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# Source code for astropy.modeling.polynomial

    # Licensed under a 3-clause BSD style license - see LICENSE.rst
    """
    This module contains models representing polynomials and polynomial series.

    """

    # pylint: disable=invalid-name
    from math import comb
    from textwrap import indent

    import numpy as np

    from .core import FittableModel, Model
    from .functional_models import Shift
    from .parameters import Parameter
    from .utils import _validate_domain_window, poly_map_domain

    __all__ = [
        "SIP",
        "Chebyshev1D",
        "Chebyshev2D",
        "Hermite1D",
        "Hermite2D",
        "InverseSIP",
        "Legendre1D",
        "Legendre2D",
        "OrthoPolynomialBase",
        "Polynomial1D",
        "Polynomial2D",
        "PolynomialModel",
    ]

    class PolynomialBase(FittableModel):
        """
        Base class for all polynomial-like models with an arbitrary number of
        parameters in the form of coefficients.

        In this case Parameter instances are returned through the class's
        ``__getattr__`` rather than through class descriptors.
        """

        # Default _param_names list; this will be filled in by the implementation's
        # __init__
        _param_names = ()

        linear = True
        col_fit_deriv = False

        @property
        def param_names(self):
            """Coefficient names generated based on the model's polynomial degree
            and number of dimensions.

            Subclasses should implement this to return parameter names in the
            desired format.

            On most `Model` classes this is a class attribute, but for polynomial
            models it is an instance attribute since each polynomial model instance
            can have different parameters depending on the degree of the polynomial
            and the number of dimensions, for example.
            """
            return self._param_names

    class PolynomialModel(PolynomialBase):
        """
        Base class for polynomial models.

        Its main purpose is to determine how many coefficients are needed
        based on the polynomial order and dimension and to provide their
        default values, names and ordering.
        """

        def __init__(
            self, degree, n_models=None, model_set_axis=None, name=None, meta=None, **params
        ):
            self._degree = degree
            self._order = self.get_num_coeff(self.n_inputs)
            self._param_names = self._generate_coeff_names(self.n_inputs)
            if n_models:
                if model_set_axis is None:
                    model_set_axis = 0
                minshape = (1,) * model_set_axis + (n_models,)
            else:
                minshape = ()
            for param_name in self._param_names:
                self._parameters_[param_name] = Parameter(
                    param_name, default=np.zeros(minshape)
                )

            super().__init__(
                n_models=n_models,
                model_set_axis=model_set_axis,
                name=name,
                meta=meta,
                **params,
            )

        @property
        def degree(self):
            """Degree of polynomial."""
            return self._degree

        def get_num_coeff(self, ndim):
            """
            Return the number of coefficients in one parameter set.
            """
            if self.degree < 0:
                raise ValueError("Degree of polynomial must be positive or null")
            # deg+1 is used to account for the difference between iraf using
            # degree and numpy using exact degree
            if ndim != 1:
                nmixed = comb(self.degree, ndim)
            else:
                nmixed = 0
            numc = self.degree * ndim + nmixed + 1
            return numc

        def _invlex(self):
            c = []
            lencoeff = self.degree + 1
            for i in range(lencoeff):
                for j in range(lencoeff):
                    if i + j <= self.degree:
                        c.append((j, i))
            return c[::-1]

        def _generate_coeff_names(self, ndim):
            names = []
            if ndim == 1:
                for n in range(self._order):
                    names.append(f"c{n}")
            else:
                for i in range(self.degree + 1):
                    names.append(f"c{i}_{0}")
                for i in range(1, self.degree + 1):
                    names.append(f"c{0}_{i}")
                for i in range(1, self.degree):
                    for j in range(1, self.degree):
                        if i + j < self.degree + 1:
                            names.append(f"c{i}_{j}")
            return tuple(names)

    class _PolyDomainWindow1D(PolynomialModel):
        """
        This class sets ``domain`` and ``window`` of 1D polynomials.
        """

        def __init__(
            self,
            degree,
            domain=None,
            window=None,
            n_models=None,
            model_set_axis=None,
            name=None,
            meta=None,
            **params,
        ):
            super().__init__(
                degree, n_models, model_set_axis, name=name, meta=meta, **params
            )

            self._set_default_domain_window(domain, window)

        @property
        def window(self):
            return self._window

        @window.setter
        def window(self, val):
            self._window = _validate_domain_window(val)

        @property
        def domain(self):
            return self._domain

        @domain.setter
        def domain(self, val):
            self._domain = _validate_domain_window(val)

        def _set_default_domain_window(self, domain, window):
            """
            This method sets the ``domain`` and ``window`` attributes on 1D subclasses.

            """
            self._default_domain_window = {"domain": None, "window": (-1, 1)}
            self.window = window or (-1, 1)
            self.domain = domain

        def __repr__(self):
            return self._format_repr(
                [self.degree],
                kwargs={"domain": self.domain, "window": self.window},
                defaults=self._default_domain_window,
            )

        def __str__(self):
            return self._format_str(
                [("Degree", self.degree), ("Domain", self.domain), ("Window", self.window)],
                self._default_domain_window,
            )

    class OrthoPolynomialBase(PolynomialBase):
        """
        This is a base class for the 2D Chebyshev and Legendre models.

        The polynomials implemented here require a maximum degree in x and y.

        For explanation of ``x_domain``, ``y_domain``, ```x_window`` and ```y_window``
        see :ref:`Notes regarding usage of domain and window <astropy:domain-window-note>`.

        Parameters
        ----------
        x_degree : int
            degree in x
        y_degree : int
            degree in y
        x_domain : tuple or None, optional
            domain of the x independent variable
        x_window : tuple or None, optional
            range of the x independent variable
        y_domain : tuple or None, optional
            domain of the y independent variable
        y_window : tuple or None, optional
            range of the y independent variable
        **params : dict
            {keyword: value} pairs, representing {parameter_name: value}
        """

        n_inputs = 2
        n_outputs = 1

        def __init__(
            self,
            x_degree,
            y_degree,
            x_domain=None,
            x_window=None,
            y_domain=None,
            y_window=None,
            n_models=None,
            model_set_axis=None,
            name=None,
            meta=None,
            **params,
        ):
            self.x_degree = x_degree
            self.y_degree = y_degree
            self._order = self.get_num_coeff()
            # Set the ``x/y_domain`` and ``x/y_wndow`` attributes in subclasses.
            self._default_domain_window = {
                "x_window": (-1, 1),
                "y_window": (-1, 1),
                "x_domain": None,
                "y_domain": None,
            }

            self.x_window = x_window or self._default_domain_window["x_window"]
            self.y_window = y_window or self._default_domain_window["y_window"]
            self.x_domain = x_domain
            self.y_domain = y_domain

            self._param_names = self._generate_coeff_names()
            if n_models:
                if model_set_axis is None:
                    model_set_axis = 0
                minshape = (1,) * model_set_axis + (n_models,)
            else:
                minshape = ()

            for param_name in self._param_names:
                self._parameters_[param_name] = Parameter(
                    param_name, default=np.zeros(minshape)
                )
            super().__init__(
                n_models=n_models,
                model_set_axis=model_set_axis,
                name=name,
                meta=meta,
                **params,
            )

        @property
        def x_domain(self):
            return self._x_domain

        @x_domain.setter
        def x_domain(self, val):
            self._x_domain = _validate_domain_window(val)

        @property
        def y_domain(self):
            return self._y_domain

        @y_domain.setter
        def y_domain(self, val):
            self._y_domain = _validate_domain_window(val)

        @property
        def x_window(self):
            return self._x_window

        @x_window.setter
        def x_window(self, val):
            self._x_window = _validate_domain_window(val)

        @property
        def y_window(self):
            return self._y_window

        @y_window.setter
        def y_window(self, val):
            self._y_window = _validate_domain_window(val)

        def __repr__(self):
            return self._format_repr(
                [self.x_degree, self.y_degree],
                kwargs={
                    "x_domain": self.x_domain,
                    "y_domain": self.y_domain,
                    "x_window": self.x_window,
                    "y_window": self.y_window,
                },
                defaults=self._default_domain_window,
            )

        def __str__(self):
            return self._format_str(
                [
                    ("X_Degree", self.x_degree),
                    ("Y_Degree", self.y_degree),
                    ("X_Domain", self.x_domain),
                    ("Y_Domain", self.y_domain),
                    ("X_Window", self.x_window),
                    ("Y_Window", self.y_window),
                ],
                self._default_domain_window,
            )

        def get_num_coeff(self):
            """
            Determine how many coefficients are needed.

            Returns
            -------
            numc : int
                number of coefficients
            """
            if self.x_degree < 0 or self.y_degree < 0:
                raise ValueError("Degree of polynomial must be positive or null")

            return (self.x_degree + 1) * (self.y_degree + 1)

        def _invlex(self):
            # TODO: This is a very slow way to do this; fix it and related methods
            # like _alpha
            c = []
            xvar = np.arange(self.x_degree + 1)
            yvar = np.arange(self.y_degree + 1)
            for j in yvar:
                for i in xvar:
                    c.append((i, j))
            return np.array(c[::-1])

        def invlex_coeff(self, coeffs):
            invlex_coeffs = []
            xvar = np.arange(self.x_degree + 1)
            yvar = np.arange(self.y_degree + 1)
            for j in yvar:
                for i in xvar:
                    name = f"c{i}_{j}"
                    coeff = coeffs[self.param_names.index(name)]
                    invlex_coeffs.append(coeff)
            return np.array(invlex_coeffs[::-1])

        def _alpha(self):
            invlexdeg = self._invlex()
            invlexdeg[:, 1] = invlexdeg[:, 1] + self.x_degree + 1
            nx = self.x_degree + 1
            ny = self.y_degree + 1
            alpha = np.zeros((ny * nx + 3, ny + nx))
            for n in range(len(invlexdeg)):
                alpha[n][invlexdeg[n]] = [1, 1]
                alpha[-2, 0] = 1
                alpha[-3, nx] = 1
            return alpha

        def imhorner(self, x, y, coeff):
            _coeff = list(coeff)
            _coeff.extend([0, 0, 0])
            alpha = self._alpha()
            r0 = _coeff[0]
            nalpha = len(alpha)

            karr = np.diff(alpha, axis=0)
            kfunc = self._fcache(x, y)
            x_terms = self.x_degree + 1
            y_terms = self.y_degree + 1
            nterms = x_terms + y_terms
            for n in range(1, nterms + 1 + 3):
                setattr(self, "r" + str(n), 0.0)

            for n in range(1, nalpha):
                k = karr[n - 1].nonzero()[0].max() + 1
                rsum = 0
                for i in range(1, k + 1):
                    rsum = rsum + getattr(self, "r" + str(i))
                val = kfunc[k - 1] * (r0 + rsum)
                setattr(self, "r" + str(k), val)
                r0 = _coeff[n]
                for i in range(1, k):
                    setattr(self, "r" + str(i), 0.0)
            result = r0
            for i in range(1, nterms + 1 + 3):
                result = result + getattr(self, "r" + str(i))
            return result

        def _generate_coeff_names(self):
            names = []
            for j in range(self.y_degree + 1):
                for i in range(self.x_degree + 1):
                    names.append(f"c{i}_{j}")
            return tuple(names)

        def _fcache(self, x, y):
            """
            Computation and store the individual functions.

            To be implemented by subclasses"
            """
            raise NotImplementedError("Subclasses should implement this")

        def evaluate(self, x, y, *coeffs):
            if self.x_domain is not None:
                x = poly_map_domain(x, self.x_domain, self.x_window)
            if self.y_domain is not None:
                y = poly_map_domain(y, self.y_domain, self.y_window)
            invcoeff = self.invlex_coeff(coeffs)
            return self.imhorner(x, y, invcoeff)

        def prepare_inputs(self, x, y, **kwargs):
            inputs, broadcasted_shapes = super().prepare_inputs(x, y, **kwargs)

            x, y = inputs

            if x.shape != y.shape:
                raise ValueError("Expected input arrays to have the same shape")

            return (x, y), broadcasted_shapes

    class Chebyshev1D(_PolyDomainWindow1D):
        r"""
        Univariate Chebyshev series.

        It is defined as:

        .. math::

            P(x) = \sum_{i=0}^{i=n}C_{i} * T_{i}(x)

        where ``T_i(x)`` is the corresponding Chebyshev polynomial of the 1st kind.

        For explanation of ```domain``, and ``window`` see
        :ref:`Notes regarding usage of domain and window <domain-window-note>`.

        Parameters
        ----------
        degree : int
            degree of the series
        domain : tuple or None, optional
        window : tuple or None, optional
            If None, it is set to (-1, 1)
            Fitters will remap the domain to this window.
        **params : dict
            keyword : value pairs, representing parameter_name: value

        Notes
        -----
        This model does not support the use of units/quantities, because each term
        in the sum of Chebyshev polynomials is a polynomial in x - since the
        coefficients within each Chebyshev polynomial are fixed, we can't use
        quantities for x since the units would not be compatible. For example, the
        third Chebyshev polynomial (T2) is 2x^2-1, but if x was specified with
        units, 2x^2 and -1 would have incompatible units.
        """

        n_inputs = 1
        n_outputs = 1

        _separable = True

        def __init__(
            self,
            degree,
            domain=None,
            window=None,
            n_models=None,
            model_set_axis=None,
            name=None,
            meta=None,
            **params,
        ):
            super().__init__(
                degree,
                domain=domain,
                window=window,
                n_models=n_models,
                model_set_axis=model_set_axis,
                name=name,
                meta=meta,
                **params,
            )

        def fit_deriv(self, x, *params):
            """
            Computes the Vandermonde matrix.

            Parameters
            ----------
            x : ndarray
                input
            *params
                throw-away parameter list returned by non-linear fitters

            Returns
            -------
            result : ndarray
                The Vandermonde matrix
            """
            x = np.array(x, dtype=float, copy=None, ndmin=1)
            v = np.empty((self.degree + 1,) + x.shape, dtype=x.dtype)
            v[0] = 1
            if self.degree > 0:
                x2 = 2 * x
                v[1] = x
                for i in range(2, self.degree + 1):
                    v[i] = v[i - 1] * x2 - v[i - 2]
            return np.rollaxis(v, 0, v.ndim)

        def prepare_inputs(self, x, **kwargs):
            inputs, broadcasted_shapes = super().prepare_inputs(x, **kwargs)

            x = inputs[0]

            return (x,), broadcasted_shapes

        def evaluate(self, x, *coeffs):
            if self.domain is not None:
                x = poly_map_domain(x, self.domain, self.window)
            return self.clenshaw(x, coeffs)

        @staticmethod
        def clenshaw(x, coeffs):
            """Evaluates the polynomial using Clenshaw's algorithm."""
            if len(coeffs) == 1:
                c0 = coeffs[0]
                c1 = 0
            elif len(coeffs) == 2:
                c0 = coeffs[0]
                c1 = coeffs[1]
            else:
                x2 = 2 * x
                c0 = coeffs[-2]
                c1 = coeffs[-1]
                for i in range(3, len(coeffs) + 1):
                    tmp = c0
                    c0 = coeffs[-i] - c1
                    c1 = tmp + c1 * x2
            return c0 + c1 * x

    class Hermite1D(_PolyDomainWindow1D):
        r"""
        Univariate Hermite series.

        It is defined as:

        .. math::

            P(x) = \sum_{i=0}^{i=n}C_{i} * H_{i}(x)

        where ``H_i(x)`` is the corresponding Hermite polynomial ("Physicist's kind").

        For explanation of ``domain``, and ``window`` see
        :ref:`Notes regarding usage of domain and window <domain-window-note>`.

        Parameters
        ----------
        degree : int
            degree of the series
        domain : tuple or None, optional
        window : tuple or None, optional
            If None, it is set to (-1, 1)
            Fitters will remap the domain to this window
        **params : dict
            keyword : value pairs, representing parameter_name: value

        Notes
        -----
        This model does not support the use of units/quantities, because each term
        in the sum of Hermite polynomials is a polynomial in x - since the
        coefficients within each Hermite polynomial are fixed, we can't use
        quantities for x since the units would not be compatible. For example, the
        third Hermite polynomial (H2) is 4x^2-2, but if x was specified with units,
        4x^2 and -2 would have incompatible units.
        """

        n_inputs = 1
        n_outputs = 1

        _separable = True

        def __init__(
            self,
            degree,
            domain=None,
            window=None,
            n_models=None,
            model_set_axis=None,
            name=None,
            meta=None,
            **params,
        ):
            super().__init__(
                degree,
                domain,
                window,
                n_models=n_models,
                model_set_axis=model_set_axis,
                name=name,
                meta=meta,
                **params,
            )

        def fit_deriv(self, x, *params):
            """
            Computes the Vandermonde matrix.

            Parameters
            ----------
            x : ndarray
                input
            *params
                throw-away parameter list returned by non-linear fitters

            Returns
            -------
            result : ndarray
                The Vandermonde matrix
            """
            x = np.array(x, dtype=float, copy=None, ndmin=1)
            v = np.empty((self.degree + 1,) + x.shape, dtype=x.dtype)
            v[0] = 1
            if self.degree > 0:
                x2 = 2 * x
                v[1] = 2 * x
                for i in range(2, self.degree + 1):
                    v[i] = x2 * v[i - 1] - 2 * (i - 1) * v[i - 2]
            return np.rollaxis(v, 0, v.ndim)

        def prepare_inputs(self, x, **kwargs):
            inputs, broadcasted_shapes = super().prepare_inputs(x, **kwargs)

            x = inputs[0]

            return (x,), broadcasted_shapes

        def evaluate(self, x, *coeffs):
            if self.domain is not None:
                x = poly_map_domain(x, self.domain, self.window)
            return self.clenshaw(x, coeffs)

        @staticmethod
        def clenshaw(x, coeffs):
            x2 = x * 2
            if len(coeffs) == 1:
                c0 = coeffs[0]
                c1 = 0
            elif len(coeffs) == 2:
                c0 = coeffs[0]
                c1 = coeffs[1]
            else:
                nd = len(coeffs)
                c0 = coeffs[-2]
                c1 = coeffs[-1]
                for i in range(3, len(coeffs) + 1):
                    temp = c0
                    nd = nd - 1
                    c0 = coeffs[-i] - c1 * (2 * (nd - 1))
                    c1 = temp + c1 * x2
            return c0 + c1 * x2

    class Hermite2D(OrthoPolynomialBase):
        r"""
        Bivariate Hermite series.

        It is defined as

        .. math:: P_{nm}(x,y) = \sum_{n,m=0}^{n=d,m=d}C_{nm} H_n(x) H_m(y)

        where ``H_n(x)`` and ``H_m(y)`` are Hermite polynomials.

        For explanation of ``x_domain``, ``y_domain``, ``x_window`` and ``y_window``
        see :ref:`Notes regarding usage of domain and window <domain-window-note>`.

        Parameters
        ----------
        x_degree : int
            degree in x
        y_degree : int
            degree in y
        x_domain : tuple or None, optional
            domain of the x independent variable
        y_domain : tuple or None, optional
            domain of the y independent variable
        x_window : tuple or None, optional
            range of the x independent variable
            If None, it is set to (-1, 1)
            Fitters will remap the domain to this window
        y_window : tuple or None, optional
            range of the y independent variable
            If None, it is set to (-1, 1)
            Fitters will remap the domain to this window
        **params : dict
            keyword: value pairs, representing parameter_name: value

        Notes
        -----
        This model does not support the use of units/quantities, because each term
        in the sum of Hermite polynomials is a polynomial in x and/or y - since the
        coefficients within each Hermite polynomial are fixed, we can't use
        quantities for x and/or y since the units would not be compatible. For
        example, the third Hermite polynomial (H2) is 4x^2-2, but if x was
        specified with units, 4x^2 and -2 would have incompatible units.
        """

        _separable = False

        def __init__(
            self,
            x_degree,
            y_degree,
            x_domain=None,
            x_window=None,
            y_domain=None,
            y_window=None,
            n_models=None,
            model_set_axis=None,
            name=None,
            meta=None,
            **params,
        ):
            super().__init__(
                x_degree,
                y_degree,
                x_domain=x_domain,
                y_domain=y_domain,
                x_window=x_window,
                y_window=y_window,
                n_models=n_models,
                model_set_axis=model_set_axis,
                name=name,
                meta=meta,
                **params,
            )

        def _fcache(self, x, y):
            """
            Calculate the individual Hermite functions once and store them in a
            dictionary to be reused.
            """
            x_terms = self.x_degree + 1
            y_terms = self.y_degree + 1
            kfunc = {}
            kfunc[0] = np.ones(x.shape)
            kfunc[1] = 2 * x.copy()
            kfunc[x_terms] = np.ones(y.shape)
            kfunc[x_terms + 1] = 2 * y.copy()
            for n in range(2, x_terms):
                kfunc[n] = 2 * x * kfunc[n - 1] - 2 * (n - 1) * kfunc[n - 2]
            for n in range(x_terms + 2, x_terms + y_terms):
                kfunc[n] = 2 * y * kfunc[n - 1] - 2 * (n - 1) * kfunc[n - 2]
            return kfunc

        def fit_deriv(self, x, y, *params):
            """
            Derivatives with respect to the coefficients.

            This is an array with Hermite polynomials:

            .. math::

                H_{x_0}H_{y_0}, H_{x_1}H_{y_0}...H_{x_n}H_{y_0}...H_{x_n}H_{y_m}

            Parameters
            ----------
            x : ndarray
                input
            y : ndarray
                input
            *params
                throw-away parameter list returned by non-linear fitters

            Returns
            -------
            result : ndarray
                The Vandermonde matrix
            """
            if x.shape != y.shape:
                raise ValueError("x and y must have the same shape")

            x = x.ravel()
            y = y.ravel()
            x_deriv = self._hermderiv1d(x, self.x_degree + 1).T
            y_deriv = self._hermderiv1d(y, self.y_degree + 1).T

            ij = []
            for i in range(self.y_degree + 1):
                for j in range(self.x_degree + 1):
                    ij.append(x_deriv[j] * y_deriv[i])

            v = np.array(ij)
            return v.T

        def _hermderiv1d(self, x, deg):
            """
            Derivative of 1D Hermite series.
            """
            x = np.array(x, dtype=float, copy=None, ndmin=1)
            d = np.empty((deg + 1, len(x)), dtype=x.dtype)
            d[0] = x * 0 + 1
            if deg > 0:
                x2 = 2 * x
                d[1] = x2
                for i in range(2, deg + 1):
                    d[i] = x2 * d[i - 1] - 2 * (i - 1) * d[i - 2]
            return np.rollaxis(d, 0, d.ndim)

    class Legendre1D(_PolyDomainWindow1D):
        r"""
        Univariate Legendre series.

        It is defined as:

        .. math::

            P(x) = \sum_{i=0}^{i=n}C_{i} * L_{i}(x)

        where ``L_i(x)`` is the corresponding Legendre polynomial.

        For explanation of ``domain``, and ``window`` see
        :ref:`Notes regarding usage of domain and window <domain-window-note>`.

        Parameters
        ----------
        degree : int
            degree of the series
        domain : tuple or None, optional
        window : tuple or None, optional
            If None, it is set to (-1, 1)
            Fitters will remap the domain to this window
        **params : dict
            keyword: value pairs, representing parameter_name: value

        Notes
        -----
        This model does not support the use of units/quantities, because each term
        in the sum of Legendre polynomials is a polynomial in x - since the
        coefficients within each Legendre polynomial are fixed, we can't use
        quantities for x since the units would not be compatible. For example, the
        third Legendre polynomial (P2) is 1.5x^2-0.5, but if x was specified with
        units, 1.5x^2 and -0.5 would have incompatible units.
        """

        n_inputs = 1
        n_outputs = 1

        _separable = True

        def __init__(
            self,
            degree,
            domain=None,
            window=None,
            n_models=None,
            model_set_axis=None,
            name=None,
            meta=None,
            **params,
        ):
            super().__init__(
                degree,
                domain,
                window,
                n_models=n_models,
                model_set_axis=model_set_axis,
                name=name,
                meta=meta,
                **params,
            )

        def prepare_inputs(self, x, **kwargs):
            inputs, broadcasted_shapes = super().prepare_inputs(x, **kwargs)

            x = inputs[0]

            return (x,), broadcasted_shapes

        def evaluate(self, x, *coeffs):
            if self.domain is not None:
                x = poly_map_domain(x, self.domain, self.window)
            return self.clenshaw(x, coeffs)

        def fit_deriv(self, x, *params):
            """
            Computes the Vandermonde matrix.

            Parameters
            ----------
            x : ndarray
                input
            *params
                throw-away parameter list returned by non-linear fitters

            Returns
            -------
            result : ndarray
                The Vandermonde matrix
            """
            x = np.array(x, dtype=float, copy=None, ndmin=1)
            v = np.empty((self.degree + 1,) + x.shape, dtype=x.dtype)
            v[0] = 1
            if self.degree > 0:
                v[1] = x
                for i in range(2, self.degree + 1):
                    v[i] = (v[i - 1] * x * (2 * i - 1) - v[i - 2] * (i - 1)) / i
            return np.rollaxis(v, 0, v.ndim)

        @staticmethod
        def clenshaw(x, coeffs):
            if len(coeffs) == 1:
                c0 = coeffs[0]
                c1 = 0
            elif len(coeffs) == 2:
                c0 = coeffs[0]
                c1 = coeffs[1]
            else:
                nd = len(coeffs)
                c0 = coeffs[-2]
                c1 = coeffs[-1]
                for i in range(3, len(coeffs) + 1):
                    tmp = c0
                    nd = nd - 1
                    c0 = coeffs[-i] - (c1 * (nd - 1)) / nd
                    c1 = tmp + (c1 * x * (2 * nd - 1)) / nd
            return c0 + c1 * x

    class Polynomial1D(_PolyDomainWindow1D):
        r"""
        1D Polynomial model.

        It is defined as:

        .. math::

            P = \sum_{i=0}^{i=n}C_{i} * x^{i}

        For explanation of ``domain``, and ``window`` see
        :ref:`Notes regarding usage of domain and window <domain-window-note>`.

        Parameters
        ----------
        degree : int
            degree of the series
        domain : tuple or None, optional
            If None, it is set to (-1, 1)
        window : tuple or None, optional
            If None, it is set to (-1, 1)
            Fitters will remap the domain to this window
        **params : dict
            keyword: value pairs, representing parameter_name: value

        """

        n_inputs = 1
        n_outputs = 1

        _separable = True

        def __init__(
            self,
            degree,
            domain=None,
            window=None,
            n_models=None,
            model_set_axis=None,
            name=None,
            meta=None,
            **params,
        ):
            super().__init__(
                degree,
                domain,
                window,
                n_models=n_models,
                model_set_axis=model_set_axis,
                name=name,
                meta=meta,
                **params,
            )

            # Set domain separately because it's different from
            # the orthogonal polynomials.
            self._default_domain_window = {
                "domain": (-1, 1),
                "window": (-1, 1),
            }
            self.domain = domain or self._default_domain_window["domain"]
            self.window = window or self._default_domain_window["window"]

        def prepare_inputs(self, x, **kwargs):
            inputs, broadcasted_shapes = super().prepare_inputs(x, **kwargs)

            x = inputs[0]
            return (x,), broadcasted_shapes

        def evaluate(self, x, *coeffs):
            if self.domain is not None:
                x = poly_map_domain(x, self.domain, self.window)
            return self.horner(x, coeffs)

        def fit_deriv(self, x, *params):
            """
            Computes the Vandermonde matrix.

            Parameters
            ----------
            x : ndarray
                input
            *params
                throw-away parameter list returned by non-linear fitters

            Returns
            -------
            result : ndarray
                The Vandermonde matrix
            """
            v = np.empty((self.degree + 1,) + x.shape, dtype=float)
            v[0] = 1
            if self.degree > 0:
                v[1] = x
                for i in range(2, self.degree + 1):
                    v[i] = v[i - 1] * x
            return np.rollaxis(v, 0, v.ndim)

        @staticmethod
        def horner(x, coeffs):
            if len(coeffs) == 1:
                c0 = coeffs[-1] * np.ones_like(x, subok=False)
            else:
                c0 = coeffs[-1]
                for i in range(2, len(coeffs) + 1):
                    c0 = coeffs[-i] + c0 * x
            return c0

        @property
        def input_units(self):
            if self.degree == 0 or self.c1.input_unit is None:
                return None
            else:
                return {self.inputs[0]: self.c0.input_unit / self.c1.input_unit}

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            mapping = {}
            for i in range(self.degree + 1):
                par = getattr(self, f"c{i}")
                mapping[par.name] = (
                    outputs_unit[self.outputs[0]] / inputs_unit[self.inputs[0]] ** i
                )
            return mapping

    class Polynomial2D(PolynomialModel):
        r"""
        2D Polynomial  model.

        Represents a general polynomial of degree n:

        .. math::

            P(x,y) = c_{00} + c_{10}x + ...+ c_{n0}x^n + c_{01}y + ...+ c_{0n}y^n
            + c_{11}xy + c_{12}xy^2 + ... + c_{1(n-1)}xy^{n-1}+ ... + c_{(n-1)1}x^{n-1}y

        For explanation of ``x_domain``, ``y_domain``, ``x_window`` and ``y_window``
        see :ref:`Notes regarding usage of domain and window <domain-window-note>`.

        Parameters
        ----------
        degree : int
            Polynomial degree: largest sum of exponents (:math:`i + j`) of
            variables in each monomial term of the form :math:`x^i y^j`. The
            number of terms in a 2D polynomial of degree ``n`` is given by binomial
            coefficient :math:`C(n + 2, 2) = (n + 2)! / (2!\,n!) = (n + 1)(n + 2) / 2`.
        x_domain : tuple or None, optional
            domain of the x independent variable
            If None, it is set to (-1, 1)
        y_domain : tuple or None, optional
            domain of the y independent variable
            If None, it is set to (-1, 1)
        x_window : tuple or None, optional
            range of the x independent variable
            If None, it is set to (-1, 1)
            Fitters will remap the x_domain to x_window
        y_window : tuple or None, optional
            range of the y independent variable
            If None, it is set to (-1, 1)
            Fitters will remap the y_domain to y_window
        **params : dict
            keyword: value pairs, representing parameter_name: value
        """

        n_inputs = 2
        n_outputs = 1

        _separable = False

        def __init__(
            self,
            degree,
            x_domain=None,
            y_domain=None,
            x_window=None,
            y_window=None,
            n_models=None,
            model_set_axis=None,
            name=None,
            meta=None,
            **params,
        ):
            super().__init__(
                degree,
                n_models=n_models,
                model_set_axis=model_set_axis,
                name=name,
                meta=meta,
                **params,
            )

            self._default_domain_window = {
                "x_domain": (-1, 1),
                "y_domain": (-1, 1),
                "x_window": (-1, 1),
                "y_window": (-1, 1),
            }

            self.x_domain = x_domain or self._default_domain_window["x_domain"]
            self.y_domain = y_domain or self._default_domain_window["y_domain"]
            self.x_window = x_window or self._default_domain_window["x_window"]
            self.y_window = y_window or self._default_domain_window["y_window"]

        def prepare_inputs(self, x, y, **kwargs):
            inputs, broadcasted_shapes = super().prepare_inputs(x, y, **kwargs)

            x, y = inputs
            return (x, y), broadcasted_shapes

        def evaluate(self, x, y, *coeffs):
            if self.x_domain is not None:
                x = poly_map_domain(x, self.x_domain, self.x_window)
            if self.y_domain is not None:
                y = poly_map_domain(y, self.y_domain, self.y_window)
            invcoeff = self.invlex_coeff(coeffs)
            result = self.multivariate_horner(x, y, invcoeff)

            # Special case for degree==0 to ensure that the shape of the output is
            # still as expected by the broadcasting rules, even though the x and y
            # inputs are not used in the evaluation
            if self.degree == 0:
                output_shape = np.broadcast_shapes(np.shape(coeffs[0]), x.shape)
                if output_shape:
                    new_result = np.empty(output_shape)
                    new_result[:] = result
                    result = new_result

            return result

        def __repr__(self):
            return self._format_repr(
                [self.degree],
                kwargs={
                    "x_domain": self.x_domain,
                    "y_domain": self.y_domain,
                    "x_window": self.x_window,
                    "y_window": self.y_window,
                },
                defaults=self._default_domain_window,
            )

        def __str__(self):
            return self._format_str(
                [
                    ("Degree", self.degree),
                    ("X_Domain", self.x_domain),
                    ("Y_Domain", self.y_domain),
                    ("X_Window", self.x_window),
                    ("Y_Window", self.y_window),
                ],
                self._default_domain_window,
            )

        def fit_deriv(self, x, y, *params):
            """
            Computes the Vandermonde matrix.

            Parameters
            ----------
            x : ndarray
                input
            y : ndarray
                input
            *params
                throw-away parameter list returned by non-linear fitters

            Returns
            -------
            result : ndarray
                The Vandermonde matrix
            """
            if x.ndim == 2:
                x = x.ravel()
            if y.ndim == 2:
                y = y.ravel()
            if x.size != y.size:
                raise ValueError("Expected x and y to be of equal size")

            designx = x[:, None] ** np.arange(self.degree + 1)
            designy = y[:, None] ** np.arange(1, self.degree + 1)

            designmixed = []
            for i in range(1, self.degree):
                for j in range(1, self.degree):
                    if i + j <= self.degree:
                        designmixed.append((x**i) * (y**j))
            designmixed = np.array(designmixed).T
            if designmixed.any():
                v = np.hstack([designx, designy, designmixed])
            else:
                v = np.hstack([designx, designy])
            return v

        def invlex_coeff(self, coeffs):
            invlex_coeffs = []
            lencoeff = range(self.degree + 1)
            for i in lencoeff:
                for j in lencoeff:
                    if i + j <= self.degree:
                        name = f"c{j}_{i}"
                        coeff = coeffs[self.param_names.index(name)]
                        invlex_coeffs.append(coeff)
            return invlex_coeffs[::-1]

        def multivariate_horner(self, x, y, coeffs):
            """
            Multivariate Horner's scheme.

            Parameters
            ----------
            x, y : array
            coeffs : array
                Coefficients in inverse lexical order.
            """
            alpha = self._invlex()
            r0 = coeffs[0]
            r1 = r0 * 0.0
            r2 = r0 * 0.0
            karr = np.diff(alpha, axis=0)

            for n in range(len(karr)):
                if karr[n, 1] != 0:
                    r2 = y * (r0 + r1 + r2)
                    r1 = np.zeros_like(coeffs[0], subok=False)
                else:
                    r1 = x * (r0 + r1)
                r0 = coeffs[n + 1]
            return r0 + r1 + r2

        @property
        def input_units(self):
            if self.degree == 0 or (
                self.c1_0.input_unit is None and self.c0_1.input_unit is None
            ):
                return None
            return {
                self.inputs[0]: self.c0_0.input_unit / self.c1_0.input_unit,
                self.inputs[1]: self.c0_0.input_unit / self.c0_1.input_unit,
            }

        def _parameter_units_for_data_units(self, inputs_unit, outputs_unit):
            mapping = {}
            for i in range(self.degree + 1):
                for j in range(self.degree + 1):
                    if i + j > 2:
                        continue
                    par = getattr(self, f"c{i}_{j}")
                    mapping[par.name] = (
                        outputs_unit[self.outputs[0]]
                        / inputs_unit[self.inputs[0]] ** i
                        / inputs_unit[self.inputs[1]] ** j
                    )
            return mapping

        @property
        def x_domain(self):
            return self._x_domain

        @x_domain.setter
        def x_domain(self, val):
            self._x_domain = _validate_domain_window(val)

        @property
        def y_domain(self):
            return self._y_domain

        @y_domain.setter
        def y_domain(self, val):
            self._y_domain = _validate_domain_window(val)

        @property
        def x_window(self):
            return self._x_window

        @x_window.setter
        def x_window(self, val):
            self._x_window = _validate_domain_window(val)

        @property
        def y_window(self):
            return self._y_window

        @y_window.setter
        def y_window(self, val):
            self._y_window = _validate_domain_window(val)

    class Chebyshev2D(OrthoPolynomialBase):
        r"""
        Bivariate Chebyshev series..

        It is defined as

        .. math:: P_{nm}(x,y) = \sum_{n,m=0}^{n=d,m=d}C_{nm}  T_n(x ) T_m(y)

        where ``T_n(x)`` and ``T_m(y)`` are Chebyshev polynomials of the first kind.

        For explanation of ``x_domain``, ``y_domain``, ``x_window`` and ``y_window``
        see :ref:`Notes regarding usage of domain and window <domain-window-note>`.

        Parameters
        ----------
        x_degree : int
            degree in x
        y_degree : int
            degree in y
        x_domain : tuple or None, optional
            domain of the x independent variable
        y_domain : tuple or None, optional
            domain of the y independent variable
        x_window : tuple or None, optional
            range of the x independent variable
            If None, it is set to (-1, 1)
            Fitters will remap the domain to this window
        y_window : tuple or None, optional
            range of the y independent variable
            If None, it is set to (-1, 1)
            Fitters will remap the domain to this window

        **params : dict
            keyword: value pairs, representing parameter_name: value

        Notes
        -----
        This model does not support the use of units/quantities, because each term
        in the sum of Chebyshev polynomials is a polynomial in x and/or y - since
        the coefficients within each Chebyshev polynomial are fixed, we can't use
        quantities for x and/or y since the units would not be compatible. For
        example, the third Chebyshev polynomial (T2) is 2x^2-1, but if x was
        specified with units, 2x^2 and -1 would have incompatible units.
        """

        _separable = False

        def __init__(
            self,
            x_degree,
            y_degree,
            x_domain=None,
            x_window=None,
            y_domain=None,
            y_window=None,
            n_models=None,
            model_set_axis=None,
            name=None,
            meta=None,
            **params,
        ):
            super().__init__(
                x_degree,
                y_degree,
                x_domain=x_domain,
                y_domain=y_domain,
                x_window=x_window,
                y_window=y_window,
                n_models=n_models,
                model_set_axis=model_set_axis,
                name=name,
                meta=meta,
                **params,
            )

        def _fcache(self, x, y):
            """
            Calculate the individual Chebyshev functions once and store them in a
            dictionary to be reused.
            """
            x_terms = self.x_degree + 1
            y_terms = self.y_degree + 1
            kfunc = {}
            kfunc[0] = np.ones(x.shape)
            kfunc[1] = x.copy()
            kfunc[x_terms] = np.ones(y.shape)
            kfunc[x_terms + 1] = y.copy()
            for n in range(2, x_terms):
                kfunc[n] = 2 * x * kfunc[n - 1] - kfunc[n - 2]
            for n in range(x_terms + 2, x_terms + y_terms):
                kfunc[n] = 2 * y * kfunc[n - 1] - kfunc[n - 2]
            return kfunc

        def fit_deriv(self, x, y, *params):
            """
            Derivatives with respect to the coefficients.

            This is an array with Chebyshev polynomials:

            .. math::

                T_{x_0}T_{y_0}, T_{x_1}T_{y_0}...T_{x_n}T_{y_0}...T_{x_n}T_{y_m}

            Parameters
            ----------
            x : ndarray
                input
            y : ndarray
                input
            *params
                throw-away parameter list returned by non-linear fitters

            Returns
            -------
            result : ndarray
                The Vandermonde matrix
            """
            if x.shape != y.shape:
                raise ValueError("x and y must have the same shape")

            x = x.ravel()
            y = y.ravel()
            x_deriv = self._chebderiv1d(x, self.x_degree + 1).T
            y_deriv = self._chebderiv1d(y, self.y_degree + 1).T

            ij = []
            for i in range(self.y_degree + 1):
                for j in range(self.x_degree + 1):
                    ij.append(x_deriv[j] * y_deriv[i])

            v = np.array(ij)
            return v.T

        def _chebderiv1d(self, x, deg):
            """
            Derivative of 1D Chebyshev series.
            """
            x = np.array(x, dtype=float, copy=None, ndmin=1)
            d = np.empty((deg + 1, len(x)), dtype=x.dtype)
            d[0] = x * 0 + 1
            if deg > 0:
                x2 = 2 * x
                d[1] = x
                for i in range(2, deg + 1):
                    d[i] = d[i - 1] * x2 - d[i - 2]
            return np.rollaxis(d, 0, d.ndim)

    class Legendre2D(OrthoPolynomialBase):
        r"""
        Bivariate Legendre series.

        Defined as:

        .. math:: P_{n_m}(x,y) = \sum_{n,m=0}^{n=d,m=d}C_{nm}  L_n(x ) L_m(y)

        where ``L_n(x)`` and ``L_m(y)`` are Legendre polynomials.

        For explanation of ``x_domain``, ``y_domain``, ``x_window`` and ``y_window``
        see :ref:`Notes regarding usage of domain and window <domain-window-note>`.

        Parameters
        ----------
        x_degree : int
            degree in x
        y_degree : int
            degree in y
        x_domain : tuple or None, optional
            domain of the x independent variable
        y_domain : tuple or None, optional
            domain of the y independent variable
        x_window : tuple or None, optional
            range of the x independent variable
            If None, it is set to (-1, 1)
            Fitters will remap the domain to this window
        y_window : tuple or None, optional
            range of the y independent variable
            If None, it is set to (-1, 1)
            Fitters will remap the domain to this window
        **params : dict
            keyword: value pairs, representing parameter_name: value

        Notes
        -----
        Model formula:

        .. math::

            P(x) = \sum_{i=0}^{i=n}C_{i} * L_{i}(x)

        where ``L_{i}`` is the corresponding Legendre polynomial.

        This model does not support the use of units/quantities, because each term
        in the sum of Legendre polynomials is a polynomial in x - since the
        coefficients within each Legendre polynomial are fixed, we can't use
        quantities for x since the units would not be compatible. For example, the
        third Legendre polynomial (P2) is 1.5x^2-0.5, but if x was specified with
        units, 1.5x^2 and -0.5 would have incompatible units.
        """

        _separable = False

        def __init__(
            self,
            x_degree,
            y_degree,
            x_domain=None,
            x_window=None,
            y_domain=None,
            y_window=None,
            n_models=None,
            model_set_axis=None,
            name=None,
            meta=None,
            **params,
        ):
            super().__init__(
                x_degree,
                y_degree,
                x_domain=x_domain,
                y_domain=y_domain,
                x_window=x_window,
                y_window=y_window,
                n_models=n_models,
                model_set_axis=model_set_axis,
                name=name,
                meta=meta,
                **params,
            )

        def _fcache(self, x, y):
            """
            Calculate the individual Legendre functions once and store them in a
            dictionary to be reused.
            """
            x_terms = self.x_degree + 1
            y_terms = self.y_degree + 1
            kfunc = {}
            kfunc[0] = np.ones(x.shape)
            kfunc[1] = x.copy()
            kfunc[x_terms] = np.ones(y.shape)
            kfunc[x_terms + 1] = y.copy()
            for n in range(2, x_terms):
                kfunc[n] = (
                    (2 * (n - 1) + 1) * x * kfunc[n - 1] - (n - 1) * kfunc[n - 2]
                ) / n
            for n in range(2, y_terms):
                kfunc[n + x_terms] = (
                    (2 * (n - 1) + 1) * y * kfunc[n + x_terms - 1]
                    - (n - 1) * kfunc[n + x_terms - 2]
                ) / (n)
            return kfunc

        def fit_deriv(self, x, y, *params):
            """Derivatives with respect to the coefficients.

            This is an array with Legendre polynomials:

            Lx0Ly0  Lx1Ly0...LxnLy0...LxnLym

            Parameters
            ----------
            x : ndarray
                input
            y : ndarray
                input
            *params
                throw-away parameter list returned by non-linear fitters

            Returns
            -------
            result : ndarray
                The Vandermonde matrix
            """
            if x.shape != y.shape:
                raise ValueError("x and y must have the same shape")
            x = x.ravel()
            y = y.ravel()
            x_deriv = self._legendderiv1d(x, self.x_degree + 1).T
            y_deriv = self._legendderiv1d(y, self.y_degree + 1).T

            ij = []
            for i in range(self.y_degree + 1):
                for j in range(self.x_degree + 1):
                    ij.append(x_deriv[j] * y_deriv[i])

            v = np.array(ij)
            return v.T

        def _legendderiv1d(self, x, deg):
            """Derivative of 1D Legendre polynomial."""
            x = np.array(x, dtype=float, copy=None, ndmin=1)
            d = np.empty((deg + 1,) + x.shape, dtype=x.dtype)
            d[0] = x * 0 + 1
            if deg > 0:
                d[1] = x
                for i in range(2, deg + 1):
                    d[i] = (d[i - 1] * x * (2 * i - 1) - d[i - 2] * (i - 1)) / i
            return np.rollaxis(d, 0, d.ndim)

    class _SIP1D(PolynomialBase):
        """
        This implements the Simple Imaging Polynomial Model (SIP) in 1D.

        It's unlikely it will be used in 1D so this class is private
        and SIP should be used instead.
        """

        n_inputs = 2
        n_outputs = 1

        _separable = False

        def __init__(
            self,
            order,
            coeff_prefix,
            n_models=None,
            model_set_axis=None,
            name=None,
            meta=None,
            **params,
        ):
            self.order = order
            self.coeff_prefix = coeff_prefix
            self._param_names = self._generate_coeff_names(coeff_prefix)

            if n_models:
                if model_set_axis is None:
                    model_set_axis = 0
                minshape = (1,) * model_set_axis + (n_models,)
            else:
                minshape = ()
            for param_name in self._param_names:
                self._parameters_[param_name] = Parameter(
                    param_name, default=np.zeros(minshape)
                )
            super().__init__(
                n_models=n_models,
                model_set_axis=model_set_axis,
                name=name,
                meta=meta,
                **params,
            )

        def __repr__(self):
            return self._format_repr(args=[self.order, self.coeff_prefix])

        def __str__(self):
            return self._format_str(
                [("Order", self.order), ("Coeff. Prefix", self.coeff_prefix)]
            )

        def evaluate(self, x, y, *coeffs):
            # TODO: Rewrite this so that it uses a simpler method of determining
            # the matrix based on the number of given coefficients.
            mcoef = self._coeff_matrix(self.coeff_prefix, coeffs)
            return self._eval_sip(x, y, mcoef)

        def get_num_coeff(self, ndim):
            """
            Return the number of coefficients in one param set.
            """
            if self.order < 2 or self.order > 9:
                raise ValueError("Degree of polynomial must be 2< deg < 9")

            nmixed = comb(self.order, ndim)
            # remove 3 terms because SIP deg >= 2
            numc = self.order * ndim + nmixed - 2
            return numc

        def _generate_coeff_names(self, coeff_prefix):
            names = []
            for i in range(2, self.order + 1):
                names.append(f"{coeff_prefix}_{i}_{0}")
            for i in range(2, self.order + 1):
                names.append(f"{coeff_prefix}_{0}_{i}")
            for i in range(1, self.order):
                for j in range(1, self.order):
                    if i + j < self.order + 1:
                        names.append(f"{coeff_prefix}_{i}_{j}")
            return tuple(names)

        def _coeff_matrix(self, coeff_prefix, coeffs):
            mat = np.zeros((self.order + 1, self.order + 1))
            for i in range(2, self.order + 1):
                attr = f"{coeff_prefix}_{i}_{0}"
                mat[i, 0] = coeffs[self.param_names.index(attr)][0]
            for i in range(2, self.order + 1):
                attr = f"{coeff_prefix}_{0}_{i}"
                mat[0, i] = coeffs[self.param_names.index(attr)][0]
            for i in range(1, self.order):
                for j in range(1, self.order):
                    if i + j < self.order + 1:
                        attr = f"{coeff_prefix}_{i}_{j}"
                        mat[i, j] = coeffs[self.param_names.index(attr)][0]
            return mat

        def _eval_sip(self, x, y, coef):
            x = np.asarray(x, dtype=np.float64)
            y = np.asarray(y, dtype=np.float64)
            if self.coeff_prefix == "A":
                result = np.zeros(x.shape)
            else:
                result = np.zeros(y.shape)

            for i in range(coef.shape[0]):
                for j in range(coef.shape[1]):
                    if 1 < i + j < self.order + 1:
                        result = result + coef[i, j] * x**i * y**j
            return result

    class SIP(Model):
        """
        Simple Imaging Polynomial (SIP) model.

        The SIP convention is used to represent distortions in FITS image headers.
        See [1]_ for a description of the SIP convention.

        Parameters
        ----------
        crpix : list or (2,) ndarray
            CRPIX values
        a_order : int
            SIP polynomial order for first axis
        b_order : int
            SIP order for second axis
        a_coeff : dict
            SIP coefficients for first axis
        b_coeff : dict
            SIP coefficients for the second axis
        ap_order : int
            order for the inverse transformation (AP coefficients)
        bp_order : int
            order for the inverse transformation (BP coefficients)
        ap_coeff : dict
            coefficients for the inverse transform
        bp_coeff : dict
            coefficients for the inverse transform

        References
        ----------
        .. [1] `David Shupe, et al, ADASS, ASP Conference Series, Vol. 347, 2005
            <https://ui.adsabs.harvard.edu/abs/2005ASPC..347..491S>`_
        """

        n_inputs = 2
        n_outputs = 2

        _separable = False

        def __init__(
            self,
            crpix,
            a_order,
            b_order,
            a_coeff={},
            b_coeff={},
            ap_order=None,
            bp_order=None,
            ap_coeff={},
            bp_coeff={},
            n_models=None,
            model_set_axis=None,
            name=None,
            meta=None,
        ):
            self._crpix = crpix
            self._a_order = a_order
            self._b_order = b_order
            self._a_coeff = a_coeff
            self._b_coeff = b_coeff
            self._ap_order = ap_order
            self._bp_order = bp_order
            self._ap_coeff = ap_coeff
            self._bp_coeff = bp_coeff
            self.shift_a = Shift(-crpix[0])
            self.shift_b = Shift(-crpix[1])
            self.sip1d_a = _SIP1D(
                a_order,
                coeff_prefix="A",
                n_models=n_models,
                model_set_axis=model_set_axis,
                **a_coeff,
            )
            self.sip1d_b = _SIP1D(
                b_order,
                coeff_prefix="B",
                n_models=n_models,
                model_set_axis=model_set_axis,
                **b_coeff,
            )
            super().__init__(
                n_models=n_models, model_set_axis=model_set_axis, name=name, meta=meta
            )
            self._inputs = ("u", "v")
            self._outputs = ("x", "y")

        def __repr__(self):
            return (
                f"<{self.__class__.__name__}"
                f"({[self.shift_a, self.shift_b, self.sip1d_a, self.sip1d_b]!r})>"
            )

        def __str__(self):
            parts = [f"Model: {self.__class__.__name__}"]
            for model in [self.shift_a, self.shift_b, self.sip1d_a, self.sip1d_b]:
                parts.append(indent(str(model), 4 * " "))
                parts.append("")

            return "\n".join(parts)

        @property
        def inverse(self):
            if self._ap_order is not None and self._bp_order is not None:
                return InverseSIP(
                    self._ap_order, self._bp_order, self._ap_coeff, self._bp_coeff
                )
            else:
                raise NotImplementedError("SIP inverse coefficients are not available.")

        def evaluate(self, x, y):
            u = self.shift_a.evaluate(x, *self.shift_a.param_sets)
            v = self.shift_b.evaluate(y, *self.shift_b.param_sets)
            f = self.sip1d_a.evaluate(u, v, *self.sip1d_a.param_sets)
            g = self.sip1d_b.evaluate(u, v, *self.sip1d_b.param_sets)
            return f, g

    class InverseSIP(Model):
        """
        Inverse Simple Imaging Polynomial.

        Parameters
        ----------
        ap_order : int
            order for the inverse transformation (AP coefficients)
        bp_order : int
            order for the inverse transformation (BP coefficients)
        ap_coeff : dict
            coefficients for the inverse transform
        bp_coeff : dict
            coefficients for the inverse transform

        """

        n_inputs = 2
        n_outputs = 2

        _separable = False

        def __init__(
            self,
            ap_order,
            bp_order,
            ap_coeff={},
            bp_coeff={},
            n_models=None,
            model_set_axis=None,
            name=None,
            meta=None,
        ):
            self._ap_order = ap_order
            self._bp_order = bp_order
            self._ap_coeff = ap_coeff
            self._bp_coeff = bp_coeff

            # define the 0th term in order to use Polynomial2D
            ap_coeff.setdefault("AP_0_0", 0)
            bp_coeff.setdefault("BP_0_0", 0)

            ap_coeff_params = {k.replace("AP_", "c"): v for k, v in ap_coeff.items()}
            bp_coeff_params = {k.replace("BP_", "c"): v for k, v in bp_coeff.items()}

            self.sip1d_ap = Polynomial2D(
                degree=ap_order, model_set_axis=model_set_axis, **ap_coeff_params
            )
            self.sip1d_bp = Polynomial2D(
                degree=bp_order, model_set_axis=model_set_axis, **bp_coeff_params
            )
            super().__init__(
                n_models=n_models, model_set_axis=model_set_axis, name=name, meta=meta
            )

        def __repr__(self):
            return f"<{self.__class__.__name__}({[self.sip1d_ap, self.sip1d_bp]!r})>"

        def __str__(self):
            parts = [f"Model: {self.__class__.__name__}"]
            for model in [self.sip1d_ap, self.sip1d_bp]:
                parts.append(indent(str(model), 4 * " "))
                parts.append("")

            return "\n".join(parts)

        def evaluate(self, x, y):
            x1 = self.sip1d_ap.evaluate(x, y, *self.sip1d_ap.param_sets)
            y1 = self.sip1d_bp.evaluate(x, y, *self.sip1d_bp.param_sets)
            return x1, y1