X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fsymmetric_pd.py;h=e396bbcba8be01a6b53354beee9e31617c49dde4;hb=7af2b9d146a6bf2fb8acc3c342983de577b417ce;hp=9c0d6bae7573a5c085a96c797b9b991b29ec0e88;hpb=e4e4685ec815a523d9ee73a61a3de8b8b2799a1b;p=sage.d.git

diff --git a/mjo/cone/symmetric_pd.py b/mjo/cone/symmetric_pd.py
index 9c0d6ba..e396bbc 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
@@ -142,60 +50,3 @@ def random_symmetric_pd(V):
     # condition ensures that we don't get the zero matrix in a
     # nontrivial space.
     return random_symmetric_psd(V, rank=V.dimension())
-
-
-def inverse_symmetric_pd(M):
-    r"""
-    Compute the inverse of a symmetric positive-definite matrix.
-
-    The symmetric positive-definite matrix ``M`` has a Cholesky
-    factorization, which can be used to invert ``M`` in a fast,
-    numerically-stable way.
-
-    INPUT:
-
-    - ``M`` -- The matrix to invert.
-
-    OUTPUT:
-
-    The inverse of ``M``. The Cholesky factorization uses roots, so the
-    returned matrix will have as its base ring the algebraic closure of
-    the base ring of ``M``.
-
-    SETUP::
-
-        sage: from mjo.cone.symmetric_pd import inverse_symmetric_pd
-
-    EXAMPLES:
-
-    If we start with a rational matrix, its inverse will be over a field
-    extension of that rationals that is necessarily contained in
-    ``QQbar`` (which contails ALL roots)::
-
-        sage: M = matrix(QQ, [[3,1],[1,5]])
-        sage: M_inv = inverse_symmetric_pd(M)
-        sage: M_inv.base_ring().is_subring(QQbar)
-        True
-
-    TESTS:
-
-    This "inverse" should actually act like an inverse::
-
-        sage: set_random_seed()
-        sage: n = ZZ.random_element(5)
-        sage: V = VectorSpace(QQ,n)
-        sage: from mjo.cone.symmetric_pd import random_symmetric_pd
-        sage: M = random_symmetric_pd(V)
-        sage: M_inv = inverse_symmetric_pd(M)
-        sage: I = identity_matrix(QQ,n)
-        sage: M_inv * M == I
-        True
-        sage: M * M_inv == I
-        True
-
-    """
-    # M = L * L.transpose()
-    # The default "echelonize" inverse() method works just fine for
-    # triangular matrices.
-    L_inverse = M.cholesky().inverse()
-    return L_inverse.transpose()*L_inverse