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 _scale(x, alpha):
   6     r"""
   7     Scale the vector, matrix, or cartesian-product-of-those-things
   8     ``x`` by ``alpha``.
   9     """
  10     if hasattr(x, 'cartesian_factors'):
  11         P = x.parent()
  12         return P(tuple( _scale(x_i, alpha)
  13                         for x_i in x.cartesian_factors() ))
  14     else:
  15         return x*alpha
  16
  17 def _all2list(x):
  18     r"""
  19     Flatten a vector, matrix, or cartesian product of those things
  20     into a long list.
  21
  22     EXAMPLES::
  23
  24         sage: from mjo.eja.eja_utils import _all2list
  25         sage: V1 = VectorSpace(QQ,2)
  26         sage: V2 = MatrixSpace(QQ,2)
  27         sage: x1 = V1([1,1])
  28         sage: x2 = V1([1,-1])
  29         sage: y1 = V2.one()
  30         sage: y2 = V2([0,1,1,0])
  31         sage: _all2list((x1,y1))
  32         [1, 1, 1, 0, 0, 1]
  33         sage: _all2list((x2,y2))
  34         [1, -1, 0, 1, 1, 0]
  35         sage: M = cartesian_product([V1,V2])
  36         sage: _all2list(M((x1,y1)))
  37         [1, 1, 1, 0, 0, 1]
  38         sage: _all2list(M((x2,y2)))
  39         [1, -1, 0, 1, 1, 0]
  40
  41     """
  42     if hasattr(x, 'list'):
  43         # Easy case...
  44         return x.list()
  45     else:
  46         # But what if it's a tuple or something else? This has to
  47         # handle cartesian products of cartesian products, too; that's
  48         # why it's recursive.
  49         return sum( map(_all2list,x), [] )
  50
  51 def _mat2vec(m):
  52         return vector(m.base_ring(), m.list())
  53
  54 def _vec2mat(v):
  55         return matrix(v.base_ring(), sqrt(v.degree()), v.list())
  56
  57 def gram_schmidt(v, inner_product=None):
  58     """
  59     Perform Gram-Schmidt on the list ``v`` which are assumed to be
  60     vectors over the same base ring. Returns a list of orthonormalized
  61     vectors over the smallest extention ring containing the necessary
  62     roots.
  63
  64     SETUP::
  65
  66         sage: from mjo.eja.eja_utils import gram_schmidt
  67
  68     EXAMPLES:
  69
  70     The usual inner-product and norm are default::
  71
  72         sage: v1 = vector(QQ,(1,2,3))
  73         sage: v2 = vector(QQ,(1,-1,6))
  74         sage: v3 = vector(QQ,(2,1,-1))
  75         sage: v = [v1,v2,v3]
  76         sage: u = gram_schmidt(v)
  77         sage: all( u_i.inner_product(u_i).sqrt() == 1 for u_i in u )
  78         True
  79         sage: bool(u[0].inner_product(u[1]) == 0)
  80         True
  81         sage: bool(u[0].inner_product(u[2]) == 0)
  82         True
  83         sage: bool(u[1].inner_product(u[2]) == 0)
  84         True
  85
  86
  87     But if you supply a custom inner product, the result is
  88     orthonormal with respect to that (and not the usual inner
  89     product)::
  90
  91         sage: v1 = vector(QQ,(1,2,3))
  92         sage: v2 = vector(QQ,(1,-1,6))
  93         sage: v3 = vector(QQ,(2,1,-1))
  94         sage: v = [v1,v2,v3]
  95         sage: B = matrix(QQ, [ [6, 4, 2],
  96         ....:                  [4, 5, 4],
  97         ....:                  [2, 4, 9] ])
  98         sage: ip = lambda x,y: (B*x).inner_product(y)
  99         sage: norm = lambda x: ip(x,x)
 100         sage: u = gram_schmidt(v,ip)
 101         sage: all( norm(u_i) == 1 for u_i in u )
 102         True
 103         sage: ip(u[0],u[1]).is_zero()
 104         True
 105         sage: ip(u[0],u[2]).is_zero()
 106         True
 107         sage: ip(u[1],u[2]).is_zero()
 108         True
 109
 110     This Gram-Schmidt routine can be used on matrices as well, so long
 111     as an appropriate inner-product is provided::
 112
 113         sage: E11 = matrix(QQ, [ [1,0],
 114         ....:                    [0,0] ])
 115         sage: E12 = matrix(QQ, [ [0,1],
 116         ....:                    [1,0] ])
 117         sage: E22 = matrix(QQ, [ [0,0],
 118         ....:                    [0,1] ])
 119         sage: I = matrix.identity(QQ,2)
 120         sage: trace_ip = lambda X,Y: (X*Y).trace()
 121         sage: gram_schmidt([E11,E12,I,E22], inner_product=trace_ip)
 122         [
 123         [1 0]  [          0 1/2*sqrt(2)]  [0 0]
 124         [0 0], [1/2*sqrt(2)           0], [0 1]
 125         ]
 126
 127     It even works on Cartesian product spaces whose factors are vector
 128     or matrix spaces::
 129
 130         sage: V1 = VectorSpace(AA,2)
 131         sage: V2 = MatrixSpace(AA,2)
 132         sage: M = cartesian_product([V1,V2])
 133         sage: x1 = V1([1,1])
 134         sage: x2 = V1([1,-1])
 135         sage: y1 = V2.one()
 136         sage: y2 = V2([0,1,1,0])
 137         sage: z1 = M((x1,y1))
 138         sage: z2 = M((x2,y2))
 139         sage: def ip(a,b):
 140         ....:     return a[0].inner_product(b[0]) + (a[1]*b[1]).trace()
 141         sage: U = gram_schmidt([z1,z2], inner_product=ip)
 142         sage: ip(U[0],U[1])
 143         0
 144         sage: ip(U[0],U[0])
 145         1
 146         sage: ip(U[1],U[1])
 147         1
 148
 149     TESTS:
 150
 151     Ensure that zero vectors don't get in the way::
 152
 153         sage: v1 = vector(QQ,(1,2,3))
 154         sage: v2 = vector(QQ,(1,-1,6))
 155         sage: v3 = vector(QQ,(0,0,0))
 156         sage: v = [v1,v2,v3]
 157         sage: len(gram_schmidt(v)) == 2
 158         True
 159     """
 160     if inner_product is None:
 161         inner_product = lambda x,y: x.inner_product(y)
 162     norm = lambda x: inner_product(x,x).sqrt()
 163
 164     v = list(v) # make a copy, don't clobber the input
 165
 166     # Drop all zero vectors before we start.
 167     v = [ v_i for v_i in v if not v_i.is_zero() ]
 168
 169     if len(v) == 0:
 170         # cool
 171         return v
 172
 173     R = v[0].base_ring()
 174
 175     # Our "zero" needs to belong to the right space for sum() to work.
 176     zero = v[0].parent().zero()
 177
 178     sc = lambda x,a: a*x
 179     if hasattr(v[0], 'cartesian_factors'):
 180         # Only use the slow implementation if necessary.
 181         sc = _scale
 182
 183     def proj(x,y):
 184         return sc(x, (inner_product(x,y)/inner_product(x,x)))
 185
 186     # First orthogonalize...
 187     for i in range(1,len(v)):
 188         # Earlier vectors can be made into zero so we have to ignore them.
 189         v[i] -= sum( (proj(v[j],v[i])
 190                       for j in range(i)
 191                       if not v[j].is_zero() ),
 192                      zero )
 193
 194     # And now drop all zero vectors again if they were "orthogonalized out."
 195     v = [ v_i for v_i in v if not v_i.is_zero() ]
 196
 197     # Just normalize. If the algebra is missing the roots, we can't add
 198     # them here because then our subalgebra would have a bigger field
 199     # than the superalgebra.
 200     for i in range(len(v)):
 201         v[i] = sc(v[i], ~norm(v[i]))
 202
 203     return v