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