X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fsymmetric_pd.py;h=607df2a726ea29890bc1ef8b056a3b954701ede1;hb=928b7d49fda98ff105c92293b5797bb7a2b9873a;hp=b6c86201ac43a55ffc9ce89e3da58e1077c219ec;hpb=93cab80b7a220a16a906be8442ba795d8146df14;p=sage.d.git

diff --git a/mjo/cone/symmetric_pd.py b/mjo/cone/symmetric_pd.py
index b6c8620..607df2a 100644
--- a/mjo/cone/symmetric_pd.py
+++ b/mjo/cone/symmetric_pd.py
@@ -1,4 +1,4 @@
-"""
+r"""
 The symmetric positive definite cone `$S^{n}_{++}$` is the cone
 consisting of all symmetric positive-definite matrices (as a subset of
 $\mathbb{R}^{n \times n}$`. It is the interior of the symmetric positive
@@ -8,98 +8,6 @@ SEMI-definite cone.
 from sage.all import *
 from mjo.cone.symmetric_psd import random_symmetric_psd
 
-def is_symmetric_pd(A):
-    """
-    Determine whether or not the matrix ``A`` is symmetric
-    positive-definite.
-
-    INPUT:
-
-    - ``A`` - The matrix in question.
-
-    OUTPUT:
-
-    Either ``True`` if ``A`` is symmetric positive-definite, or
-    ``False`` otherwise.
-
-    SETUP::
-
-        sage: from mjo.cone.symmetric_pd import (is_symmetric_pd,
-        ....:                                    random_symmetric_pd)
-
-    EXAMPLES:
-
-    The identity matrix is obviously symmetric and positive-definite::
-
-        sage: A = identity_matrix(ZZ, ZZ.random_element(10))
-        sage: is_symmetric_pd(A)
-        True
-
-    The following matrix is symmetric but not positive definite::
-
-        sage: A = matrix(ZZ, [[1, 2], [2, 1]])
-        sage: is_symmetric_pd(A)
-        False
-
-    This matrix isn't even symmetric::
-
-        sage: A = matrix(ZZ, [[1, 2], [3, 4]])
-        sage: is_symmetric_pd(A)
-        False
-
-    The trivial matrix in a trivial space is trivially symmetric and
-    positive-definite::
-
-        sage: A = matrix(QQ, 0,0)
-        sage: is_symmetric_pd(A)
-        True
-
-    The implementation of "is_positive_definite" in Sage leaves a bit to
-    be desired. This matrix is technically positive definite over the
-    real numbers, but isn't symmetric::
-
-        sage: A = matrix(RR,[[1,1],[-1,1]])
-        sage: A.is_positive_definite()
-        Traceback (most recent call last):
-        ...
-        ValueError: Could not see Real Field with 53 bits of precision as a
-        subring of the real or complex numbers
-        sage: is_symmetric_pd(A)
-        False
-
-    TESTS:
-
-    Every gram matrix is positive-definite, and we can sum two
-    positive-definite matrices (``A`` and its transpose) to get a new
-    positive-definite matrix that happens to be symmetric::
-
-        sage: set_random_seed()
-        sage: V = VectorSpace(QQ, ZZ.random_element(5))
-        sage: A = random_symmetric_pd(V)
-        sage: is_symmetric_pd(A)
-        True
-
-    """
-
-    if A.base_ring() == SR:
-        msg = 'The matrix ``A`` cannot be symbolic.'
-        raise ValueError.new(msg)
-
-    # First make sure that ``A`` is symmetric.
-    if not A.is_symmetric():
-        return False
-
-    # If ``A`` is symmetric, we only need to check that it is positive
-    # definite. For that we can consult its minimum eigenvalue, which
-    # should be greater than zero. Since ``A`` is symmetric, its
-    # eigenvalues are guaranteed to be real.
-    if A.is_zero():
-        # A is trivial... so trivially positive-definite.
-        return True
-    else:
-        return min(A.eigenvalues()) > 0
-
-
 def random_symmetric_pd(V):
     r"""
     Generate a random symmetric positive-definite matrix over the
@@ -110,7 +18,7 @@ def random_symmetric_pd(V):
 
     INPUT:
 
-    - ``V`` - The vector space on which the returned matrix will act.
+    - ``V`` -- The vector space on which the returned matrix will act.
 
     OUTPUT:
 
@@ -130,7 +38,6 @@ def random_symmetric_pd(V):
 
     Well, it doesn't crash at least::
 
-        sage: set_random_seed()
         sage: V = VectorSpace(QQ, 2)
         sage: A = random_symmetric_pd(V)
         sage: A.matrix_space()