# Licensed under a 3-clause BSD style license - see LICENSE.rst
"""
Convenience functions for `astropy.cosmology`.
"""
import warnings
import numpy as np
from astropy.units import Quantity
from astropy.utils.exceptions import AstropyUserWarning
from .core import CosmologyError
__all__ = ['z_at_value']
__doctest_requires__ = {'*': ['scipy']}
[ドキュメント]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` instance
The (scalar) value of ``func(z)`` to recover.
zmin : float, 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, optional
The upper search limit for ``z`` (default 1000).
ztol : float, optional
The relative error in ``z`` acceptable for convergence.
maxfun : int, 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, 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-monotone 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 : float
The redshift ``z`` satisfying ``zmin < z < zmax`` and ``func(z) =
fval`` within ``ztol``.
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.logspace(np.log10(zmin), np.log10(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
3.19812268
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
0.68044452
>>> z_at_value(Planck18.angular_diameter_distance,
... 1500 * u.Mpc, zmin=2.5) # doctest: +FLOAT_CMP
3.7823268
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, bracket=(1.0, 1.2)) # doctest: +FLOAT_CMP +IGNORE_WARNINGS
0.68044452
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
3.7823268 # 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
3.7823268 # doctest: +SKIP
>>> z_at_value(Planck18.angular_diameter_distance,
... 1500 * u.Mpc, bracket=(2.0, 2.5)) # doctest: +SKIP
0.68044452 # 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, bracket=(0.1, 1.0, 1.5)) # doctest: +FLOAT_CMP
0.68044452
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.
"""
from scipy.optimize import minimize_scalar
opt = {'maxiter': maxfun}
# Assume custom methods support the same options as default; otherwise user will see warnings.
if str(method).lower() == 'bounded':
opt['xatol'] = ztol
if bracket is not None:
warnings.warn(f"Option 'bracket' is ignored by method {method}.")
bracket = None
else:
opt['xtol'] = ztol
# 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, zmax))
nobracket = False
if np.sign(fval - fval_zmin) != np.sign(fval_zmax - fval):
if bracket is None:
nobracket = True
else:
fval_brac = func(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 func(zmax)="
f"{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
# 'Brent' and 'Golden' ignore `bounds`, force solution inside zlim
def f(z):
if z > zmax:
return 1.e300 * (1.0 + z - zmax)
elif z < zmin:
return 1.e300 * (1.0 + zmin - z)
elif isinstance(fval_zmin, Quantity):
return abs(func(z).value - val)
else:
return abs(func(z) - val)
res = minimize_scalar(f, method=method, bounds=(zmin, zmax), 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')}: {res.get('message', 'Unsuccessful')}\n"
f"Precision {res.fun} reached after {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}.\n"
"Try 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}.\n"
"Try re-running with a different zmin.")
return res.x