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