-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
+from sage.structure.element import is_Matrix
-def _mat2vec(m):
- return vector(m.base_ring(), m.list())
+def _scale(x, alpha):
+ r"""
+ Scale the vector, matrix, or cartesian-product-of-those-things
+ ``x`` by ``alpha``.
-def gram_schmidt(v):
+ 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.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])
+ 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: _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, '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()
+
+ 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 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, (1,0,0))
+ 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 )
True
- sage: u[0].inner_product(u[1]) == 0
+ 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(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: u[0].inner_product(u[2]) == 0
+ sage: ip(u[0],u[2]).is_zero()
True
- sage: u[1].inner_product(u[2]) == 0
+ 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
"""
+ if len(v) == 0:
+ # cool
+ return v
+
+ V = v[0].parent()
+
+ if inner_product is None:
+ inner_product = lambda x,y: x.inner_product(y)
+
+ sc = lambda x,a: a*x
+ if hasattr(V, 'cartesian_factors'):
+ # Only use the slow implementation if necessary.
+ sc = _scale
+
def proj(x,y):
- return (y.inner_product(x)/x.inner_product(x))*x
+ # project y onto the span of {x}
+ return sc(x, (inner_product(x,y)/inner_product(x,x)))
- v = list(v) # make a copy, don't clobber the input
+ 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())
- # Drop all zero vectors before we start.
- v = [ v_i for v_i in v if not v_i.is_zero() ]
+ v_out = [] # make a copy, don't clobber the input
- if len(v) == 0:
- # cool
- return v
+ 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))
- R = v[0].base_ring()
-
- # First orthogonalize...
- for i in xrange(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() )
-
- # 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() ]
-
- # Now pretend to normalize, building a new ring R that contains
- # all of the necessary square roots.
- norms_squared = [0]*len(v)
-
- for i in xrange(len(v)):
- norms_squared[i] = v[i].inner_product(v[i])
- ns = [norms_squared[i].numerator(), norms_squared[i].denominator()]
-
- # Do the numerator and denominator separately so that we
- # adjoin e.g. sqrt(2) and sqrt(3) instead of sqrt(2/3).
- for j in xrange(len(ns)):
- PR = PolynomialRing(R, 'z')
- z = PR.gen()
- p = z**2 - ns[j]
- if p.is_irreducible():
- R = NumberField(p,
- 'sqrt' + str(ns[j]),
- embedding=RLF(ns[j]).sqrt())
-
- # When we're done, we have to change every element's ring to the
- # extension that we wound up with, and then normalize it (which
- # should work, since "R" contains its norm now).
- for i in xrange(len(v)):
- v[i] = v[i].change_ring(R) / R(norms_squared[i]).sqrt()
-
- return v
+ return v_out
projects / sage.d.git / blobdiff