X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Feja%2Feja_utils.py;h=c25b81921e1be4f0d6a77580227cb8692e21605f;hb=9efefa3e54fc3e69e3f2c78457d50127a7a10131;hp=701d5f561f6ef2418cee5f914ceb030ee9a6f5cb;hpb=2f0d9b59cd2184614bbbce803f78a40778e812ef;p=sage.d.git diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py index 701d5f5..c25b819 100644 --- a/mjo/eja/eja_utils.py +++ b/mjo/eja/eja_utils.py @@ -1,6 +1,203 @@ +from sage.functions.other import sqrt from sage.matrix.constructor import matrix from sage.modules.free_module_element import vector -from sage.functions.other import sqrt + +def _scale(x, alpha): + r""" + Scale the vector, matrix, or cartesian-product-of-those-things + ``x`` by ``alpha``. + """ + 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()) + +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. + + SETUP:: + + sage: from mjo.eja.eja_utils import gram_schmidt + + EXAMPLES: + + 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: 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 + + + 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(QQ,(1,2,3)) + sage: v2 = vector(QQ,(1,-1,6)) + sage: v3 = vector(QQ,(2,1,-1)) + sage: v = [v1,v2,v3] + sage: B = matrix(QQ, [ [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(QQ, [ [1,0], + ....: [0,0] ]) + sage: E12 = matrix(QQ, [ [0,1], + ....: [1,0] ]) + sage: E22 = matrix(QQ, [ [0,0], + ....: [0,1] ]) + sage: I = matrix.identity(QQ,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] + ] + + 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: 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() + + # 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() ), + 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 range(len(v)): + v[i] = sc(v[i], ~norm(v[i])) + + return v