dx[(slice(None),)+rest]**s
if extrapolate is None: extrapolate = bp.extrapolate
return cls.construct_fast(c, bp.x, extrapolate, bp.axis)
"""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-dimensional 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. Here's a non-exhaustive list:
.. [1] http://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
"""
_ppoly.evaluate_bernstein( self.c.reshape(self.c.shape[0], self.c.shape[1], -1), self.x, x, nu, bool(extrapolate), out)
""" 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.
""" if nu < 0: return self.antiderivative(-nu)
if nu > 1: bp = self for k in range(nu): bp = bp.derivative() return bp
# reduce order if nu == 0: c2 = self.c.copy() else: # For a polynomial # B(x) = \sum_{a=0}^{k} c_a b_{a, k}(x), # we use the fact that # b'_{a, k} = k ( b_{a-1, k-1} - b_{a, k-1} ), # which leads to # B'(x) = \sum_{a=0}^{k-1} (c_{a+1} - c_a) b_{a, k-1} # # finally, for an interval [y, y + dy] with dy != 1, # we need to correct for an extra power of dy
rest = (None,)*(self.c.ndim-2)
k = self.c.shape[0] - 1 dx = np.diff(self.x)[(None, slice(None))+rest] c2 = k * np.diff(self.c, axis=0) / dx
if c2.shape[0] == 0: # derivative of order 0 is zero c2 = np.zeros((1,) + c2.shape[1:], dtype=c2.dtype)
# construct a compatible polynomial return self.construct_fast(c2, self.x, self.extrapolate, self.axis)
""" 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. """ if nu <= 0: return self.derivative(-nu)
if nu > 1: bp = self for k in range(nu): bp = bp.antiderivative() return bp
# Construct the indefinite integrals on individual intervals c, x = self.c, self.x k = c.shape[0] c2 = np.zeros((k+1,) + c.shape[1:], dtype=c.dtype)
c2[1:, ...] = np.cumsum(c, axis=0) / k delta = x[1:] - x[:-1] c2 *= delta[(None, slice(None)) + (None,)*(c.ndim-2)]
# Now fix continuity: on the very first interval, take the integration # constant to be zero; on an interval [x_j, x_{j+1}) with j>0, # the integration constant is then equal to the jump of the `bp` at x_j. # The latter is given by the coefficient of B_{n+1, n+1} # *on the previous interval* (other B. polynomials are zero at the # breakpoint). Finally, use the fact that BPs form a partition of unity. c2[:,1:] += np.cumsum(c2[k, :], axis=0)[:-1]
if self.extrapolate == 'periodic': extrapolate = False else: extrapolate = self.extrapolate
return self.construct_fast(c2, x, extrapolate, axis=self.axis)
""" Compute a definite integral over a piecewise polynomial.
Parameters ---------- a : float Lower integration bound b : float Upper integration bound extrapolate : {bool, 'periodic', None}, optional 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 ------- array_like Definite integral of the piecewise polynomial over [a, b]
""" # XXX: can probably use instead the fact that # \int_0^{1} B_{j, n}(x) \dx = 1/(n+1) ib = self.antiderivative() if extrapolate is None: extrapolate = self.extrapolate
# ib.extrapolate shouldn't be 'periodic', it is converted to # False for 'periodic. in antiderivative() call. if extrapolate != 'periodic': ib.extrapolate = extrapolate
if extrapolate == 'periodic': # Split the integral into the part over period (can be several # of them) and the remaining part.
# For simplicity and clarity convert to a <= b case. if a <= b: sign = 1 else: a, b = b, a sign = -1
xs, xe = self.x[0], self.x[-1] period = xe - xs interval = b - a n_periods, left = divmod(interval, period) res = n_periods * (ib(xe) - ib(xs))
# Map a and b to [xs, xe]. a = xs + (a - xs) % period b = a + left
# If b <= xe then we need to integrate over [a, b], otherwise # over [a, xe] and from xs to what is remained. if b <= xe: res += ib(b) - ib(a) else: res += ib(xe) - ib(a) + ib(xs + left + a - xe) - ib(xs)
return sign * res else: return ib(b) - ib(a)
k = max(self.c.shape[0], c.shape[0]) self.c = self._raise_degree(self.c, k - self.c.shape[0]) c = self._raise_degree(c, k - c.shape[0]) return _PPolyBase.extend(self, c, x, right)
""" 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. """ dx = np.diff(pp.x) k = pp.c.shape[0] - 1 # polynomial order
rest = (None,)*(pp.c.ndim-2)
c = np.zeros_like(pp.c) for a in range(k+1): factor = pp.c[a] / comb(k, k-a) * dx[(slice(None),)+rest]**(k-a) for j in range(k-a, k+1): c[j] += factor * comb(j, k-a)
if extrapolate is None: extrapolate = pp.extrapolate
return cls.construct_fast(c, pp.x, extrapolate, pp.axis)
"""Construct a piecewise polynomial in the Bernstein basis, compatible with the specified values and derivatives at breakpoints.
Parameters ---------- xi : array_like sorted 1D 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
""" xi = np.asarray(xi) if len(xi) != len(yi): raise ValueError("xi and yi need to have the same length") if np.any(xi[1:] - xi[:1] <= 0): raise ValueError("x coordinates are not in increasing order")
# number of intervals m = len(xi) - 1
# global poly order is k-1, local orders are <=k and can vary try: k = max(len(yi[i]) + len(yi[i+1]) for i in range(m)) except TypeError: raise ValueError("Using a 1D array for y? Please .reshape(-1, 1).")
if orders is None: orders = [None] * m else: if isinstance(orders, (integer_types, np.integer)): orders = [orders] * m k = max(k, max(orders))
if any(o <= 0 for o in orders): raise ValueError("Orders must be positive.")
c = [] for i in range(m): y1, y2 = yi[i], yi[i+1] if orders[i] is None: n1, n2 = len(y1), len(y2) else: n = orders[i]+1 n1 = min(n//2, len(y1)) n2 = min(n - n1, len(y2)) n1 = min(n - n2, len(y2)) if n1+n2 != n: mesg = ("Point %g has %d derivatives, point %g" " has %d derivatives, but order %d requested" % ( xi[i], len(y1), xi[i+1], len(y2), orders[i])) raise ValueError(mesg)
if not (n1 <= len(y1) and n2 <= len(y2)): raise ValueError("`order` input incompatible with" " length y1 or y2.")
b = BPoly._construct_from_derivatives(xi[i], xi[i+1], y1[:n1], y2[:n2]) if len(b) < k: b = BPoly._raise_degree(b, k - len(b)) c.append(b)
c = np.asarray(c) return cls(c.swapaxes(0, 1), xi, extrapolate)
def _construct_from_derivatives(xa, xb, ya, yb): r"""Compute 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`.
""" ya, yb = np.asarray(ya), np.asarray(yb) if ya.shape[1:] != yb.shape[1:]: raise ValueError('ya and yb have incompatible dimensions.')
dta, dtb = ya.dtype, yb.dtype if (np.issubdtype(dta, np.complexfloating) or np.issubdtype(dtb, np.complexfloating)): dt = np.complex_ else: dt = np.float_
na, nb = len(ya), len(yb) n = na + nb
c = np.empty((na+nb,) + ya.shape[1:], dtype=dt)
# compute coefficients of a polynomial degree na+nb-1 # walk left-to-right for q in range(0, na): c[q] = ya[q] / spec.poch(n - q, q) * (xb - xa)**q for j in range(0, q): c[q] -= (-1)**(j+q) * comb(q, j) * c[j]
# now walk right-to-left for q in range(0, nb): c[-q-1] = yb[q] / spec.poch(n - q, q) * (-1)**q * (xb - xa)**q for j in range(0, q): c[-q-1] -= (-1)**(j+1) * comb(q, j+1) * c[-q+j]
return c
def _raise_degree(c, d): r"""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, 1D d : integer
Returns ------- array coefficient array, 1D 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)
""" if d == 0: return c
k = c.shape[0] - 1 out = np.zeros((c.shape[0] + d,) + c.shape[1:], dtype=c.dtype)
for a in range(c.shape[0]): f = c[a] * comb(k, a) for j in range(d+1): out[a+j] += f * comb(d, j) / comb(k+d, a+j) return out
""" 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__ construct_fast
See also -------- PPoly : piecewise polynomials in 1D
Notes ----- High-order polynomials in the power basis can be numerically unstable.
"""
self.x = tuple(np.ascontiguousarray(v, dtype=np.float64) for v in x) self.c = np.asarray(c) if extrapolate is None: extrapolate = True self.extrapolate = bool(extrapolate)
ndim = len(self.x) if any(v.ndim != 1 for v in self.x): raise ValueError("x arrays must all be 1-dimensional") if any(v.size < 2 for v in self.x): raise ValueError("x arrays must all contain at least 2 points") if c.ndim < 2*ndim: raise ValueError("c must have at least 2*len(x) dimensions") if any(np.any(v[1:] - v[:-1] < 0) for v in self.x): raise ValueError("x-coordinates are not in increasing order") if any(a != b.size - 1 for a, b in zip(c.shape[ndim:2*ndim], self.x)): raise ValueError("x and c do not agree on the number of intervals")
dtype = self._get_dtype(self.c.dtype) self.c = np.ascontiguousarray(self.c, dtype=dtype)
""" 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.
""" self = object.__new__(cls) self.c = c self.x = x if extrapolate is None: extrapolate = True self.extrapolate = extrapolate return self
if np.issubdtype(dtype, np.complexfloating) \ or np.issubdtype(self.c.dtype, np.complexfloating): return np.complex_ else: return np.float_
if not self.c.flags.c_contiguous: self.c = self.c.copy() if not isinstance(self.x, tuple): self.x = tuple(self.x)
""" 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]``.
""" if extrapolate is None: extrapolate = self.extrapolate else: extrapolate = bool(extrapolate)
ndim = len(self.x)
x = _ndim_coords_from_arrays(x) x_shape = x.shape x = np.ascontiguousarray(x.reshape(-1, x.shape[-1]), dtype=np.float_)
if nu is None: nu = np.zeros((ndim,), dtype=np.intc) else: nu = np.asarray(nu, dtype=np.intc) if nu.ndim != 1 or nu.shape[0] != ndim: raise ValueError("invalid number of derivative orders nu")
dim1 = prod(self.c.shape[:ndim]) dim2 = prod(self.c.shape[ndim:2*ndim]) dim3 = prod(self.c.shape[2*ndim:]) ks = np.array(self.c.shape[:ndim], dtype=np.intc)
out = np.empty((x.shape[0], dim3), dtype=self.c.dtype) self._ensure_c_contiguous()
_ppoly.evaluate_nd(self.c.reshape(dim1, dim2, dim3), self.x, ks, x, nu, bool(extrapolate), out)
return out.reshape(x_shape[:-1] + self.c.shape[2*ndim:])
""" Compute 1D derivative along a selected dimension in-place May result to non-contiguous c array. """ if nu < 0: return self._antiderivative_inplace(-nu, axis)
ndim = len(self.x) axis = axis % ndim
# reduce order if nu == 0: # noop return else: sl = [slice(None)]*ndim sl[axis] = slice(None, -nu, None) c2 = self.c[sl]
if c2.shape[axis] == 0: # derivative of order 0 is zero shp = list(c2.shape) shp[axis] = 1 c2 = np.zeros(shp, dtype=c2.dtype)
# multiply by the correct rising factorials factor = spec.poch(np.arange(c2.shape[axis], 0, -1), nu) sl = [None]*c2.ndim sl[axis] = slice(None) c2 *= factor[sl]
self.c = c2
""" Compute 1D antiderivative along a selected dimension May result to non-contiguous c array. """ if nu <= 0: return self._derivative_inplace(-nu, axis)
ndim = len(self.x) axis = axis % ndim
perm = list(range(ndim)) perm[0], perm[axis] = perm[axis], perm[0] perm = perm + list(range(ndim, self.c.ndim))
c = self.c.transpose(perm)
c2 = np.zeros((c.shape[0] + nu,) + c.shape[1:], dtype=c.dtype) c2[:-nu] = c
# divide by the correct rising factorials factor = spec.poch(np.arange(c.shape[0], 0, -1), nu) c2[:-nu] /= factor[(slice(None),) + (None,)*(c.ndim-1)]
# fix continuity of added degrees of freedom perm2 = list(range(c2.ndim)) perm2[1], perm2[ndim+axis] = perm2[ndim+axis], perm2[1]
c2 = c2.transpose(perm2) c2 = c2.copy() _ppoly.fix_continuity(c2.reshape(c2.shape[0], c2.shape[1], -1), self.x[axis], nu-1)
c2 = c2.transpose(perm2) c2 = c2.transpose(perm)
# Done self.c = c2
""" 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]``.
""" p = self.construct_fast(self.c.copy(), self.x, self.extrapolate)
for axis, n in enumerate(nu): p._derivative_inplace(n, axis)
p._ensure_c_contiguous() return p
""" 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.
""" p = self.construct_fast(self.c.copy(), self.x, self.extrapolate)
for axis, n in enumerate(nu): p._antiderivative_inplace(n, axis)
p._ensure_c_contiguous() return p
r""" 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 1D 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 1-dimensional, an array is returned, otherwise, an NdPPoly object.
""" if extrapolate is None: extrapolate = self.extrapolate else: extrapolate = bool(extrapolate)
ndim = len(self.x) axis = int(axis) % ndim
# reuse 1D integration routines c = self.c swap = list(range(c.ndim)) swap.insert(0, swap[axis]) del swap[axis + 1] swap.insert(1, swap[ndim + axis]) del swap[ndim + axis + 1]
c = c.transpose(swap) p = PPoly.construct_fast(c.reshape(c.shape[0], c.shape[1], -1), self.x[axis], extrapolate=extrapolate) out = p.integrate(a, b, extrapolate=extrapolate)
# Construct result if ndim == 1: return out.reshape(c.shape[2:]) else: c = out.reshape(c.shape[2:]) x = self.x[:axis] + self.x[axis+1:] return self.construct_fast(c, x, extrapolate=extrapolate)
""" 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]]
"""
ndim = len(self.x)
if extrapolate is None: extrapolate = self.extrapolate else: extrapolate = bool(extrapolate)
if not hasattr(ranges, '__len__') or len(ranges) != ndim: raise ValueError("Range not a sequence of correct length")
self._ensure_c_contiguous()
# Reuse 1D integration routine c = self.c for n, (a, b) in enumerate(ranges): swap = list(range(c.ndim)) swap.insert(1, swap[ndim - n]) del swap[ndim - n + 1]
c = c.transpose(swap)
p = PPoly.construct_fast(c, self.x[n], extrapolate=extrapolate) out = p.integrate(a, b, extrapolate=extrapolate) c = out.reshape(c.shape[2:])
return c
""" 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-neighbour 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 3D 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) >>> data = f(*np.meshgrid(x, y, z, indexing='ij', sparse=True))
``data`` is now a 3D 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 neighbour 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] Trilinear interpolation. (2013, January 17). In Wikipedia, The Free Encyclopedia. Retrieved 27 Feb 2013 01:28. http://en.wikipedia.org/w/index.php?title=Trilinear_interpolation&oldid=533448871 .. [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. http://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917826-0/S0025-5718-1988-0917826-0.pdf
""" # this class is based on code originally programmed by Johannes Buchner, # see https://github.com/JohannesBuchner/regulargrid
fill_value=np.nan): if method not in ["linear", "nearest"]: raise ValueError("Method '%s' is not defined" % method) self.method = method self.bounds_error = bounds_error
if not hasattr(values, 'ndim'): # allow reasonable duck-typed values values = np.asarray(values)
if len(points) > values.ndim: raise ValueError("There are %d point arrays, but values has %d " "dimensions" % (len(points), values.ndim))
if hasattr(values, 'dtype') and hasattr(values, 'astype'): if not np.issubdtype(values.dtype, np.inexact): values = values.astype(float)
self.fill_value = fill_value if fill_value is not None: fill_value_dtype = np.asarray(fill_value).dtype if (hasattr(values, 'dtype') and not np.can_cast(fill_value_dtype, values.dtype, casting='same_kind')): raise ValueError("fill_value must be either 'None' or " "of a type compatible with values")
for i, p in enumerate(points): if not np.all(np.diff(p) > 0.): raise ValueError("The points in dimension %d must be strictly " "ascending" % i) if not np.asarray(p).ndim == 1: raise ValueError("The points in dimension %d must be " "1-dimensional" % i) if not values.shape[i] == len(p): raise ValueError("There are %d points and %d values in " "dimension %d" % (len(p), values.shape[i], i)) self.grid = tuple([np.asarray(p) for p in points]) self.values = values
""" 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. Supported are "linear" and "nearest".
""" method = self.method if method is None else method if method not in ["linear", "nearest"]: raise ValueError("Method '%s' is not defined" % method)
ndim = len(self.grid) xi = _ndim_coords_from_arrays(xi, ndim=ndim) if xi.shape[-1] != len(self.grid): raise ValueError("The requested sample points xi have dimension " "%d, but this RegularGridInterpolator has " "dimension %d" % (xi.shape[1], ndim))
xi_shape = xi.shape xi = xi.reshape(-1, xi_shape[-1])
if self.bounds_error: for i, p in enumerate(xi.T): if not np.logical_and(np.all(self.grid[i][0] <= p), np.all(p <= self.grid[i][-1])): raise ValueError("One of the requested xi is out of bounds " "in dimension %d" % i)
indices, norm_distances, out_of_bounds = self._find_indices(xi.T) if method == "linear": result = self._evaluate_linear(indices, norm_distances, out_of_bounds) elif method == "nearest": result = self._evaluate_nearest(indices, norm_distances, out_of_bounds) if not self.bounds_error and self.fill_value is not None: result[out_of_bounds] = self.fill_value
return result.reshape(xi_shape[:-1] + self.values.shape[ndim:])
# slice for broadcasting over trailing dimensions in self.values vslice = (slice(None),) + (None,)*(self.values.ndim - len(indices))
# find relevant values # each i and i+1 represents a edge edges = itertools.product(*[[i, i + 1] for i in indices]) values = 0. for edge_indices in edges: weight = 1. for ei, i, yi in zip(edge_indices, indices, norm_distances): weight *= np.where(ei == i, 1 - yi, yi) values += np.asarray(self.values[edge_indices]) * weight[vslice] return values
idx_res = [] for i, yi in zip(indices, norm_distances): idx_res.append(np.where(yi <= .5, i, i + 1)) return self.values[idx_res]
# find relevant edges between which xi are situated indices = [] # compute distance to lower edge in unity units norm_distances = [] # check for out of bounds xi out_of_bounds = np.zeros((xi.shape[1]), dtype=bool) # iterate through dimensions for x, grid in zip(xi, self.grid): i = np.searchsorted(grid, x) - 1 i[i < 0] = 0 i[i > grid.size - 2] = grid.size - 2 indices.append(i) norm_distances.append((x - grid[i]) / (grid[i + 1] - grid[i])) if not self.bounds_error: out_of_bounds += x < grid[0] out_of_bounds += x > grid[-1] return indices, norm_distances, out_of_bounds
fill_value=np.nan): """ 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
See also -------- NearestNDInterpolator : Nearest neighbour 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
""" # sanity check 'method' kwarg if method not in ["linear", "nearest", "splinef2d"]: raise ValueError("interpn only understands the methods 'linear', " "'nearest', and 'splinef2d'. You provided %s." % method)
if not hasattr(values, 'ndim'): values = np.asarray(values)
ndim = values.ndim if ndim > 2 and method == "splinef2d": raise ValueError("The method spline2fd can only be used for " "2-dimensional input data") if not bounds_error and fill_value is None and method == "splinef2d": raise ValueError("The method spline2fd does not support extrapolation.")
# sanity check consistency of input dimensions if len(points) > ndim: raise ValueError("There are %d point arrays, but values has %d " "dimensions" % (len(points), ndim)) if len(points) != ndim and method == 'splinef2d': raise ValueError("The method spline2fd can only be used for " "scalar data with one point per coordinate")
# sanity check input grid for i, p in enumerate(points): if not np.all(np.diff(p) > 0.): raise ValueError("The points in dimension %d must be strictly " "ascending" % i) if not np.asarray(p).ndim == 1: raise ValueError("The points in dimension %d must be " "1-dimensional" % i) if not values.shape[i] == len(p): raise ValueError("There are %d points and %d values in " "dimension %d" % (len(p), values.shape[i], i)) grid = tuple([np.asarray(p) for p in points])
# sanity check requested xi xi = _ndim_coords_from_arrays(xi, ndim=len(grid)) if xi.shape[-1] != len(grid): raise ValueError("The requested sample points xi have dimension " "%d, but this RegularGridInterpolator has " "dimension %d" % (xi.shape[1], len(grid)))
for i, p in enumerate(xi.T): if bounds_error and not np.logical_and(np.all(grid[i][0] <= p), np.all(p <= grid[i][-1])): raise ValueError("One of the requested xi is out of bounds " "in dimension %d" % i)
# perform interpolation if method == "linear": interp = RegularGridInterpolator(points, values, method="linear", bounds_error=bounds_error, fill_value=fill_value) return interp(xi) elif method == "nearest": interp = RegularGridInterpolator(points, values, method="nearest", bounds_error=bounds_error, fill_value=fill_value) return interp(xi) elif method == "splinef2d": xi_shape = xi.shape xi = xi.reshape(-1, xi.shape[-1])
# RectBivariateSpline doesn't support fill_value; we need to wrap here idx_valid = np.all((grid[0][0] <= xi[:, 0], xi[:, 0] <= grid[0][-1], grid[1][0] <= xi[:, 1], xi[:, 1] <= grid[1][-1]), axis=0) result = np.empty_like(xi[:, 0])
# make a copy of values for RectBivariateSpline interp = RectBivariateSpline(points[0], points[1], values[:]) result[idx_valid] = interp.ev(xi[idx_valid, 0], xi[idx_valid, 1]) result[np.logical_not(idx_valid)] = fill_value
return result.reshape(xi_shape[:-1])
# backward compatibility wrapper """ Deprecated piecewise polynomial class.
New code should use the `PPoly` class instead.
"""
warnings.warn("_ppform is deprecated -- use PPoly instead", category=DeprecationWarning)
if sort: breaks = np.sort(breaks) else: breaks = np.asarray(breaks)
PPoly.__init__(self, coeffs, breaks)
self.coeffs = self.c self.breaks = self.x self.K = self.coeffs.shape[0] self.fill = fill self.a = self.breaks[0] self.b = self.breaks[-1]
return PPoly.__call__(self, x, 0, False)
PPoly._evaluate(self, x, nu, extrapolate, out) out[~((x >= self.a) & (x <= self.b))] = self.fill return out
# Note: this spline representation is incompatible with FITPACK N = len(xk)-1 sivals = np.empty((order+1, N), dtype=float) for m in xrange(order, -1, -1): fact = spec.gamma(m+1) res = _fitpack._bspleval(xk[:-1], xk, cvals, order, m) res /= fact sivals[order-m, :] = res return cls(sivals, xk, fill=fill)
# The 3 private functions below can be called by splmake().
"""Similar to numpy.dot, but sum over last axis of a and 1st axis of b""" if b.ndim <= 2: return dot(a, b) else: axes = list(range(b.ndim)) axes.insert(-1, 0) axes.pop(0) return dot(a, b.transpose(axes))
# construct Bmatrix, and Jmatrix # e = J*c # minimize norm(e,2) given B*c=yk # if desired B can be given # conds is ignored N = len(xk)-1 K = order if B is None: B = _fitpack._bsplmat(order, xk) J = _fitpack._bspldismat(order, xk) u, s, vh = scipy.linalg.svd(B) ind = K-1 V2 = vh[-ind:,:].T V1 = vh[:-ind,:].T A = dot(J.T,J) tmp = dot(V2.T,A) Q = dot(tmp,V2) p = scipy.linalg.solve(Q, tmp) tmp = dot(V2,p) tmp = np.eye(N+K) - tmp tmp = dot(tmp,V1) tmp = dot(tmp,np.diag(1.0/s)) tmp = dot(tmp,u.T) return _dot0(tmp, yk)
# conds is a tuple of an array and a vector # giving the left-hand and the right-hand side # of the additional equations to add to B
lh = conds[0] rh = conds[1] B = np.concatenate((B, lh), axis=0) w = np.concatenate((yk, rh), axis=0) M, N = B.shape if (M > N): raise ValueError("over-specification of conditions") elif (M < N): return _find_smoothest(xk, yk, order, None, B) else: return scipy.linalg.solve(B, w)
# Remove the 3 private functions above as well when removing splmake "use make_interp_spline instead.") """ Return a representation of a spline given data-points at internal knots
Parameters ---------- xk : array_like The input array of x values of rank 1 yk : array_like The input array of y values of rank N. `yk` can be an N-d array to represent more than one curve, through the same `xk` points. The first dimension is assumed to be the interpolating dimension and is the same length of `xk`. order : int, optional Order of the spline kind : str, optional Can be 'smoothest', 'not_a_knot', 'fixed', 'clamped', 'natural', 'periodic', 'symmetric', 'user', 'mixed' and it is ignored if order < 2 conds : optional Conds
Returns ------- splmake : tuple Return a (`xk`, `cvals`, `k`) representation of a spline given data-points where the (internal) knots are at the data-points.
""" yk = np.asanyarray(yk)
order = int(order) if order < 0: raise ValueError("order must not be negative") if order == 0: return xk, yk[:-1], order elif order == 1: return xk, yk, order
try: func = eval('_find_%s' % kind) except: raise NotImplementedError
# the constraint matrix B = _fitpack._bsplmat(order, xk) coefs = func(xk, yk, order, conds, B) return xk, coefs, order
"use BSpline instead.") """ Evaluate a fixed spline represented by the given tuple at the new x-values
The `xj` values are the interior knot points. The approximation region is `xj[0]` to `xj[-1]`. If N+1 is the length of `xj`, then `cvals` should have length N+k where `k` is the order of the spline.
Parameters ---------- (xj, cvals, k) : tuple Parameters that define the fixed spline xj : array_like Interior knot points cvals : array_like Curvature k : int Order of the spline xnew : array_like Locations to calculate spline deriv : int Deriv
Returns ------- spleval : ndarray If `cvals` represents more than one curve (`cvals.ndim` > 1) and/or `xnew` is N-d, then the result is `xnew.shape` + `cvals.shape[1:]` providing the interpolation of multiple curves.
Notes ----- Internally, an additional `k`-1 knot points are added on either side of the spline.
""" (xj, cvals, k) = xck oldshape = np.shape(xnew) xx = np.ravel(xnew) sh = cvals.shape[1:] res = np.empty(xx.shape + sh, dtype=cvals.dtype) for index in np.ndindex(*sh): sl = (slice(None),) + index if issubclass(cvals.dtype.type, np.complexfloating): res[sl].real = _fitpack._bspleval(xx,xj, cvals.real[sl], k, deriv) res[sl].imag = _fitpack._bspleval(xx,xj, cvals.imag[sl], k, deriv) else: res[sl] = _fitpack._bspleval(xx, xj, cvals[sl], k, deriv) res.shape = oldshape + sh return res
# When `spltopp` gets removed, also remove the _ppform class. "use PPoly.from_spline instead.") def spltopp(xk, cvals, k): """Return a piece-wise polynomial object from a fixed-spline tuple.""" return _ppform.fromspline(xk, cvals, k)
"use Bspline class instead.") """ Interpolate a curve at new points using a spline fit
Parameters ---------- xk, yk : array_like The x and y values that define the curve. xnew : array_like The x values where spline should estimate the y values. order : int Default is 3. kind : string One of {'smoothest'} conds : Don't know Don't know
Returns ------- spline : ndarray An array of y values; the spline evaluated at the positions `xnew`.
""" return spleval(splmake(xk, yk, order=order, kind=kind, conds=conds), xnew) |