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     This works around the inability to scale certain elements of
  11     Cartesian product spaces, as reported in
  12
  13       https://trac.sagemath.org/ticket/31435
  14
  15     ..WARNING:
  16
  17         This will do the wrong thing if you feed it a tuple or list.
  18
  19     SETUP::
  20
  21         sage: from mjo.eja.eja_utils import _scale
  22
  23     EXAMPLES::
  24
  25         sage: v = vector(QQ, (1,2,3))
  26         sage: _scale(v,2)
  27         (2, 4, 6)
  28         sage: m = matrix(QQ, [[1,2],[3,4]])
  29         sage: M = cartesian_product([m.parent(), m.parent()])
  30         sage: _scale(M((m,m)), 2)
  31         ([2 4]
  32         [6 8], [2 4]
  33         [6 8])
  34
  35     """
  36     if hasattr(x, 'cartesian_factors'):
  37         P = x.parent()
  38         return P(tuple( _scale(x_i, alpha)
  39                         for x_i in x.cartesian_factors() ))
  40     else:
  41         return x*alpha
  42
  43
  44 def _all2list(x):
  45     r"""
  46     Flatten a vector, matrix, or cartesian product of those things
  47     into a long list.
  48
  49     EXAMPLES::
  50
  51         sage: from mjo.eja.eja_utils import _all2list
  52         sage: V1 = VectorSpace(QQ,2)
  53         sage: V2 = MatrixSpace(QQ,2)
  54         sage: x1 = V1([1,1])
  55         sage: x2 = V1([1,-1])
  56         sage: y1 = V2.one()
  57         sage: y2 = V2([0,1,1,0])
  58         sage: _all2list((x1,y1))
  59         [1, 1, 1, 0, 0, 1]
  60         sage: _all2list((x2,y2))
  61         [1, -1, 0, 1, 1, 0]
  62         sage: M = cartesian_product([V1,V2])
  63         sage: _all2list(M((x1,y1)))
  64         [1, 1, 1, 0, 0, 1]
  65         sage: _all2list(M((x2,y2)))
  66         [1, -1, 0, 1, 1, 0]
  67
  68     """
  69     if hasattr(x, 'list'):
  70         # Easy case...
  71         return x.list()
  72     else:
  73         # But what if it's a tuple or something else? This has to
  74         # handle cartesian products of cartesian products, too; that's
  75         # why it's recursive.
  76         return sum( map(_all2list,x), [] )
  77
  78 def _mat2vec(m):
  79         return vector(m.base_ring(), m.list())
  80
  81 def _vec2mat(v):
  82         return matrix(v.base_ring(), sqrt(v.degree()), v.list())
  83
  84 def gram_schmidt(v, inner_product=None):
  85     """
  86     Perform Gram-Schmidt on the list ``v`` which are assumed to be
  87     vectors over the same base ring. Returns a list of orthonormalized
  88     vectors over the smallest extention ring containing the necessary
  89     roots.
  90
  91     SETUP::
  92
  93         sage: from mjo.eja.eja_utils import gram_schmidt
  94
  95     EXAMPLES:
  96
  97     The usual inner-product and norm are default::
  98
  99         sage: v1 = vector(QQ,(1,2,3))
 100         sage: v2 = vector(QQ,(1,-1,6))
 101         sage: v3 = vector(QQ,(2,1,-1))
 102         sage: v = [v1,v2,v3]
 103         sage: u = gram_schmidt(v)
 104         sage: all( u_i.inner_product(u_i).sqrt() == 1 for u_i in u )
 105         True
 106         sage: bool(u[0].inner_product(u[1]) == 0)
 107         True
 108         sage: bool(u[0].inner_product(u[2]) == 0)
 109         True
 110         sage: bool(u[1].inner_product(u[2]) == 0)
 111         True
 112
 113
 114     But if you supply a custom inner product, the result is
 115     orthonormal with respect to that (and not the usual inner
 116     product)::
 117
 118         sage: v1 = vector(QQ,(1,2,3))
 119         sage: v2 = vector(QQ,(1,-1,6))
 120         sage: v3 = vector(QQ,(2,1,-1))
 121         sage: v = [v1,v2,v3]
 122         sage: B = matrix(QQ, [ [6, 4, 2],
 123         ....:                  [4, 5, 4],
 124         ....:                  [2, 4, 9] ])
 125         sage: ip = lambda x,y: (B*x).inner_product(y)
 126         sage: norm = lambda x: ip(x,x)
 127         sage: u = gram_schmidt(v,ip)
 128         sage: all( norm(u_i) == 1 for u_i in u )
 129         True
 130         sage: ip(u[0],u[1]).is_zero()
 131         True
 132         sage: ip(u[0],u[2]).is_zero()
 133         True
 134         sage: ip(u[1],u[2]).is_zero()
 135         True
 136
 137     This Gram-Schmidt routine can be used on matrices as well, so long
 138     as an appropriate inner-product is provided::
 139
 140         sage: E11 = matrix(QQ, [ [1,0],
 141         ....:                    [0,0] ])
 142         sage: E12 = matrix(QQ, [ [0,1],
 143         ....:                    [1,0] ])
 144         sage: E22 = matrix(QQ, [ [0,0],
 145         ....:                    [0,1] ])
 146         sage: I = matrix.identity(QQ,2)
 147         sage: trace_ip = lambda X,Y: (X*Y).trace()
 148         sage: gram_schmidt([E11,E12,I,E22], inner_product=trace_ip)
 149         [
 150         [1 0]  [          0 1/2*sqrt(2)]  [0 0]
 151         [0 0], [1/2*sqrt(2)           0], [0 1]
 152         ]
 153
 154     It even works on Cartesian product spaces whose factors are vector
 155     or matrix spaces::
 156
 157         sage: V1 = VectorSpace(AA,2)
 158         sage: V2 = MatrixSpace(AA,2)
 159         sage: M = cartesian_product([V1,V2])
 160         sage: x1 = V1([1,1])
 161         sage: x2 = V1([1,-1])
 162         sage: y1 = V2.one()
 163         sage: y2 = V2([0,1,1,0])
 164         sage: z1 = M((x1,y1))
 165         sage: z2 = M((x2,y2))
 166         sage: def ip(a,b):
 167         ....:     return a[0].inner_product(b[0]) + (a[1]*b[1]).trace()
 168         sage: U = gram_schmidt([z1,z2], inner_product=ip)
 169         sage: ip(U[0],U[1])
 170         0
 171         sage: ip(U[0],U[0])
 172         1
 173         sage: ip(U[1],U[1])
 174         1
 175
 176     TESTS:
 177
 178     Ensure that zero vectors don't get in the way::
 179
 180         sage: v1 = vector(QQ,(1,2,3))
 181         sage: v2 = vector(QQ,(1,-1,6))
 182         sage: v3 = vector(QQ,(0,0,0))
 183         sage: v = [v1,v2,v3]
 184         sage: len(gram_schmidt(v)) == 2
 185         True
 186     """
 187     if inner_product is None:
 188         inner_product = lambda x,y: x.inner_product(y)
 189     norm = lambda x: inner_product(x,x).sqrt()
 190
 191     v = list(v) # make a copy, don't clobber the input
 192
 193     # Drop all zero vectors before we start.
 194     v = [ v_i for v_i in v if not v_i.is_zero() ]
 195
 196     if len(v) == 0:
 197         # cool
 198         return v
 199
 200     R = v[0].base_ring()
 201
 202     # Our "zero" needs to belong to the right space for sum() to work.
 203     zero = v[0].parent().zero()
 204
 205     sc = lambda x,a: a*x
 206     if hasattr(v[0], 'cartesian_factors'):
 207         # Only use the slow implementation if necessary.
 208         sc = _scale
 209
 210     def proj(x,y):
 211         return sc(x, (inner_product(x,y)/inner_product(x,x)))
 212
 213     # First orthogonalize...
 214     for i in range(1,len(v)):
 215         # Earlier vectors can be made into zero so we have to ignore them.
 216         v[i] -= sum( (proj(v[j],v[i])
 217                       for j in range(i)
 218                       if not v[j].is_zero() ),
 219                      zero )
 220
 221     # And now drop all zero vectors again if they were "orthogonalized out."
 222     v = [ v_i for v_i in v if not v_i.is_zero() ]
 223
 224     # Just normalize. If the algebra is missing the roots, we can't add
 225     # them here because then our subalgebra would have a bigger field
 226     # than the superalgebra.
 227     for i in range(len(v)):
 228         v[i] = sc(v[i], ~norm(v[i]))
 229
 230     return v