X-Git-Url: http://gitweb.michael.orlitzky.com/?p=sage.d.git;a=blobdiff_plain;f=mjo%2Feja%2Feja_utils.py;h=a8abeff6be073b2b0aea3dd4f5af933c251666a5;hp=832dcef1fac0baa573b4883bc4e2ddd3fbfd55a8;hb=HEAD;hpb=ff8c9b19da5ed821366a491a95b4f6c946f315ae diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py index 832dcef..a8abeff 100644 --- a/mjo/eja/eja_utils.py +++ b/mjo/eja/eja_utils.py @@ -1,6 +1,4 @@ -from sage.functions.other import sqrt -from sage.matrix.constructor import matrix -from sage.modules.free_module_element import vector +from sage.structure.element import is_Matrix def _scale(x, alpha): r""" @@ -46,9 +44,25 @@ def _all2list(x): Flatten a vector, matrix, or cartesian product of those things into a long list. - EXAMPLES:: + If the entries of the matrix themselves belong to a real vector + space (such as the complex numbers which can be thought of as + pairs of real numbers), they will also be expanded in vector form + and flattened into the list. + + SETUP:: sage: from mjo.eja.eja_utils import _all2list + sage: from mjo.hurwitz import (QuaternionMatrixAlgebra, + ....: Octonions, + ....: OctonionMatrixAlgebra) + + EXAMPLES:: + + sage: _all2list([[1]]) + [1] + + :: + sage: V1 = VectorSpace(QQ,2) sage: V2 = MatrixSpace(QQ,2) sage: x1 = V1([1,1]) @@ -65,28 +79,65 @@ def _all2list(x): sage: _all2list(M((x2,y2))) [1, -1, 0, 1, 1, 0] + :: + + sage: _all2list(Octonions().one()) + [1, 0, 0, 0, 0, 0, 0, 0] + sage: _all2list(OctonionMatrixAlgebra(1).one()) + [1, 0, 0, 0, 0, 0, 0, 0] + + :: + + sage: _all2list(QuaternionAlgebra(QQ, -1, -1).one()) + [1, 0, 0, 0] + sage: _all2list(QuaternionMatrixAlgebra(1).one()) + [1, 0, 0, 0] + + :: + + sage: V1 = VectorSpace(QQ,2) + sage: V2 = OctonionMatrixAlgebra(1,field=QQ) + sage: C = cartesian_product([V1,V2]) + sage: x1 = V1([3,4]) + sage: y1 = V2.one() + sage: _all2list(C( (x1,y1) )) + [3, 4, 1, 0, 0, 0, 0, 0, 0, 0] + """ - if hasattr(x, 'list'): - # Easy case... + if hasattr(x, 'to_vector'): + # This works on matrices of e.g. octonions directly, without + # first needing to convert them to a list of octonions and + # then recursing down into the list. It also avoids the wonky + # list(x) when x is an element of a CFM. I don't know what it + # returns but it aint the coordinates. We don't recurse + # because vectors can only contain ring elements as entries. + return x.to_vector().list() + + if is_Matrix(x): + # This sucks, but for performance reasons we don't want to + # call _all2list recursively on the contents of a matrix + # when we don't have to (they only contain ring elements + # as entries) 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()) + try: + xl = list(x) + except TypeError: # x is not iterable + return [x] + + if xl == [x]: + # Avoid the retardation of list(QQ(1)) == [1]. + return [x] + + return sum( map(_all2list, xl) , []) -def _vec2mat(v): - return matrix(v.base_ring(), sqrt(v.degree()), v.list()) def gram_schmidt(v, inner_product=None): """ 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. + vectors over the same base ring, which means that your base ring + needs to contain the appropriate roots. SETUP:: @@ -94,11 +145,21 @@ def gram_schmidt(v, inner_product=None): EXAMPLES: + If you start with an orthonormal set, you get it back. We can use + the rationals here because we don't need any square roots:: + + sage: v1 = vector(QQ, (1,0,0)) + sage: v2 = vector(QQ, (0,1,0)) + sage: v3 = vector(QQ, (0,0,1)) + sage: v = [v1,v2,v3] + sage: gram_schmidt(v) == v + True + The usual inner-product and norm are default:: - sage: v1 = vector(QQ,(1,2,3)) - sage: v2 = vector(QQ,(1,-1,6)) - sage: v3 = vector(QQ,(2,1,-1)) + sage: v1 = vector(AA,(1,2,3)) + sage: v2 = vector(AA,(1,-1,6)) + sage: v3 = vector(AA,(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 ) @@ -115,11 +176,11 @@ def gram_schmidt(v, inner_product=None): orthonormal with respect to that (and not the usual inner product):: - sage: v1 = vector(QQ,(1,2,3)) - sage: v2 = vector(QQ,(1,-1,6)) - sage: v3 = vector(QQ,(2,1,-1)) + sage: v1 = vector(AA,(1,2,3)) + sage: v2 = vector(AA,(1,-1,6)) + sage: v3 = vector(AA,(2,1,-1)) sage: v = [v1,v2,v3] - sage: B = matrix(QQ, [ [6, 4, 2], + sage: B = matrix(AA, [ [6, 4, 2], ....: [4, 5, 4], ....: [2, 4, 9] ]) sage: ip = lambda x,y: (B*x).inner_product(y) @@ -137,18 +198,18 @@ def gram_schmidt(v, inner_product=None): This Gram-Schmidt routine can be used on matrices as well, so long as an appropriate inner-product is provided:: - sage: E11 = matrix(QQ, [ [1,0], + sage: E11 = matrix(AA, [ [1,0], ....: [0,0] ]) - sage: E12 = matrix(QQ, [ [0,1], + sage: E12 = matrix(AA, [ [0,1], ....: [1,0] ]) - sage: E22 = matrix(QQ, [ [0,0], + sage: E22 = matrix(AA, [ [0,0], ....: [0,1] ]) - sage: I = matrix.identity(QQ,2) + sage: I = matrix.identity(AA,2) sage: trace_ip = lambda X,Y: (X*Y).trace() sage: gram_schmidt([E11,E12,I,E22], inner_product=trace_ip) [ - [1 0] [ 0 1/2*sqrt(2)] [0 0] - [0 0], [1/2*sqrt(2) 0], [0 1] + [1 0] [ 0 0.7071067811865475?] [0 0] + [0 0], [0.7071067811865475? 0], [0 1] ] It even works on Cartesian product spaces whose factors are vector @@ -177,54 +238,45 @@ def gram_schmidt(v, inner_product=None): 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: v1 = vector(AA,(1,2,3)) + sage: v2 = vector(AA,(1,-1,6)) + sage: v3 = vector(AA,(0,0,0)) 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() - - 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() + V = v[0].parent() - # Our "zero" needs to belong to the right space for sum() to work. - zero = v[0].parent().zero() + if inner_product is None: + inner_product = lambda x,y: x.inner_product(y) sc = lambda x,a: a*x - if hasattr(v[0], 'cartesian_factors'): + if hasattr(V, 'cartesian_factors'): # Only use the slow implementation if necessary. sc = _scale def proj(x,y): + # project y onto the span of {x} 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() ), - zero ) + def normalize(x): + # Don't extend the given field with the necessary + # square roots. This will probably throw weird + # errors about the symbolic ring if you e.g. try + # to use it on a set of rational vectors that isn't + # already orthonormalized. + return sc(x, ~inner_product(x,x).sqrt()) - # 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() ] + v_out = [] # make a copy, don't clobber the input - # 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 range(len(v)): - v[i] = sc(v[i], ~norm(v[i])) + for (i, v_i) in enumerate(v): + ortho_v_i = v_i - V.sum( proj(v_out[j],v_i) for j in range(i) ) + if not ortho_v_i.is_zero(): + v_out.append(normalize(ortho_v_i)) - return v + return v_out