From: Michael Orlitzky <michael@orlitzky.com>
Date: Mon, 15 Mar 2021 22:16:17 +0000 (-0400)
Subject: eja: try to speed up gram_schmidt(), but all my sages are broken.
X-Git-Url: http://gitweb.michael.orlitzky.com/?p=sage.d.git;a=commitdiff_plain;h=dc92538d3fc92d16c9b6432ad17c37cb0d6b2be9

eja: try to speed up gram_schmidt(), but all my sages are broken.
---

diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py
index 8422fbf..8334f51 100644
--- a/mjo/eja/eja_utils.py
+++ b/mjo/eja/eja_utils.py
@@ -245,52 +245,40 @@ def gram_schmidt(v, inner_product=None):
         sage: len(gram_schmidt(v)) == 2
         True
     """
+    if len(v) == 0:
+        # cool
+        return v
+
+    V = v[0].parent()
+
     if inner_product is None:
         inner_product = lambda x,y: x.inner_product(y)
+
     def norm(x):
-        ip = inner_product(x,x)
         # Don't expand the given field; the inner-product's codomain
         # is already correct. For example QQ(2).sqrt() returns sqrt(2)
         # in SR, and that will give you weird errors about symbolics
         # when what's really going wrong is that you're trying to
         # orthonormalize in QQ.
-        return ip.parent()(ip.sqrt())
-
-    v = list(v) # make a copy, don't clobber the input
-
-    # Drop all zero vectors before we start.
-    v = [ v_i for v_i in v if not v_i.is_zero() ]
-
-    if len(v) == 0:
-        # cool
-        return v
-
-    # Our "zero" needs to belong to the right space for sum() to work.
-    zero = v[0].parent().zero()
+        return V.base_ring()(inner_product(x,x).sqrt())
 
     sc = lambda x,a: a*x
-    if hasattr(v[0], 'cartesian_factors'):
+    if hasattr(V, 'cartesian_factors'):
         # Only use the slow implementation if necessary.
         sc = _scale
 
     def proj(x,y):
+        # project y onto the span of {x}
         return sc(x, (inner_product(x,y)/inner_product(x,x)))
 
-    # First orthogonalize...
-    for i in range(1,len(v)):
-        # Earlier vectors can be made into zero so we have to ignore them.
-        v[i] -= sum( (proj(v[j],v[i])
-                      for j in range(i)
-                      if not v[j].is_zero() ),
-                     zero )
+    def normalize(x):
+        return sc(x, ~norm(x))
 
-    # And now drop all zero vectors again if they were "orthogonalized out."
-    v = [ v_i for v_i in v if not v_i.is_zero() ]
+    v_out = [] # make a copy, don't clobber the input
 
-    # Just normalize. If the algebra is missing the roots, we can't add
-    # them here because then our subalgebra would have a bigger field
-    # than the superalgebra.
-    for i in range(len(v)):
-        v[i] = sc(v[i], ~norm(v[i]))
+    for (i, v_i) in enumerate(v):
+        ortho_v_i = v_i - V.sum( proj(v_out[j],v_i) for j in range(i) )
+        if not ortho_v_i.is_zero():
+            v_out.append(normalize(ortho_v_i))
 
-    return v
+    return v_out