class_docstr: Radial basis function (RBF) interpolation in N dimensions.
  
  Parameters
  ----------
  y : (P, N) array_like
      Data point coordinates.
  d : (P, ...) array_like
      Data values at `y`.
  neighbors : int, optional
      If specified, the value of the interpolant at each evaluation point
      will be computed using only this many nearest data points. All the data
      points are used by default.
  smoothing : float or (P,) array_like, optional
      Smoothing parameter. The interpolant perfectly fits the data when this
      is set to 0. For large values, the interpolant approaches a least
      squares fit of a polynomial with the specified degree. Default is 0.
  kernel : str, optional
      Type of RBF. This should be one of
  
          - 'linear'               : ``-r``
          - 'thin_plate_spline'    : ``r**2 * log(r)``
          - 'cubic'                : ``r**3``
          - 'quintic'              : ``-r**5``
          - 'multiquadric'         : ``-sqrt(1 + r**2)``
          - 'inverse_multiquadric' : ``1/sqrt(1 + r**2)``
          - 'inverse_quadratic'    : ``1/(1 + r**2)``
          - 'gaussian'             : ``exp(-r**2)``
  
      Default is 'thin_plate_spline'.
  epsilon : float, optional
      Shape parameter that scales the input to the RBF. If `kernel` is
      'linear', 'thin_plate_spline', 'cubic', or 'quintic', this defaults to
      1 and can be ignored because it has the same effect as scaling the
      smoothing parameter. Otherwise, this must be specified.
  degree : int, optional
      Degree of the added polynomial. For some RBFs the interpolant may not
      be well-posed if the polynomial degree is too small. Those RBFs and
      their corresponding minimum degrees are
  
          - 'multiquadric'      : 0
          - 'linear'            : 0
          - 'thin_plate_spline' : 1
          - 'cubic'             : 1
          - 'quintic'           : 2
  
      The default value is the minimum degree for `kernel` or 0 if there is
      no minimum degree. Set this to -1 for no added polynomial.
  
  Notes
  -----
  An RBF is a scalar valued function in N-dimensional space whose value at
  :math:`x` can be expressed in terms of :math:`r=||x - c||`, where :math:`c`
  is the center of the RBF.
  
  An RBF interpolant for the vector of data values :math:`d`, which are from
  locations :math:`y`, is a linear combination of RBFs centered at :math:`y`
  plus a polynomial with a specified degree. The RBF interpolant is written
  as
  
  .. math::
      f(x) = K(x, y) a + P(x) b,
  
  where :math:`K(x, y)` is a matrix of RBFs with centers at :math:`y`
  evaluated at the points :math:`x`, and :math:`P(x)` is a matrix of
  monomials, which span polynomials with the specified degree, evaluated at
  :math:`x`. The coefficients :math:`a` and :math:`b` are the solution to the
  linear equations
  
  .. math::
      (K(y, y) + \lambda I) a + P(y) b = d
  
  and
  
  .. math::
      P(y)^T a = 0,
  
  where :math:`\lambda` is a non-negative smoothing parameter that controls
  how well we want to fit the data. The data are fit exactly when the
  smoothing parameter is 0.
  
  The above system is uniquely solvable if the following requirements are
  met:
  
      - :math:`P(y)` must have full column rank. :math:`P(y)` always has full
        column rank when `degree` is -1 or 0. When `degree` is 1,
        :math:`P(y)` has full column rank if the data point locations are not
        all collinear (N=2), coplanar (N=3), etc.
      - If `kernel` is 'multiquadric', 'linear', 'thin_plate_spline',
        'cubic', or 'quintic', then `degree` must not be lower than the
        minimum value listed above.
      - If `smoothing` is 0, then each data point location must be distinct.
  
  When using an RBF that is not scale invariant ('multiquadric',
  'inverse_multiquadric', 'inverse_quadratic', or 'gaussian'), an appropriate
  shape parameter must be chosen (e.g., through cross validation). Smaller
  values for the shape parameter correspond to wider RBFs. The problem can
  become ill-conditioned or singular when the shape parameter is too small.
  
  The memory required to solve for the RBF interpolation coefficients
  increases quadratically with the number of data points, which can become
  impractical when interpolating more than about a thousand data points.
  To overcome memory limitations for large interpolation problems, the
  `neighbors` argument can be specified to compute an RBF interpolant for
  each evaluation point using only the nearest data points.
  
  .. versionadded:: 1.7.0
  
  See Also
  --------
  NearestNDInterpolator
  LinearNDInterpolator
  CloughTocher2DInterpolator
  
  References
  ----------
  .. [1] Fasshauer, G., 2007. Meshfree Approximation Methods with Matlab.
      World Scientific Publishing Co.
  
  .. [2] http://amadeus.math.iit.edu/~fass/603_ch3.pdf
  
  .. [3] Wahba, G., 1990. Spline Models for Observational Data. SIAM.
  
  .. [4] http://pages.stat.wisc.edu/~wahba/stat860public/lect/lect8/lect8.pdf
  
  Examples
  --------
  Demonstrate interpolating scattered data to a grid in 2-D.
  
  >>> import matplotlib.pyplot as plt
  >>> from scipy.interpolate import RBFInterpolator
  >>> from scipy.stats.qmc import Halton
  
  >>> rng = np.random.default_rng()
  >>> xobs = 2*Halton(2, seed=rng).random(100) - 1
  >>> yobs = np.sum(xobs, axis=1)*np.exp(-6*np.sum(xobs**2, axis=1))
  
  >>> xgrid = np.mgrid[-1:1:50j, -1:1:50j]
  >>> xflat = xgrid.reshape(2, -1).T
  >>> yflat = RBFInterpolator(xobs, yobs)(xflat)
  >>> ygrid = yflat.reshape(50, 50)
  
  >>> fig, ax = plt.subplots()
  >>> ax.pcolormesh(*xgrid, ygrid, vmin=-0.25, vmax=0.25, shading='gouraud')
  >>> p = ax.scatter(*xobs.T, c=yobs, s=50, ec='k', vmin=-0.25, vmax=0.25)
  >>> fig.colorbar(p)
  >>> plt.show()
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code: def __init__(
      self,
      y,
      d,
      neighbors=None,
      smoothing=0.0,
      kernel="thin_plate_spline",
      epsilon=None,
      degree=None,
  ):
      y = np.asarray(y, dtype=float, order="C")
      if y.ndim != 2:
          raise ValueError("`y` must be a 2-dimensional array.")
  
      ny, ndim = y.shape
  
      d_dtype = complex if np.iscomplexobj(d) else float
      d = np.asarray(d, dtype=d_dtype, order="C")
      if d.shape[0] != ny:
          raise ValueError(
              f"Expected the first axis of `d` to have length {ny}."
          )
  
      d_shape = d.shape[1:]
      d = d.reshape((ny, -1))
      # If `d` is complex, convert it to a float array with twice as many
      # columns. Otherwise, the LHS matrix would need to be converted to
      # complex and take up 2x more memory than necessary.
      d = d.view(float)
  
      if np.isscalar(smoothing):
          smoothing = np.full(ny, smoothing, dtype=float)
      else:
          smoothing = np.asarray(smoothing, dtype=float, order="C")
          if smoothing.shape != (ny,):
              raise ValueError(
                  "Expected `smoothing` to be a scalar or have shape "
                  f"({ny},)."
              )
  
      kernel = kernel.lower()
      if kernel not in _AVAILABLE:
          raise ValueError(f"`kernel` must be one of {_AVAILABLE}.")
  
      if epsilon is None:
          if kernel in _SCALE_INVARIANT:
              epsilon = 1.0
          else:
              raise ValueError(
                  "`epsilon` must be specified if `kernel` is not one of "
                  f"{_SCALE_INVARIANT}."
              )
      else:
          epsilon = float(epsilon)
  
      min_degree = _NAME_TO_MIN_DEGREE.get(kernel, -1)
      if degree is None:
          degree = max(min_degree, 0)
      else:
          degree = int(degree)
          if degree < -1:
              raise ValueError("`degree` must be at least -1.")
          elif degree < min_degree:
              warnings.warn(
                  f"`degree` should not be below {min_degree} when `kernel` "
                  f"is '{kernel}'. The interpolant may not be uniquely "
                  "solvable, and the smoothing parameter may have an "
                  "unintuitive effect.",
                  UserWarning,
              )
  
      if neighbors is None:
          nobs = ny
      else:
          # Make sure the number of nearest neighbors used for interpolation
          # does not exceed the number of observations.
          neighbors = int(min(neighbors, ny))
          nobs = neighbors
  
      powers = _monomial_powers(ndim, degree)
      # The polynomial matrix must have full column rank in order for the
      # interpolant to be well-posed, which is not possible if there are
      # fewer observations than monomials.
      if powers.shape[0] > nobs:
          raise ValueError(
              f"At least {powers.shape[0]} data points are required when "
              f"`degree` is {degree} and the number of dimensions is {ndim}."
          )
  
      if neighbors is None:
          shift, scale, coeffs = _build_and_solve_system(
              y, d, smoothing, kernel, epsilon, powers
          )
  
          # Make these attributes private since they do not always exist.
          self._shift = shift
          self._scale = scale
          self._coeffs = coeffs
  
      else:
          self._tree = KDTree(y)
  
      self.y = y
      self.d = d
      self.d_shape = d_shape
      self.d_dtype = d_dtype
      self.neighbors = neighbors
      self.smoothing = smoothing
      self.kernel = kernel
      self.epsilon = epsilon
      self.powers = powers
  
file_path: catboost/contrib/python/scipy/py3/scipy/interpolate/_rbfinterp.py
imports: import warnings
  from itertools import combinations_with_replacement
  import numpy
  from numpy.linalg import LinAlgError
  from scipy.spatial import KDTree
  from scipy.special import comb
  from scipy.linalg.lapack import dgesv
  from ._rbfinterp_pythran import _build_system, _evaluate, _polynomial_matrix
label_desc: Use image node_id 1 to create a new RBFInterpolator object  with example: obj = RBFInterpolator(y, d, neighbors, smoothing, kernel, epsilon, degree)