X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_utils.py;h=0b2d2a315989949c2431641c8f82dea9b576f9b8;hb=5154ccb39a8fd2d69330ae440bd6d92a12f67e7c;hp=81b5634bc921736b0d007f3df04013d0dc6df265;hpb=2108ec57e32e5417f92c4ccbff9df6fdad8a819d;p=sage.d.git diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py index 81b5634..0b2d2a3 100644 --- a/mjo/eja/eja_utils.py +++ b/mjo/eja/eja_utils.py @@ -2,57 +2,6 @@ from sage.functions.other import sqrt from sage.matrix.constructor import matrix from sage.modules.free_module_element import vector -def _change_ring(x, R): - r""" - Change the ring of a vector, matrix, or a cartesian product of - those things. - - SETUP:: - - sage: from mjo.eja.eja_utils import _change_ring - - EXAMPLES:: - - sage: v = vector(QQ, (1,2,3)) - sage: m = matrix(QQ, [[1,2],[3,4]]) - sage: _change_ring(v, RDF) - (1.0, 2.0, 3.0) - sage: _change_ring(m, RDF) - [1.0 2.0] - [3.0 4.0] - sage: _change_ring((v,m), RDF) - ( - [1.0 2.0] - (1.0, 2.0, 3.0), [3.0 4.0] - ) - sage: V1 = cartesian_product([v.parent(), v.parent()]) - sage: V = cartesian_product([v.parent(), V1]) - sage: V((v, (v, v))) - ((1, 2, 3), ((1, 2, 3), (1, 2, 3))) - sage: _change_ring(V((v, (v, v))), RDF) - ((1.0, 2.0, 3.0), ((1.0, 2.0, 3.0), (1.0, 2.0, 3.0))) - - """ - try: - return x.change_ring(R) - except AttributeError: - try: - from sage.categories.sets_cat import cartesian_product - if hasattr(x, 'element_class'): - # x is a parent and we're in a recursive call. - return cartesian_product( [_change_ring(x_i, R) - for x_i in x.cartesian_factors()] ) - else: - # x is an element, and we want to change the ring - # of its parent. - P = x.parent() - Q = cartesian_product( [_change_ring(P_i, R) - for P_i in P.cartesian_factors()] ) - return Q(x) - except AttributeError: - # No parent for x - return x.__class__( _change_ring(x_i, R) for x_i in x ) - def _scale(x, alpha): r""" Scale the vector, matrix, or cartesian-product-of-those-things @@ -97,9 +46,23 @@ 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.octonions import Octonions, OctonionMatrixAlgebra + + EXAMPLES:: + + sage: _all2list([[1]]) + [1] + + :: + sage: V1 = VectorSpace(QQ,2) sage: V2 = MatrixSpace(QQ,2) sage: x1 = V1([1,1]) @@ -116,15 +79,45 @@ 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: 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... - 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), [] ) + 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. This will fall through + # to the iterable case the next time around. + return _all2list(x.to_vector()) + + 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(list( map(_all2list, xl) ), []) + + def _mat2vec(m): return vector(m.base_ring(), m.list()) @@ -136,8 +129,8 @@ 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:: @@ -145,11 +138,21 @@ 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 ) @@ -166,11 +169,11 @@ 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) @@ -188,18 +191,18 @@ 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 @@ -228,16 +231,23 @@ 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() + def norm(x): + ip = inner_product(x,x) + # Don't expand the given field; the inner-product's codomain + # is already correct. For example QQ(2).sqrt() returns sqrt(2) + # in SR, and that will give you weird errors about symbolics + # when what's really going wrong is that you're trying to + # orthonormalize in QQ. + return ip.parent()(ip.sqrt()) v = list(v) # make a copy, don't clobber the input