From 3f2f1e2cc17d8796878a6971050f7aed7a37f737 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Tue, 14 Feb 2017 09:29:24 -0500 Subject: [PATCH] Add polynomial ring examples for is_positive_on. --- mjo/cone/cone.py | 28 ++++++++++++++++++++++++++++ 1 file changed, 28 insertions(+) diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index 0158757..4221566 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -51,6 +51,34 @@ def is_positive_on(L,K): sage: is_positive_on(L,K) True + Your matrix can be over any exact ring, but you may get unexpected + answers with weirder rings. For example, any rational matrix is + positive on the plane, but if your matrix contains polynomial + variables, the answer will be negative:: + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: K.is_full_space() + True + sage: L = matrix(QQ[x], [[x,0],[0,1]]) + sage: is_positive_on(L,K) + False + + The previous example is "unexpected" because it depends on how we + check whether or not ``L`` is positive. For exact base rings, we + check whether or not ``L*z`` belongs to ``K`` for each ``z in K``. + If ``K`` is closed, then an equally-valid test would be to check + whether the inner product of ``L*z`` and ``s`` is nonnegative for + every ``z`` in ``K`` and ``s`` in ``K.dual()``. In fact, that is + what we do over inexact rings. In the previous example, that test + would return an affirmative answer:: + + sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)]) + sage: L = matrix(QQ[x], [[x,0],[0,1]]) + sage: all([ (L*z).inner_product(s) for z in K for s in K.dual() ]) + True + sage: is_positive_on(L.change_ring(SR), K) + True + TESTS: The identity operator is always positive:: -- 2.43.2