mjo/eja/eja_utils.py

   1 from sage.modules.free_module_element import vector
   2 from sage.rings.number_field.number_field import NumberField
   3 from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
   4 from sage.rings.real_lazy import RLF
   5
   6 def _mat2vec(m):
   7         return vector(m.base_ring(), m.list())
   8
   9 def gram_schmidt(v):
  10     """
  11     Perform Gram-Schmidt on the list ``v`` which are assumed to be
  12     vectors over the same base ring. Returns a list of orthonormalized
  13     vectors over the smallest extention ring containing the necessary
  14     roots.
  15
  16     SETUP::
  17
  18         sage: from mjo.eja.eja_utils import gram_schmidt
  19
  20     EXAMPLES::
  21
  22         sage: v1 = vector(QQ,(1,2,3))
  23         sage: v2 = vector(QQ,(1,-1,6))
  24         sage: v3 = vector(QQ,(2,1,-1))
  25         sage: v = [v1,v2,v3]
  26         sage: u = gram_schmidt(v)
  27         sage: [ u_i.inner_product(u_i).sqrt() == 1 for u_i in u ]
  28         True
  29         sage: u[0].inner_product(u[1]) == 0
  30         True
  31         sage: u[0].inner_product(u[2]) == 0
  32         True
  33         sage: u[1].inner_product(u[2]) == 0
  34         True
  35
  36     TESTS:
  37
  38     Ensure that zero vectors don't get in the way::
  39
  40         sage: v1 = vector(QQ,(1,2,3))
  41         sage: v2 = vector(QQ,(1,-1,6))
  42         sage: v3 = vector(QQ,(0,0,0))
  43         sage: v = [v1,v2,v3]
  44         sage: len(gram_schmidt(v)) == 2
  45         True
  46
  47     """
  48     def proj(x,y):
  49         return (y.inner_product(x)/x.inner_product(x))*x
  50
  51     v = list(v) # make a copy, don't clobber the input
  52
  53     # Drop all zero vectors before we start.
  54     v = [ v_i for v_i in v if not v_i.is_zero() ]
  55
  56     if len(v) == 0:
  57         # cool
  58         return v
  59
  60     R = v[0].base_ring()
  61
  62     # First orthogonalize...
  63     for i in xrange(1,len(v)):
  64         # Earlier vectors can be made into zero so we have to ignore them.
  65         v[i] -= sum( proj(v[j],v[i]) for j in range(i) if not v[j].is_zero() )
  66
  67     # And now drop all zero vectors again if they were "orthogonalized out."
  68     v = [ v_i for v_i in v if not v_i.is_zero() ]
  69
  70     # Now pretend to normalize, building a new ring R that contains
  71     # all of the necessary square roots.
  72     norms_squared = [0]*len(v)
  73
  74     for i in xrange(len(v)):
  75         norms_squared[i] = v[i].inner_product(v[i])
  76         ns = [norms_squared[i].numerator(), norms_squared[i].denominator()]
  77
  78         # Do the numerator and denominator separately so that we
  79         # adjoin e.g. sqrt(2) and sqrt(3) instead of sqrt(2/3).
  80         for j in xrange(len(ns)):
  81             PR = PolynomialRing(R, 'z')
  82             z = PR.gen()
  83             p = z**2 - ns[j]
  84             if p.is_irreducible():
  85                 R = NumberField(p,
  86                                 'sqrt' + str(ns[j]),
  87                                 embedding=RLF(ns[j]).sqrt())
  88
  89     # When we're done, we have to change every element's ring to the
  90     # extension that we wound up with, and then normalize it (which
  91     # should work, since "R" contains its norm now).
  92     for i in xrange(len(v)):
  93         v[i] = v[i].change_ring(R) / R(norms_squared[i]).sqrt()
  94
  95     return v