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+ GNU AFFERO GENERAL PUBLIC LICENSE
+ Version 3, 19 November 2007
+
+ Copyright (C) 2007 Free Software Foundation, Inc. <https://fsf.org/>
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+ How to Apply These Terms to Your New Programs
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+ If you develop a new program, and you want it to be of the greatest
+possible use to the public, the best way to achieve this is to make it
+free software which everyone can redistribute and change under these terms.
+
+ To do so, attach the following notices to the program. It is safest
+to attach them to the start of each source file to most effectively
+state the exclusion of warranty; and each file should have at least
+the "copyright" line and a pointer to where the full notice is found.
+
+ <one line to give the program's name and a brief idea of what it does.>
+ Copyright (C) <year> <name of author>
+
+ This program is free software: you can redistribute it and/or modify
+ it under the terms of the GNU Affero General Public License as published by
+ the Free Software Foundation, either version 3 of the License, or
+ (at your option) any later version.
+
+ This program is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ GNU Affero General Public License for more details.
+
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+ along with this program. If not, see <https://www.gnu.org/licenses/>.
+
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+if any, to sign a "copyright disclaimer" for the program, if necessary.
+For more information on this, and how to apply and follow the GNU AGPL, see
+<https://www.gnu.org/licenses/>.
+++ /dev/null
-from sage.all import *
-
-def rearrangement_cone(p,n,lattice=None):
- r"""
- Return the rearrangement cone of order ``p`` in ``n`` dimensions.
-
- The rearrangement cone in ``n`` dimensions has as its elements
- vectors of length ``n``. For inclusion in the cone, the smallest
- ``p`` components of a vector must sum to a nonnegative number.
-
- For example, the rearrangement cone of order ``p == 1`` has its
- single smallest component nonnegative. This implies that all
- components are nonnegative, and that therefore the rearrangement
- cone of order one is the nonnegative orthant.
-
- When ``p == n``, the sum of all components of a vector must be
- nonnegative for inclusion in the cone. That is, the cone is a
- half-space in ``n`` dimensions.
-
- INPUT:
-
- - ``p`` -- The number of components to "rearrange."
-
- - ``n`` -- The dimension of the ambient space for the resulting cone.
-
- - ``lattice`` -- (default: ``None``) an ambient lattice of rank ``n``
- to be passed to the :func:`Cone` constructor.
-
- OUTPUT:
-
- A polyhedral closed convex cone object representing a rearrangement
- cone of order ``p`` in ``n`` dimensions. Each generating ray will
- have the integer ring as its base ring.
-
- If a ``lattice`` was specified, then the resulting cone will live in
- that lattice unless its rank is incompatible with the dimension
- ``n`` (in which case a ``ValueError`` is raised).
-
- ALGORITHM:
-
- The generators for the rearrangement cone are given by [Jeong]_
- Theorem 5.2.3.
-
- REFERENCES:
-
- .. [GowdaJeong] Muddappa Seetharama Gowda and Juyoung Jeong.
- Spectral cones in Euclidean Jordan algebras.
- Linear Algebra and its Applications, 509, 286-305.
- doi:10.1016/j.laa.2016.08.004
-
- .. [HenrionSeeger] Rene Henrion and Alberto Seeger.
- Inradius and Circumradius of Various Convex Cones Arising in
- Applications. Set-Valued and Variational Analysis, 18(3-4),
- 483-511, 2010. doi:10.1007/s11228-010-0150-z
-
- .. [Jeong] Juyoung Jeong.
- Spectral sets and functions on Euclidean Jordan algebras.
- University of Maryland, Baltimore County, Ph.D. thesis, 2017.
-
- SETUP::
-
- sage: from mjo.cone.rearrangement import rearrangement_cone
-
- EXAMPLES:
-
- The rearrangement cones of order one are nonnegative orthants::
-
- sage: rearrangement_cone(1,1) == Cone([(1,)])
- True
- sage: rearrangement_cone(1,2) == Cone([(0,1),(1,0)])
- True
- sage: rearrangement_cone(1,3) == Cone([(0,0,1),(0,1,0),(1,0,0)])
- True
-
- When ``p == n``, the resulting cone will be a half-space, so we
- expect its lineality to be one less than ``n`` because it will
- contain a hyperplane but is not the entire space::
-
- sage: rearrangement_cone(5,5).lineality()
- 4
-
- All rearrangement cones are proper when ``p`` is less than ``n`` by
- [Jeong]_ Proposition 5.2.1::
-
- sage: all( rearrangement_cone(p,n).is_proper()
- ....: for n in range(10)
- ....: for p in range(1, n) )
- True
-
- The Lyapunov rank of the rearrangement cone of order ``p`` in ``n``
- dimensions is ``n`` for ``p == 1`` or ``p == n`` and one otherwise,
- by [Jeong]_ Corollary 5.2.4::
-
- sage: all( rearrangement_cone(p,n).lyapunov_rank() == n
- ....: for n in range(2, 10)
- ....: for p in [1, n-1] )
- True
- sage: all( rearrangement_cone(p,n).lyapunov_rank() == 1
- ....: for n in range(3, 10)
- ....: for p in range(2, n-1) )
- True
-
- TESTS:
-
- All rearrangement cones are permutation-invariant by [Jeong]_
- Proposition 5.2.1::
-
- sage: n = ZZ.random_element(2,10).abs()
- sage: p = ZZ.random_element(1,n)
- sage: K = rearrangement_cone(p,n)
- sage: P = SymmetricGroup(n).random_element().matrix()
- sage: all( K.contains(P*r) for r in K )
- True
-
- The smallest ``p`` components of every element of the rearrangement
- cone should sum to a nonnegative number (this tests that the
- generators really are what we think they are)::
-
- sage: def _has_rearrangement_property(v,p):
- ....: return sum( sorted(v)[0:p] ) >= 0
- sage: all( _has_rearrangement_property(
- ....: rearrangement_cone(p,n).random_element(),
- ....: p
- ....: )
- ....: for n in range(2, 10)
- ....: for p in range(1, n-1)
- ....: )
- True
-
- The rearrangenent cone of order ``p`` is contained in the rearrangement
- cone of order ``p + 1`` by [Jeong]_ Proposition 5.2.1::
-
- sage: n = ZZ.random_element(2,10)
- sage: p = ZZ.random_element(1,n)
- sage: K1 = rearrangement_cone(p,n)
- sage: K2 = rearrangement_cone(p+1,n)
- sage: all( x in K2 for x in K1 )
- True
-
- The rearrangement cone of order ``p`` is linearly isomorphic to the
- rearrangement cone of order ``n - p`` when ``p`` is less than ``n``,
- by [Jeong]_ Proposition 5.2.1::
-
- sage: n = ZZ.random_element(2,10)
- sage: p = ZZ.random_element(1,n)
- sage: K1 = rearrangement_cone(p,n)
- sage: K2 = rearrangement_cone(n-p, n)
- sage: Mp = (1/p)*matrix.ones(QQ,n) - identity_matrix(QQ,n)
- sage: Cone( (Mp*K2.rays()).columns() ).is_equivalent(K1)
- True
-
- The order ``p`` should be between ``1`` and ``n``, inclusive::
-
- sage: rearrangement_cone(0,3)
- Traceback (most recent call last):
- ...
- ValueError: order p=0 should be between 1 and n=3, inclusive
- sage: rearrangement_cone(5,3)
- Traceback (most recent call last):
- ...
- ValueError: order p=5 should be between 1 and n=3, inclusive
-
- If a ``lattice`` was given, it is actually used::
-
- sage: L = ToricLattice(3, 'M')
- sage: rearrangement_cone(2, 3, lattice=L)
- 3-d cone in 3-d lattice M
-
- Unless the rank of the lattice disagrees with ``n``::
-
- sage: L = ToricLattice(1, 'M')
- sage: rearrangement_cone(2, 3, lattice=L)
- Traceback (most recent call last):
- ...
- ValueError: lattice rank=1 and dimension n=3 are incompatible
-
- """
- if p < 1 or p > n:
- raise ValueError('order p=%d should be between 1 and n=%d, inclusive'
- %
- (p,n))
-
- if lattice is None:
- lattice = ToricLattice(n)
-
- if lattice.rank() != n:
- raise ValueError('lattice rank=%d and dimension n=%d are incompatible'
- %
- (lattice.rank(), n))
-
- I = identity_matrix(ZZ,n)
- M = matrix.ones(ZZ,n) - p*I
- G = identity_matrix(ZZ,n).rows() + M.rows()
- return Cone(G, lattice=lattice)
If computing my determinant will be fast, we do so and compare
with zero (Proposition II.2.4 in Faraut and
- Koranyi). Otherwise, Proposition II.3.2 in Faraut and Koranyi
+ Korányi). Otherwise, Proposition II.3.2 in Faraut and Korányi
reduces the problem to the invertibility of my quadratic
representation.
# we want the negative of THAT for the trace.
return -p(*self.to_vector())
- def operator_inner_product(self, other):
+
+ def trace_inner_product(self, other):
+ """
+ Return the trace inner product of myself and ``other``.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import random_eja
+
+ TESTS:
+
+ The trace inner product is commutative, bilinear, and associative::
+
+ sage: J = random_eja()
+ sage: x,y,z = J.random_elements(3)
+ sage: # commutative
+ sage: x.trace_inner_product(y) == y.trace_inner_product(x)
+ True
+ sage: # bilinear
+ sage: a = J.base_ring().random_element()
+ sage: actual = (a*(x+z)).trace_inner_product(y)
+ sage: expected = ( a*x.trace_inner_product(y) +
+ ....: a*z.trace_inner_product(y) )
+ sage: actual == expected
+ True
+ sage: actual = x.trace_inner_product(a*(y+z))
+ sage: expected = ( a*x.trace_inner_product(y) +
+ ....: a*x.trace_inner_product(z) )
+ sage: actual == expected
+ True
+ sage: # associative
+ sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
+ True
+
+ """
+ if not other in self.parent():
+ raise TypeError("'other' must live in the same algebra")
+
+ return (self*other).trace()
+
+
+ def trace_norm(self):
+ """
+ The norm of this element with respect to :meth:`trace_inner_product`.
+
+ SETUP::
+
+ sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
+ ....: HadamardEJA)
+
+ EXAMPLES::
+
+ sage: J = HadamardEJA(2)
+ sage: x = sum(J.gens())
+ sage: x.trace_norm()
+ 1.414213562373095?
+
+ ::
+
+ sage: J = JordanSpinEJA(4)
+ sage: x = sum(J.gens())
+ sage: x.trace_norm()
+ 2.828427124746190?
+
+ """
+ return self.trace_inner_product(self).sqrt()
+
+
+ def operator_trace_inner_product(self, other):
r"""
Return the operator inner product of myself and ``other``.
Euclidean Jordan algebra, this is another associative inner
product under which the cone of squares is symmetric.
- This *probably* works even if the basis hasn't been
- orthonormalized because the eigenvalues of the corresponding
- matrix don't change when the basis does (they're preserved by
- any similarity transformation).
+ This works even if the basis hasn't been orthonormalized
+ because the eigenvalues of the corresponding matrix don't
+ change when the basis does (they're preserved by any
+ similarity transformation).
SETUP::
sage: x,y = J.random_elements(2)
sage: n = J.dimension()
sage: r = J.rank()
- sage: actual = x.operator_inner_product(y)
+ sage: actual = x.operator_trace_inner_product(y)
sage: expected = (n/r)*x.trace_inner_product(y)
sage: actual == expected
True
sage: x,y = J.random_elements(2)
sage: n = J.dimension()
sage: r = J.rank()
- sage: actual = x.operator_inner_product(y)
+ sage: actual = x.operator_trace_inner_product(y)
sage: expected = (n/r)*x.trace_inner_product(y)
sage: actual == expected
True
sage: x,y = J.random_elements(2)
sage: n = J.dimension()
sage: r = J.rank()
- sage: actual = x.operator_inner_product(y)
+ sage: actual = x.operator_trace_inner_product(y)
sage: expected = (n/r)*x.trace_inner_product(y)
sage: actual == expected
True
sage: J = random_eja()
sage: x,y,z = J.random_elements(3)
sage: # commutative
- sage: x.operator_inner_product(y) == y.operator_inner_product(x)
+ sage: actual = x.operator_trace_inner_product(y)
+ sage: expected = y.operator_trace_inner_product(x)
+ sage: actual == expected
True
sage: # bilinear
sage: a = J.base_ring().random_element()
- sage: actual = (a*(x+z)).operator_inner_product(y)
- sage: expected = ( a*x.operator_inner_product(y) +
- ....: a*z.operator_inner_product(y) )
+ sage: actual = (a*(x+z)).operator_trace_inner_product(y)
+ sage: expected = ( a*x.operator_trace_inner_product(y) +
+ ....: a*z.operator_trace_inner_product(y) )
sage: actual == expected
True
- sage: actual = x.operator_inner_product(a*(y+z))
- sage: expected = ( a*x.operator_inner_product(y) +
- ....: a*x.operator_inner_product(z) )
+ sage: actual = x.operator_trace_inner_product(a*(y+z))
+ sage: expected = ( a*x.operator_trace_inner_product(y) +
+ ....: a*x.operator_trace_inner_product(z) )
sage: actual == expected
True
sage: # associative
- sage: actual = (x*y).operator_inner_product(z)
- sage: expected = y.operator_inner_product(x*z)
+ sage: actual = (x*y).operator_trace_inner_product(z)
+ sage: expected = y.operator_trace_inner_product(x*z)
sage: actual == expected
True
- """
- if not other in self.parent():
- raise TypeError("'other' must live in the same algebra")
-
- return (self*other).operator().matrix().trace()
+ Despite the fact that the implementation uses a matrix representation,
+ the answer is independent of the basis used::
-
- def trace_inner_product(self, other):
- """
- Return the trace inner product of myself and ``other``.
-
- SETUP::
-
- sage: from mjo.eja.eja_algebra import random_eja
-
- TESTS:
-
- The trace inner product is commutative, bilinear, and associative::
-
- sage: J = random_eja()
- sage: x,y,z = J.random_elements(3)
- sage: # commutative
- sage: x.trace_inner_product(y) == y.trace_inner_product(x)
- True
- sage: # bilinear
- sage: a = J.base_ring().random_element()
- sage: actual = (a*(x+z)).trace_inner_product(y)
- sage: expected = ( a*x.trace_inner_product(y) +
- ....: a*z.trace_inner_product(y) )
- sage: actual == expected
- True
- sage: actual = x.trace_inner_product(a*(y+z))
- sage: expected = ( a*x.trace_inner_product(y) +
- ....: a*x.trace_inner_product(z) )
+ sage: J = RealSymmetricEJA(3, field=QQ, orthonormalize=False)
+ sage: V = RealSymmetricEJA(3)
+ sage: x,y = J.random_elements(2)
+ sage: w = V(x.to_matrix())
+ sage: z = V(y.to_matrix())
+ sage: expected = x.operator_trace_inner_product(y)
+ sage: actual = w.operator_trace_inner_product(z)
sage: actual == expected
True
- sage: # associative
- sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
- True
"""
if not other in self.parent():
raise TypeError("'other' must live in the same algebra")
- return (self*other).trace()
+ return (self*other).operator().matrix().trace()
- def trace_norm(self):
+ def operator_trace_norm(self):
"""
- The norm of this element with respect to :meth:`trace_inner_product`.
+ The norm of this element with respect to
+ :meth:`operator_trace_inner_product`.
SETUP::
sage: from mjo.eja.eja_algebra import (JordanSpinEJA,
....: HadamardEJA)
- EXAMPLES::
+ EXAMPLES:
+
+ On a simple algebra, this will differ from :meth:`trace_norm`
+ by the scalar factor ``(n/r).sqrt()``, where `n` is the
+ dimension of the algebra and `r` its rank. This follows from
+ the corresponding result (Proposition III.4.2 of Faraut and
+ Korányi) for the trace inner product::
sage: J = HadamardEJA(2)
sage: x = sum(J.gens())
- sage: x.trace_norm()
+ sage: x.operator_trace_norm()
1.414213562373095?
::
sage: J = JordanSpinEJA(4)
sage: x = sum(J.gens())
- sage: x.trace_norm()
- 2.828427124746190?
+ sage: x.operator_trace_norm()
+ 4
"""
- return self.trace_inner_product(self).sqrt()
+ return self.operator_trace_inner_product(self).sqrt()
class CartesianProductParentEJAElement(EJAElement):
projects / sage.d.git / commitdiff