astro py:docs

- Index
- Modules

### Navigation

- Astropy v5.2.1 »
- Module code »

# Source code for astropy.cosmology.funcs.optimize

    # Licensed under a 3-clause BSD style license - see LICENSE.rst
    """
    Convenience functions for `astropy.cosmology`.
    """

    import warnings

    import numpy as np

    from astropy.cosmology import units as cu
    from astropy.cosmology.core import CosmologyError
    from astropy.units import Quantity
    from astropy.utils.exceptions import AstropyUserWarning

    __all__ = ["z_at_value"]

    __doctest_requires__ = {"*": ["scipy"]}

    def _z_at_scalar_value(
        func,
        fval,
        zmin=1e-8,
        zmax=1000,
        ztol=1e-8,
        maxfun=500,
        method="Brent",
        bracket=None,
        verbose=False,
    ):
        """
        Find the redshift ``z`` at which ``func(z) = fval``.
        See :func:`astropy.cosmology.funcs.z_at_value`.
        """
        from scipy.optimize import minimize_scalar

        opt = {"maxiter": maxfun, "xtol": ztol}
        # Assume custom methods support the same options as default; otherwise user
        # will see warnings.
        if callable(method):  # can skip callables
            pass
        elif str(method).lower() == "bounded":
            opt["xatol"] = opt.pop("xtol")
            if bracket is not None:
                warnings.warn(f"Option 'bracket' is ignored by method {method}.")
                bracket = None

        # fval falling inside the interval of bracketing function values does not
        # guarantee it has a unique solution, but for Standard Cosmological
        # quantities normally should (being monotonic or having a single extremum).
        # In these cases keep solver from returning solutions outside of bracket.
        fval_zmin, fval_zmax = func(zmin), func(zmax)
        nobracket = False
        if np.sign(fval - fval_zmin) != np.sign(fval_zmax - fval):
            if bracket is None:
                nobracket = True
            else:
                fval_brac = func(np.asanyarray(bracket))
                if np.sign(fval - fval_brac[0]) != np.sign(fval_brac[-1] - fval):
                    nobracket = True
                else:
                    zmin, zmax = bracket[0], bracket[-1]
                    fval_zmin, fval_zmax = fval_brac[[0, -1]]
        if nobracket:
            warnings.warn(
                f"fval is not bracketed by func(zmin)={fval_zmin} and "
                f"func(zmax)={fval_zmax}. This means either there is no "
                "solution, or that there is more than one solution "
                "between zmin and zmax satisfying fval = func(z).",
                AstropyUserWarning,
            )

        if isinstance(fval_zmin, Quantity):
            val = fval.to_value(fval_zmin.unit)
        else:
            val = fval

        # Construct bounds (Brent and Golden fail if bounds are not None)
        if callable(method) or str(method).lower() not in {"brent", "golden"}:
            bounds = (zmin, zmax)
        else:
            bounds = None

        # Objective function to minimize.
        # 'Brent' and 'Golden' ignore `bounds` but this keeps the domain witihin the bounds.
        def f(z):
            if z > zmax:
                return 1.0e300 * (1.0 + z - zmax)
            elif z < zmin:
                return 1.0e300 * (1.0 + zmin - z)
            elif isinstance(fval_zmin, Quantity):
                return abs(func(z).value - val)
            else:
                return abs(func(z) - val)

        # Perform the minimization
        res = minimize_scalar(f, method=method, bounds=bounds, bracket=bracket, options=opt)

        # Scipy docs state that `OptimizeResult` always has 'status' and 'message'
        # attributes, but only `_minimize_scalar_bounded()` seems to have really
        # implemented them.
        if not res.success:
            warnings.warn(
                f"Solver returned {res.get('status')}:"
                f" {res.get('message', 'Unsuccessful')}\nPrecision {res.fun} reached after"
                f" {res.nfev} function calls.",
                AstropyUserWarning,
            )

        if verbose:
            print(res)

        if np.allclose(res.x, zmax):
            raise CosmologyError(
                f"Best guess z={res.x} is very close to the upper z limit {zmax}."
                "\nTry re-running with a different zmax."
            )
        elif np.allclose(res.x, zmin):
            raise CosmologyError(
                f"Best guess z={res.x} is very close to the lower z limit {zmin}."
                "\nTry re-running with a different zmin."
            )
        return res.x

    def z_at_value(
        func,
        fval,
        zmin=1e-8,
        zmax=1000,
        ztol=1e-8,
        maxfun=500,
        method="Brent",
        bracket=None,
        verbose=False,
    ):
        """Find the redshift ``z`` at which ``func(z) = fval``.

        This finds the redshift at which one of the cosmology functions or
        methods (for example Planck13.distmod) is equal to a known value.

        .. warning::
           Make sure you understand the behavior of the function that you are
           trying to invert! Depending on the cosmology, there may not be a
           unique solution. For example, in the standard Lambda CDM cosmology,
           there are two redshifts which give an angular diameter distance of
           1500 Mpc, z ~ 0.7 and z ~ 3.8. To force ``z_at_value`` to find the
           solution you are interested in, use the ``zmin`` and ``zmax`` keywords
           to limit the search range (see the example below).

        Parameters
        ----------
        func : function or method
            A function that takes a redshift as input.

        fval : `~astropy.units.Quantity`
            The (scalar or array) value of ``func(z)`` to recover.

        zmin : float or array-like['dimensionless'] or quantity-like, optional
            The lower search limit for ``z``.  Beware of divergences
            in some cosmological functions, such as distance moduli,
            at z=0 (default 1e-8).

        zmax : float or array-like['dimensionless'] or quantity-like, optional
            The upper search limit for ``z`` (default 1000).

        ztol : float or array-like['dimensionless'], optional
            The relative error in ``z`` acceptable for convergence.

        maxfun : int or array-like, optional
            The maximum number of function evaluations allowed in the
            optimization routine (default 500).

        method : str or callable, optional
            Type of solver to pass to the minimizer. The built-in options provided
            by :func:`~scipy.optimize.minimize_scalar` are 'Brent' (default),
            'Golden' and 'Bounded' with names case insensitive - see documentation
            there for details. It also accepts a custom solver by passing any
            user-provided callable object that meets the requirements listed
            therein under the Notes on "Custom minimizers" - or in more detail in
            :doc:`scipy:tutorial/optimize` - although their use is currently
            untested.

            .. versionadded:: 4.3

        bracket : sequence or object array[sequence], optional
            For methods 'Brent' and 'Golden', ``bracket`` defines the bracketing
            interval and can either have three items (z1, z2, z3) so that
            z1 < z2 < z3 and ``func(z2) < func (z1), func(z3)`` or two items z1
            and z3 which are assumed to be a starting interval for a downhill
            bracket search. For non-monotonic functions such as angular diameter
            distance this may be used to start the search on the desired side of
            the maximum, but see Examples below for usage notes.

            .. versionadded:: 4.3

        verbose : bool, optional
            Print diagnostic output from solver (default `False`).

            .. versionadded:: 4.3

        Returns
        -------
        z : `~astropy.units.Quantity` ['redshift']
            The redshift ``z`` satisfying ``zmin < z < zmax`` and ``func(z) =
            fval`` within ``ztol``. Has units of cosmological redshift.

        Warns
        -----
        :class:`~astropy.utils.exceptions.AstropyUserWarning`
            If ``fval`` is not bracketed by ``func(zmin)=fval(zmin)`` and
            ``func(zmax)=fval(zmax)``.

            If the solver was not successful.

        Raises
        ------
        :class:`astropy.cosmology.CosmologyError`
            If the result is very close to either ``zmin`` or ``zmax``.
        ValueError
            If ``bracket`` is not an array nor a 2 (or 3) element sequence.
        TypeError
            If ``bracket`` is not an object array. 2 (or 3) element sequences will
            be turned into object arrays, so this error should only occur if a
            non-object array is used for ``bracket``.

        Notes
        -----
        This works for any arbitrary input cosmology, but is inefficient if you
        want to invert a large number of values for the same cosmology. In this
        case, it is faster to instead generate an array of values at many
        closely-spaced redshifts that cover the relevant redshift range, and then
        use interpolation to find the redshift at each value you are interested
        in. For example, to efficiently find the redshifts corresponding to 10^6
        values of the distance modulus in a Planck13 cosmology, you could do the
        following:

        >>> import astropy.units as u
        >>> from astropy.cosmology import Planck13, z_at_value

        Generate 10^6 distance moduli between 24 and 44 for which we
        want to find the corresponding redshifts:

        >>> Dvals = (24 + np.random.rand(1000000) * 20) * u.mag

        Make a grid of distance moduli covering the redshift range we
        need using 50 equally log-spaced values between zmin and
        zmax. We use log spacing to adequately sample the steep part of
        the curve at low distance moduli:

        >>> zmin = z_at_value(Planck13.distmod, Dvals.min())
        >>> zmax = z_at_value(Planck13.distmod, Dvals.max())
        >>> zgrid = np.geomspace(zmin, zmax, 50)
        >>> Dgrid = Planck13.distmod(zgrid)

        Finally interpolate to find the redshift at each distance modulus:

        >>> zvals = np.interp(Dvals.value, Dgrid.value, zgrid)

        Examples
        --------
        >>> import astropy.units as u
        >>> from astropy.cosmology import Planck13, Planck18, z_at_value

        The age and lookback time are monotonic with redshift, and so a
        unique solution can be found:

        >>> z_at_value(Planck13.age, 2 * u.Gyr)               # doctest: +FLOAT_CMP
        <Quantity 3.19812268 redshift>

        The angular diameter is not monotonic however, and there are two
        redshifts that give a value of 1500 Mpc. You can use the zmin and
        zmax keywords to find the one you are interested in:

        >>> z_at_value(Planck18.angular_diameter_distance,
        ...            1500 * u.Mpc, zmax=1.5)                # doctest: +FLOAT_CMP
        <Quantity 0.68044452 redshift>
        >>> z_at_value(Planck18.angular_diameter_distance,
        ...            1500 * u.Mpc, zmin=2.5)                # doctest: +FLOAT_CMP
        <Quantity 3.7823268 redshift>

        Alternatively the ``bracket`` option may be used to initialize the
        function solver on a desired region, but one should be aware that this
        does not guarantee it will remain close to this starting bracket.
        For the example of angular diameter distance, which has a maximum near
        a redshift of 1.6 in this cosmology, defining a bracket on either side
        of this maximum will often return a solution on the same side:

        >>> z_at_value(Planck18.angular_diameter_distance, 1500 * u.Mpc,
        ...            method="Brent", bracket=(1.0, 1.2))  # doctest: +FLOAT_CMP +IGNORE_WARNINGS
        <Quantity 0.68044452 redshift>

        But this is not ascertained especially if the bracket is chosen too wide
        and/or too close to the turning point:

        >>> z_at_value(Planck18.angular_diameter_distance,
        ...            1500 * u.Mpc, bracket=(0.1, 1.5))           # doctest: +SKIP
        <Quantity 3.7823268 redshift>                              # doctest: +SKIP

        Likewise, even for the same minimizer and same starting conditions different
        results can be found depending on architecture or library versions:

        >>> z_at_value(Planck18.angular_diameter_distance,
        ...            1500 * u.Mpc, bracket=(2.0, 2.5))           # doctest: +SKIP
        <Quantity 3.7823268 redshift>                              # doctest: +SKIP

        >>> z_at_value(Planck18.angular_diameter_distance,
        ...            1500 * u.Mpc, bracket=(2.0, 2.5))           # doctest: +SKIP
        <Quantity 0.68044452 redshift>                             # doctest: +SKIP

        It is therefore generally safer to use the 3-parameter variant to ensure
        the solution stays within the bracketing limits:

        >>> z_at_value(Planck18.angular_diameter_distance, 1500 * u.Mpc, method="Brent",
        ...            bracket=(0.1, 1.0, 1.5))               # doctest: +FLOAT_CMP
        <Quantity 0.68044452 redshift>

        Also note that the luminosity distance and distance modulus (two
        other commonly inverted quantities) are monotonic in flat and open
        universes, but not in closed universes.

        All the arguments except ``func``, ``method`` and ``verbose`` accept array
        inputs. This does NOT use interpolation tables or any method to speed up
        evaluations, rather providing a convenient means to broadcast arguments
        over an element-wise scalar evaluation.

        The most common use case for non-scalar input is to evaluate 'func' for an
        array of ``fval``:

        >>> z_at_value(Planck13.age, [2, 7] * u.Gyr)          # doctest: +FLOAT_CMP
        <Quantity [3.19812061, 0.75620443] redshift>

        ``fval`` can be any shape:

        >>> z_at_value(Planck13.age, [[2, 7], [1, 3]]*u.Gyr)  # doctest: +FLOAT_CMP
        <Quantity [[3.19812061, 0.75620443],
                   [5.67661227, 2.19131955]] redshift>

        Other arguments can be arrays. For non-monotic functions  -- for example,
        the angular diameter distance -- this can be useful to find all solutions.

        >>> z_at_value(Planck13.angular_diameter_distance, 1500 * u.Mpc,
        ...            zmin=[0, 2.5], zmax=[2, 4])            # doctest: +FLOAT_CMP
        <Quantity [0.68127747, 3.79149062] redshift>

        The ``bracket`` argument can likewise be be an array. However, since
        bracket must already be a sequence (or None), it MUST be given as an
        object `numpy.ndarray`. Importantly, the depth of the array must be such
        that each bracket subsequence is an object. Errors or unexpected results
        will happen otherwise. A convenient means to ensure the right depth is by
        including a length-0 tuple as a bracket and then truncating the object
        array to remove the placeholder. This can be seen in the following
        example:

        >>> bracket=np.array([(1.0, 1.2),(2.0, 2.5), ()], dtype=object)[:-1]
        >>> z_at_value(Planck18.angular_diameter_distance, 1500 * u.Mpc,
        ...            bracket=bracket)  # doctest: +SKIP
        <Quantity [0.68044452, 3.7823268] redshift>
        """
        # `fval` can be a Quantity, which isn't (yet) compatible w/ `numpy.nditer`
        # so we strip it of units for broadcasting and restore the units when
        # passing the elements to `_z_at_scalar_value`.
        fval = np.asanyarray(fval)
        unit = getattr(fval, "unit", 1)  # can be unitless
        zmin = Quantity(zmin, cu.redshift).value  # must be unitless
        zmax = Quantity(zmax, cu.redshift).value

        # bracket must be an object array (assumed to be correct) or a 'scalar'
        # bracket: 2 or 3 elt sequence
        if not isinstance(bracket, np.ndarray):  # 'scalar' bracket
            if bracket is not None and len(bracket) not in (2, 3):
                raise ValueError(
                    "`bracket` is not an array nor a 2 (or 3) element sequence."
                )
            else:  # munge bracket into a 1-elt object array
                bracket = np.array([bracket, ()], dtype=object)[:1].squeeze()
        if bracket.dtype != np.object_:
            raise TypeError(f"`bracket` has dtype {bracket.dtype}, not 'O'")

        # make multi-dimensional iterator for all but `method`, `verbose`
        with np.nditer(
            [fval, zmin, zmax, ztol, maxfun, bracket, None],
            flags=["refs_ok"],
            op_flags=[
                *[["readonly"]] * 6,  # ← inputs  output ↓
                ["writeonly", "allocate", "no_subtype"],
            ],
            op_dtypes=(*(None,) * 6, fval.dtype),
            casting="no",
        ) as it:
            for fv, zmn, zmx, zt, mfe, bkt, zs in it:  # ← eltwise unpack & eval ↓
                zs[...] = _z_at_scalar_value(
                    func,
                    fv * unit,
                    zmin=zmn,
                    zmax=zmx,
                    ztol=zt,
                    maxfun=mfe,
                    bracket=bkt.item(),
                    # not broadcasted
                    method=method,
                    verbose=verbose,
                )
            # since bracket is an object array, the output will be too, so it is
            # cast to the same type as the function value.
            result = it.operands[-1]  # zs

        return result << cu.redshift