mjo/eja/eja_utils.py

   1 from sage.functions.other import sqrt
   2 from sage.matrix.constructor import matrix
   3 from sage.modules.free_module_element import vector
   4
   5 def _all2list(x):
   6     r"""
   7     Flatten a vector, matrix, or cartesian product of those things
   8     into a long list.
   9     """
  10     if hasattr(x, 'list'):
  11         # Easy case...
  12         return x.list()
  13     if hasattr(x, 'cartesian_factors'):
  14         # If it's a formal cartesian product space element, then
  15         # we also know what to do...
  16         return sum(( x_i.list() for x_i in x ), [])
  17     else:
  18         # But what if it's a tuple or something else?
  19         return sum( map(_all2list,x), [] )
  20
  21 def _mat2vec(m):
  22         return vector(m.base_ring(), m.list())
  23
  24 def _vec2mat(v):
  25         return matrix(v.base_ring(), sqrt(v.degree()), v.list())
  26
  27 def gram_schmidt(v, inner_product=None):
  28     """
  29     Perform Gram-Schmidt on the list ``v`` which are assumed to be
  30     vectors over the same base ring. Returns a list of orthonormalized
  31     vectors over the smallest extention ring containing the necessary
  32     roots.
  33
  34     SETUP::
  35
  36         sage: from mjo.eja.eja_utils import gram_schmidt
  37
  38     EXAMPLES:
  39
  40     The usual inner-product and norm are default::
  41
  42         sage: v1 = vector(QQ,(1,2,3))
  43         sage: v2 = vector(QQ,(1,-1,6))
  44         sage: v3 = vector(QQ,(2,1,-1))
  45         sage: v = [v1,v2,v3]
  46         sage: u = gram_schmidt(v)
  47         sage: all( u_i.inner_product(u_i).sqrt() == 1 for u_i in u )
  48         True
  49         sage: bool(u[0].inner_product(u[1]) == 0)
  50         True
  51         sage: bool(u[0].inner_product(u[2]) == 0)
  52         True
  53         sage: bool(u[1].inner_product(u[2]) == 0)
  54         True
  55
  56
  57     But if you supply a custom inner product, the result is
  58     orthonormal with respect to that (and not the usual inner
  59     product)::
  60
  61         sage: v1 = vector(QQ,(1,2,3))
  62         sage: v2 = vector(QQ,(1,-1,6))
  63         sage: v3 = vector(QQ,(2,1,-1))
  64         sage: v = [v1,v2,v3]
  65         sage: B = matrix(QQ, [ [6, 4, 2],
  66         ....:                  [4, 5, 4],
  67         ....:                  [2, 4, 9] ])
  68         sage: ip = lambda x,y: (B*x).inner_product(y)
  69         sage: norm = lambda x: ip(x,x)
  70         sage: u = gram_schmidt(v,ip)
  71         sage: all( norm(u_i) == 1 for u_i in u )
  72         True
  73         sage: ip(u[0],u[1]).is_zero()
  74         True
  75         sage: ip(u[0],u[2]).is_zero()
  76         True
  77         sage: ip(u[1],u[2]).is_zero()
  78         True
  79
  80     This Gram-Schmidt routine can be used on matrices as well, so long
  81     as an appropriate inner-product is provided::
  82
  83         sage: E11 = matrix(QQ, [ [1,0],
  84         ....:                    [0,0] ])
  85         sage: E12 = matrix(QQ, [ [0,1],
  86         ....:                    [1,0] ])
  87         sage: E22 = matrix(QQ, [ [0,0],
  88         ....:                    [0,1] ])
  89         sage: I = matrix.identity(QQ,2)
  90         sage: trace_ip = lambda X,Y: (X*Y).trace()
  91         sage: gram_schmidt([E11,E12,I,E22], inner_product=trace_ip)
  92         [
  93         [1 0]  [          0 1/2*sqrt(2)]  [0 0]
  94         [0 0], [1/2*sqrt(2)           0], [0 1]
  95         ]
  96
  97     TESTS:
  98
  99     Ensure that zero vectors don't get in the way::
 100
 101         sage: v1 = vector(QQ,(1,2,3))
 102         sage: v2 = vector(QQ,(1,-1,6))
 103         sage: v3 = vector(QQ,(0,0,0))
 104         sage: v = [v1,v2,v3]
 105         sage: len(gram_schmidt(v)) == 2
 106         True
 107
 108     """
 109     if inner_product is None:
 110         inner_product = lambda x,y: x.inner_product(y)
 111     norm = lambda x: inner_product(x,x).sqrt()
 112
 113     v = list(v) # make a copy, don't clobber the input
 114
 115     # Drop all zero vectors before we start.
 116     v = [ v_i for v_i in v if not v_i.is_zero() ]
 117
 118     if len(v) == 0:
 119         # cool
 120         return v
 121
 122     R = v[0].base_ring()
 123
 124     # Define a scaling operation that can be used on tuples.
 125     # Oh and our "zero" needs to belong to the right space.
 126     scale = lambda x,alpha: x*alpha
 127     zero = v[0].parent().zero()
 128     if hasattr(v[0], 'cartesian_factors'):
 129         P = v[0].parent()
 130         scale = lambda x,alpha: P(tuple( x_i*alpha
 131                                          for x_i in x.cartesian_factors() ))
 132
 133
 134     def proj(x,y):
 135         return scale(x, (inner_product(x,y)/inner_product(x,x)))
 136
 137     # First orthogonalize...
 138     for i in range(1,len(v)):
 139         # Earlier vectors can be made into zero so we have to ignore them.
 140         v[i] -= sum( (proj(v[j],v[i])
 141                       for j in range(i)
 142                       if not v[j].is_zero() ),
 143                      zero )
 144
 145     # And now drop all zero vectors again if they were "orthogonalized out."
 146     v = [ v_i for v_i in v if not v_i.is_zero() ]
 147
 148     # Just normalize. If the algebra is missing the roots, we can't add
 149     # them here because then our subalgebra would have a bigger field
 150     # than the superalgebra.
 151     for i in range(len(v)):
 152         v[i] = scale(v[i], ~norm(v[i]))
 153
 154     return v