* Module wpylib.math.linalg.gram_schmidt to orthonormalize a set of

columns vectors.

math/linalg/gram_schmidt.py

@@ -0,0 +1,91 @@
+#
+# wpylib.math.linalg.gram_schmidt module
+# Created: 20150401
+# Wirawan Purwanto
+#
+"""
+wpylib.math.linalg.gram_schmidt
+
+Provides Gram-Schmidt orthogonalization of a vector set (a 2-D array).
+
+We will use the modified algorithm that is more stable numerically.
+
+Reference: http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process
+"""
+
+
+def modgs_classic(V):
+  """Gram-Schmidt orthonormalization for the column vectors in matrix V.
+
+  Classic routine, all hand-written, no acceleration.
+  This is the reference implementation.
+  """
+  from numpy import array, vdot, empty, outer, sqrt, real
+  V = array(V, copy=True)
+  assert len(V.shape) == 2
+  numcols = V.shape[1]
+  N_orig = empty((numcols,), dtype=V.dtype)
+  for i in xrange(numcols):
+    Vi = V[:,i]
+    N_orig[i] = real(vdot(Vi, Vi))  # always a real number
+    # FIXME: below could blow up if the norm is exactly zero
+    Vi /= sqrt(N_orig[i])
+    Ui = Vi  # now Ui has been orthonormalized
+    for j in xrange(i+1, numcols):
+      Vj = V[:,j]
+      proj_Ui_Vj = vdot(Ui, Vj)
+      Vj -= proj_Ui_Vj * Ui
+
+  return V, N_orig.real
+
+
+def modgs_fast1(V):
+  """Gram-Schmidt orthonormalization for the column vectors in matrix V.
+
+  Fast(er) routine, replaced inner loop with mat-vec and outer products.
+  """
+  from numpy import array, dot, vdot, empty, outer, sqrt, real
+  V = array(V, copy=True)
+  assert len(V.shape) == 2
+  numcols = V.shape[1]
+  N_orig = empty((numcols,), dtype=V.dtype)
+  for i in xrange(numcols):
+    Vi = V[:,i]
+    N_orig[i] = real(vdot(Vi, Vi))  # always a real number
+    # FIXME: below could blow up if the norm is exactly zero
+    Vi /= sqrt(N_orig[i])
+    Ui = Vi  # now Ui has been orthonormalized
+    # Now Vi is normalized -> renamed to Ui
+    if i+1 < numcols:
+      Vjj = V[:,i+1:]
+      proj_u_Vjj = dot(Ui.conj(), Vjj)
+      Vjj -= outer(Ui, proj_u_Vjj)
+
+  return V, N_orig.real
+
+
+def modgs_einsum(V):
+  """Gram-Schmidt orthonormalization for the column vectors in matrix V.
+
+  Fast(er) routine, using einsum.
+  """
+  from numpy import array, vdot, einsum, empty, sqrt, real
+  V = array(V, copy=True)
+  assert len(V.shape) == 2
+  numcols = V.shape[1]
+  N_orig = empty((numcols,), dtype=V.dtype)
+  for i in xrange(numcols):
+    Vi = V[:,i]
+    N_orig[i] = real(vdot(Vi, Vi))  # always a real number
+    # FIXME: below could blow up if the norm is exactly zero
+    Vi /= sqrt(N_orig[i])
+    Ui = Vi  # now Ui has been orthonormalized
+    # Now Vi is normalized -> renamed to Ui
+    if i+1 < numcols:
+      Vjj = V[:,i+1:]
+      Vjj -= einsum('i,j,jk', Ui, Ui.conj(), Vjj)
+
+  return V, N_orig.real
+
+
+modgs = modgs_fast1