X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_utils.py;h=0b2d2a315989949c2431641c8f82dea9b576f9b8;hb=5154ccb39a8fd2d69330ae440bd6d92a12f67e7c;hp=cf75e325697dcefb3bf682b855f8d83e3e4f89e2;hpb=e529e0e2775cf50207c7d01d5907214d03cdff5c;p=sage.d.git diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py index cf75e32..0b2d2a3 100644 --- a/mjo/eja/eja_utils.py +++ b/mjo/eja/eja_utils.py @@ -1,27 +1,158 @@ +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 _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. + + 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]) + 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] + + :: + + 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, '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()) -def gram_schmidt(v): +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:: sage: from mjo.eja.eja_utils import gram_schmidt - EXAMPLES:: + 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,(1,2,3)) - sage: v2 = vector(QQ,(1,-1,6)) - sage: v3 = vector(QQ,(2,1,-1)) + 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(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 ) @@ -33,20 +164,90 @@ def gram_schmidt(v): sage: bool(u[1].inner_product(u[2]) == 0) True + + But if you supply a custom inner product, the result is + orthonormal with respect to that (and not the usual inner + product):: + + 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(AA, [ [6, 4, 2], + ....: [4, 5, 4], + ....: [2, 4, 9] ]) + sage: ip = lambda x,y: (B*x).inner_product(y) + sage: norm = lambda x: ip(x,x) + sage: u = gram_schmidt(v,ip) + sage: all( norm(u_i) == 1 for u_i in u ) + True + sage: ip(u[0],u[1]).is_zero() + True + sage: ip(u[0],u[2]).is_zero() + True + sage: ip(u[1],u[2]).is_zero() + True + + This Gram-Schmidt routine can be used on matrices as well, so long + as an appropriate inner-product is provided:: + + sage: E11 = matrix(AA, [ [1,0], + ....: [0,0] ]) + sage: E12 = matrix(AA, [ [0,1], + ....: [1,0] ]) + sage: E22 = matrix(AA, [ [0,0], + ....: [0,1] ]) + 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 0.7071067811865475?] [0 0] + [0 0], [0.7071067811865475? 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:: - 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 - """ - def proj(x,y): - return (y.inner_product(x)/x.inner_product(x))*x + if inner_product is None: + inner_product = lambda x,y: x.inner_product(y) + 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 @@ -59,10 +260,24 @@ def gram_schmidt(v): 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 xrange(1,len(v)): + 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() ] @@ -70,7 +285,7 @@ def gram_schmidt(v): # 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() + for i in range(len(v)): + v[i] = sc(v[i], ~norm(v[i])) return v