X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_utils.py;h=832dcef1fac0baa573b4883bc4e2ddd3fbfd55a8;hb=8059516ad9df112ac18c740af7d6f856639d4b8d;hp=b6e0c7d38eaf1d9114973bba4f5ad0674cb8a8a2;hpb=6d875d564f6d962f855c520c21cf539bdd4a3c1d;p=sage.d.git diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py index b6e0c7d..832dcef 100644 --- a/mjo/eja/eja_utils.py +++ b/mjo/eja/eja_utils.py @@ -2,6 +2,79 @@ from sage.functions.other import sqrt from sage.matrix.constructor import matrix from sage.modules.free_module_element import vector +def _scale(x, alpha): + r""" + Scale the vector, matrix, or cartesian-product-of-those-things + ``x`` by ``alpha``. + + This works around the inability to scale certain elements of + Cartesian product spaces, as reported in + + https://trac.sagemath.org/ticket/31435 + + ..WARNING: + + This will do the wrong thing if you feed it a tuple or list. + + SETUP:: + + sage: from mjo.eja.eja_utils import _scale + + EXAMPLES:: + + sage: v = vector(QQ, (1,2,3)) + sage: _scale(v,2) + (2, 4, 6) + sage: m = matrix(QQ, [[1,2],[3,4]]) + sage: M = cartesian_product([m.parent(), m.parent()]) + sage: _scale(M((m,m)), 2) + ([2 4] + [6 8], [2 4] + [6 8]) + + """ + if hasattr(x, 'cartesian_factors'): + P = x.parent() + return P(tuple( _scale(x_i, alpha) + for x_i in x.cartesian_factors() )) + else: + return x*alpha + + +def _all2list(x): + r""" + Flatten a vector, matrix, or cartesian product of those things + into a long list. + + EXAMPLES:: + + sage: from mjo.eja.eja_utils import _all2list + sage: V1 = VectorSpace(QQ,2) + sage: V2 = MatrixSpace(QQ,2) + sage: x1 = V1([1,1]) + sage: x2 = V1([1,-1]) + sage: y1 = V2.one() + sage: y2 = V2([0,1,1,0]) + sage: _all2list((x1,y1)) + [1, 1, 1, 0, 0, 1] + sage: _all2list((x2,y2)) + [1, -1, 0, 1, 1, 0] + sage: M = cartesian_product([V1,V2]) + sage: _all2list(M((x1,y1))) + [1, 1, 1, 0, 0, 1] + sage: _all2list(M((x2,y2))) + [1, -1, 0, 1, 1, 0] + + """ + if hasattr(x, 'list'): + # Easy case... + return x.list() + else: + # But what if it's a tuple or something else? This has to + # handle cartesian products of cartesian products, too; that's + # why it's recursive. + return sum( map(_all2list,x), [] ) + def _mat2vec(m): return vector(m.base_ring(), m.list()) @@ -78,6 +151,28 @@ def gram_schmidt(v, inner_product=None): [0 0], [1/2*sqrt(2) 0], [0 1] ] + It even works on Cartesian product spaces whose factors are vector + or matrix spaces:: + + sage: V1 = VectorSpace(AA,2) + sage: V2 = MatrixSpace(AA,2) + sage: M = cartesian_product([V1,V2]) + sage: x1 = V1([1,1]) + sage: x2 = V1([1,-1]) + sage: y1 = V2.one() + sage: y2 = V2([0,1,1,0]) + sage: z1 = M((x1,y1)) + sage: z2 = M((x2,y2)) + sage: def ip(a,b): + ....: return a[0].inner_product(b[0]) + (a[1]*b[1]).trace() + sage: U = gram_schmidt([z1,z2], inner_product=ip) + sage: ip(U[0],U[1]) + 0 + sage: ip(U[0],U[0]) + 1 + sage: ip(U[1],U[1]) + 1 + TESTS: Ensure that zero vectors don't get in the way:: @@ -88,15 +183,11 @@ def gram_schmidt(v, inner_product=None): sage: v = [v1,v2,v3] sage: len(gram_schmidt(v)) == 2 True - """ if inner_product is None: inner_product = lambda x,y: x.inner_product(y) norm = lambda x: inner_product(x,x).sqrt() - def proj(x,y): - return (inner_product(x,y)/inner_product(x,x))*x - v = list(v) # make a copy, don't clobber the input # Drop all zero vectors before we start. @@ -108,10 +199,24 @@ def gram_schmidt(v, inner_product=None): R = v[0].base_ring() + # Our "zero" needs to belong to the right space for sum() to work. + zero = v[0].parent().zero() + + sc = lambda x,a: a*x + if hasattr(v[0], 'cartesian_factors'): + # Only use the slow implementation if necessary. + sc = _scale + + def proj(x,y): + return sc(x, (inner_product(x,y)/inner_product(x,x))) + # First orthogonalize... for i in range(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() ) + v[i] -= sum( (proj(v[j],v[i]) + for j in range(i) + if not v[j].is_zero() ), + 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() ] @@ -120,6 +225,6 @@ def gram_schmidt(v, inner_product=None): # them here because then our subalgebra would have a bigger field # than the superalgebra. for i in range(len(v)): - v[i] = v[i] / norm(v[i]) + v[i] = sc(v[i], ~norm(v[i])) return v