eja: use cached charpoly for element inverse() if possible.

[sage.d.git] / mjo / eja / TODO

diff --git a/mjo/eja/TODO b/mjo/eja/TODO
index dd671c5fd7ab847a4c635748923bf0cba12a63ad..87cdcbcb57c781449590f999b0f14589da705d0e 100644 (file)
--- a/mjo/eja/TODO
+++ b/mjo/eja/TODO
@@ -1,44 +1,36 @@
-Trace inner product tests:
-
-            TESTS:
-
-            The trace inner product is commutative::
-
-            sage: set_random_seed()
-            sage: J = random_eja()
-            sage: x = J.random_element(); y = J.random_element()
-            sage: x.trace_inner_product(y) == y.trace_inner_product(x)
-            True
-
-            The trace inner product is bilinear::
-
-            sage: set_random_seed()
-            sage: J = random_eja()
-            sage: x = J.random_element()
-            sage: y = J.random_element()
-            sage: z = J.random_element()
-            sage: a = QQ.random_element();
-            sage: actual = (a*(x+z)).trace_inner_product(y)
-            sage: expected = a*x.trace_inner_product(y) + a*z.trace_inner_product(y)
-            sage: actual == expected
-            True
-            sage: actual = x.trace_inner_product(a*(y+z))
-            sage: expected = a*x.trace_inner_product(y) +  a*x.trace_inner_product(z)
-            sage: actual == expected
-            True
-
-            The trace inner product is associative::
-
-            sage: pass
-
-            The trace inner product satisfies the compatibility
-            condition in the definition of a Euclidean Jordan algebra:
-
-            sage: set_random_seed()
-            sage: J = random_eja()
-            sage: x = J.random_element()
-            sage: y = J.random_element()
-            sage: z = J.random_element()
-            sage: (x*y).trace_inner_product(z) == y.trace_inner_product(x*z)
-            True
-           
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+1. Add CartesianProductEJA.
+
+2. Add references and start citing them.
+
+3. Implement the octonion simple EJA.
+
+4. Override random_instance(), one(), et cetera in DirectSumEJA.
+
+5. Switch to QQ in *all* algebras for _charpoly_coefficients().
+   This only works when we know that the basis can be rationalized...
+   which is the case at least for the concrete EJAs we provide,
+   but not in general.
+
+6. Pass already_echelonized (default: False) and echelon_basis
+   (default: None) into the subalgebra constructor. The value of
+   already_echelonized can be passed to V.span_of_basis() to save
+   some time, and usinf e.g. FreeModule_submodule_with_basis_field
+   we may somehow be able to pass the echelon basis straight in to
+   save time.
+
+   This may require supporting "basis" as a list of basis vectors
+   (as opposed to superalgebra elements) in the subalgebra constructor.
+
+7. The inner product should be an *argument* to the main EJA
+   constructor.  Afterwards, the basis normalization step should be
+   optional (and enabled by default) for ALL algebras, since any
+   algebra can have a nonstandard inner-product and its basis can be
+   normalized with respect to that inner- product. For example, the
+   HadamardEJA could be equipped with an inner- product that is twice
+   the usual one. Then for the basis to be orthonormal, we would need
+   to divide e.g. (1,0,0) by <(1,0,0),(1,0,0)> = 2 to normalize it.
+
+8. Pre-cache charpoly for some small algebras?
+
+9. Compute the scalar in the general natural_inner_product() for
+   matrices, so no overrides are necessary.
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