From bc66cbb7a3683e000be25bf9c289e397b8ac959c Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Wed, 10 Mar 2021 19:06:46 -0500 Subject: [PATCH] eja: rework the random_eja() bounds in terms of dimension. --- mjo/eja/eja_algebra.py | 161 +++++++++++++++++++++++++++-------------- 1 file changed, 108 insertions(+), 53 deletions(-) diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py index a2adbb2..70a150a 100644 --- a/mjo/eja/eja_algebra.py +++ b/mjo/eja/eja_algebra.py @@ -1715,25 +1715,35 @@ class ConcreteEJA(FiniteDimensionalEJA): """ @staticmethod - def _max_random_instance_size(): + def _max_random_instance_dimension(): + r""" + The maximum dimension of any random instance. Ten dimensions seems + to be about the point where everything takes a turn for the + worse. And dimension ten (but not nine) allows the 4-by-4 real + Hermitian matrices, the 2-by-2 quaternion Hermitian matrices, + and the 2-by-2 octonion Hermitian matrices. + """ + return 10 + + @staticmethod + def _max_random_instance_size(max_dimension): """ Return an integer "size" that is an upper bound on the size of - this algebra when it is used in a random test - case. Unfortunately, the term "size" is ambiguous -- when - dealing with `R^n` under either the Hadamard or Jordan spin - product, the "size" refers to the dimension `n`. When dealing - with a matrix algebra (real symmetric or complex/quaternion - Hermitian), it refers to the size of the matrix, which is far - less than the dimension of the underlying vector space. + this algebra when it is used in a random test case. This size + (which can be passed to the algebra's constructor) is itself + based on the ``max_dimension`` parameter. This method must be implemented in each subclass. """ raise NotImplementedError @classmethod - def random_instance(cls, *args, **kwargs): + def random_instance(cls, max_dimension=None, *args, **kwargs): """ - Return a random instance of this type of algebra. + Return a random instance of this type of algebra whose dimension + is less than or equal to ``max_dimension``. If the dimension bound + is omitted, then the ``_max_random_instance_dimension()`` is used + to get a suitable bound. This method should be implemented in each subclass. """ @@ -1969,15 +1979,19 @@ class RealSymmetricEJA(MatrixEJA, RationalBasisEJA, ConcreteEJA): """ @staticmethod - def _max_random_instance_size(): - return 4 # Dimension 10 + def _max_random_instance_size(max_dimension): + # Obtained by solving d = (n^2 + n)/2. + return int(ZZ(8*max_dimension + 1).sqrt()/2 - 1/2) @classmethod - def random_instance(cls, **kwargs): + def random_instance(cls, max_dimension=None, **kwargs): """ Return a random instance of this type of algebra. """ - n = ZZ.random_element(cls._max_random_instance_size() + 1) + if max_dimension is None: + max_dimension = cls._max_random_instance_dimension() + max_size = cls._max_random_instance_size(max_dimension) + 1 + n = ZZ.random_element(max_size) return cls(n, **kwargs) def __init__(self, n, field=AA, **kwargs): @@ -2068,6 +2082,7 @@ class ComplexHermitianEJA(MatrixEJA, RationalBasisEJA, ConcreteEJA): sage: ComplexHermitianEJA(0) Euclidean Jordan algebra of dimension 0 over Algebraic Real Field + """ def __init__(self, n, field=AA, **kwargs): # We know this is a valid EJA, but will double-check @@ -2087,15 +2102,18 @@ class ComplexHermitianEJA(MatrixEJA, RationalBasisEJA, ConcreteEJA): self._rational_algebra._charpoly_coefficients.set_cache(a) @staticmethod - def _max_random_instance_size(): - return 3 # Dimension 9 + def _max_random_instance_size(max_dimension): + # Obtained by solving d = n^2. + return int(ZZ(max_dimension).sqrt()) @classmethod - def random_instance(cls, **kwargs): + def random_instance(cls, max_dimension=None, **kwargs): """ Return a random instance of this type of algebra. """ - n = ZZ.random_element(cls._max_random_instance_size() + 1) + if max_dimension is None: + max_dimension = cls._max_random_instance_dimension() + n = ZZ.random_element(cls._max_random_instance_size(max_dimension) + 1) return cls(n, **kwargs) @@ -2176,18 +2194,21 @@ class QuaternionHermitianEJA(MatrixEJA, RationalBasisEJA, ConcreteEJA): @staticmethod - def _max_random_instance_size(): + def _max_random_instance_size(max_dimension): r""" The maximum rank of a random QuaternionHermitianEJA. """ - return 2 # Dimension 6 + # Obtained by solving d = 2n^2 - n. + return int(ZZ(8*max_dimension + 1).sqrt()/4 + 1/4) @classmethod - def random_instance(cls, **kwargs): + def random_instance(cls, max_dimension=None, **kwargs): """ Return a random instance of this type of algebra. """ - n = ZZ.random_element(cls._max_random_instance_size() + 1) + if max_dimension is None: + max_dimension = cls._max_random_instance_dimension() + n = ZZ.random_element(cls._max_random_instance_size(max_dimension) + 1) return cls(n, **kwargs) class OctonionHermitianEJA(MatrixEJA, RationalBasisEJA, ConcreteEJA): @@ -2278,18 +2299,29 @@ class OctonionHermitianEJA(MatrixEJA, RationalBasisEJA, ConcreteEJA): """ @staticmethod - def _max_random_instance_size(): + def _max_random_instance_size(max_dimension): r""" The maximum rank of a random QuaternionHermitianEJA. """ - return 1 # Dimension 1 + # There's certainly a formula for this, but with only four + # cases to worry about, I'm not that motivated to derive it. + if max_dimension >= 27: + return 3 + elif max_dimension >= 10: + return 2 + elif max_dimension >= 1: + return 1 + else: + return 0 @classmethod - def random_instance(cls, **kwargs): + def random_instance(cls, max_dimension=None, **kwargs): """ Return a random instance of this type of algebra. """ - n = ZZ.random_element(cls._max_random_instance_size() + 1) + if max_dimension is None: + max_dimension = cls._max_random_instance_dimension() + n = ZZ.random_element(cls._max_random_instance_size(max_dimension) + 1) return cls(n, **kwargs) def __init__(self, n, field=AA, **kwargs): @@ -2410,18 +2442,28 @@ class HadamardEJA(RationalBasisEJA, ConcreteEJA): self.one.set_cache( self.sum(self.gens()) ) @staticmethod - def _max_random_instance_size(): + def _max_random_instance_dimension(): r""" - The maximum dimension of a random HadamardEJA. + There's no reason to go higher than five here. That's + enough to get the point across. """ return 5 + @staticmethod + def _max_random_instance_size(max_dimension): + r""" + The maximum size (=dimension) of a random HadamardEJA. + """ + return max_dimension + @classmethod - def random_instance(cls, **kwargs): + def random_instance(cls, max_dimension=None, **kwargs): """ Return a random instance of this type of algebra. """ - n = ZZ.random_element(cls._max_random_instance_size() + 1) + if max_dimension is None: + max_dimension = cls._max_random_instance_dimension() + n = ZZ.random_element(cls._max_random_instance_size(max_dimension) + 1) return cls(n, **kwargs) @@ -2561,18 +2603,28 @@ class BilinearFormEJA(RationalBasisEJA, ConcreteEJA): self.one.set_cache( self.monomial(0) ) @staticmethod - def _max_random_instance_size(): + def _max_random_instance_dimension(): r""" - The maximum dimension of a random BilinearFormEJA. + There's no reason to go higher than five here. That's + enough to get the point across. """ return 5 + @staticmethod + def _max_random_instance_size(max_dimension): + r""" + The maximum size (=dimension) of a random BilinearFormEJA. + """ + return max_dimension + @classmethod - def random_instance(cls, **kwargs): + def random_instance(cls, max_dimension=None, **kwargs): """ Return a random instance of this algebra. """ - n = ZZ.random_element(cls._max_random_instance_size() + 1) + if max_dimension is None: + max_dimension = cls._max_random_instance_dimension() + n = ZZ.random_element(cls._max_random_instance_size(max_dimension) + 1) if n.is_zero(): B = matrix.identity(ZZ, n) return cls(B, **kwargs) @@ -2655,21 +2707,16 @@ class JordanSpinEJA(BilinearFormEJA): # can pass in a field! super().__init__(B, *args, **kwargs) - @staticmethod - def _max_random_instance_size(): - r""" - The maximum dimension of a random JordanSpinEJA. - """ - return 5 - @classmethod - def random_instance(cls, **kwargs): + def random_instance(cls, max_dimension=None, **kwargs): """ Return a random instance of this type of algebra. Needed here to override the implementation for ``BilinearFormEJA``. """ - n = ZZ.random_element(cls._max_random_instance_size() + 1) + if max_dimension is None: + max_dimension = cls._max_random_instance_dimension() + n = ZZ.random_element(cls._max_random_instance_size(max_dimension) + 1) return cls(n, **kwargs) @@ -3273,15 +3320,23 @@ class RationalBasisCartesianProductEJA(CartesianProductEJA, RationalBasisEJA.CartesianProduct = RationalBasisCartesianProductEJA -def random_eja(*args, **kwargs): - J1 = ConcreteEJA.random_instance(*args, **kwargs) +def random_eja(max_dimension=None, *args, **kwargs): + # Use the ConcreteEJA default as the total upper bound (regardless + # of any whether or not any individual factors set a lower limit). + if max_dimension is None: + max_dimension = ConcreteEJA._max_random_instance_dimension() + J1 = ConcreteEJA.random_instance(max_dimension, *args, **kwargs) - # This might make Cartesian products appear roughly as often as - # any other ConcreteEJA. - if ZZ.random_element(len(ConcreteEJA.__subclasses__()) + 1) == 0: - # Use random_eja() again so we can get more than two factors. - J2 = random_eja(*args, **kwargs) - J = cartesian_product([J1,J2]) - return J - else: + + # Roll the dice to see if we attempt a Cartesian product. + dice_roll = ZZ.random_element(len(ConcreteEJA.__subclasses__()) + 1) + new_max_dimension = max_dimension - J1.dimension() + if new_max_dimension == 0 or dice_roll != 0: + # If it's already as big as we're willing to tolerate, just + # return it and don't worry about Cartesian products. return J1 + else: + # Use random_eja() again so we can get more than two factors + # if the sub-call also Decides on a cartesian product. + J2 = random_eja(new_max_dimension, *args, **kwargs) + return cartesian_product([J1,J2]) -- 2.43.2