From fcf647efc97b96655b0ca34326488bb0d978fce3 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 27 Nov 2020 08:03:38 -0500 Subject: [PATCH] Revert "eja: drop custom gram_schmidt() routine that isn't noticeably faster." This reverts commit 1e9700cdd04434465ffcad148d078f7fa361e426. We only bring back the gram_schmidt() function (but don't use it yet) because the plan is to modify it to take a custom inner-product. --- mjo/eja/eja_utils.py | 72 ++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 72 insertions(+) diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py index cbbf1e3..49e3078 100644 --- a/mjo/eja/eja_utils.py +++ b/mjo/eja/eja_utils.py @@ -1,9 +1,81 @@ from sage.functions.other import sqrt from sage.matrix.constructor import matrix from sage.modules.free_module_element import vector +from sage.rings.number_field.number_field import NumberField +from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing +from sage.rings.real_lazy import RLF def _mat2vec(m): return vector(m.base_ring(), m.list()) def _vec2mat(v): return matrix(v.base_ring(), sqrt(v.degree()), v.list()) + +def gram_schmidt(v): + """ + Perform Gram-Schmidt on the list ``v`` which are assumed to be + vectors over the same base ring. Returns a list of orthonormalized + vectors over the smallest extention ring containing the necessary + roots. + + SETUP:: + + sage: from mjo.eja.eja_utils import gram_schmidt + + EXAMPLES:: + + sage: v1 = vector(QQ,(1,2,3)) + sage: v2 = vector(QQ,(1,-1,6)) + sage: v3 = vector(QQ,(2,1,-1)) + sage: v = [v1,v2,v3] + sage: u = gram_schmidt(v) + sage: all( u_i.inner_product(u_i).sqrt() == 1 for u_i in u ) + True + sage: bool(u[0].inner_product(u[1]) == 0) + True + sage: bool(u[0].inner_product(u[2]) == 0) + True + sage: bool(u[1].inner_product(u[2]) == 0) + True + + TESTS: + + Ensure that zero vectors don't get in the way:: + + sage: v1 = vector(QQ,(1,2,3)) + sage: v2 = vector(QQ,(1,-1,6)) + sage: v3 = vector(QQ,(0,0,0)) + sage: v = [v1,v2,v3] + sage: len(gram_schmidt(v)) == 2 + True + + """ + def proj(x,y): + return (y.inner_product(x)/x.inner_product(x))*x + + v = list(v) # make a copy, don't clobber the input + + # Drop all zero vectors before we start. + v = [ v_i for v_i in v if not v_i.is_zero() ] + + if len(v) == 0: + # cool + return v + + R = v[0].base_ring() + + # First orthogonalize... + for i in xrange(1,len(v)): + # Earlier vectors can be made into zero so we have to ignore them. + v[i] -= sum( proj(v[j],v[i]) for j in range(i) if not v[j].is_zero() ) + + # And now drop all zero vectors again if they were "orthogonalized out." + v = [ v_i for v_i in v if not v_i.is_zero() ] + + # Just normalize. If the algebra is missing the roots, we can't add + # them here because then our subalgebra would have a bigger field + # than the superalgebra. + for i in xrange(len(v)): + v[i] = v[i] / v[i].norm() + + return v -- 2.44.2