operator_pd_vdvt_update.py 文件源码

def _sqrt_solve(self, rhs):
    # Recall the square root of this operator is M + VDV^T.
    # The Woodbury formula gives:
    # (M + VDV^T)^{-1}
    # = M^{-1} - M^{-1} V (D^{-1} + V^T M^{-1} V)^{-1} V^T M^{-1}
    # = M^{-1} - M^{-1} V C^{-1} V^T M^{-1}
    # where C is the capacitance matrix.
    # TODO(jvdillon) Determine if recursively applying rank-1 updates is more
    # efficient.  May not be possible because a general n x n matrix can be
    # represeneted as n rank-1 updates, and solving with this matrix is always
    # done in O(n^3) time.
    m = self._operator
    v = self._v
    cchol = self._chol_capacitance(batch_mode=False)

    # The operators will use batch/singleton mode automatically.  We don't
    # override.
    # M^{-1} rhs
    minv_rhs = m.solve(rhs)
    # V^T M^{-1} rhs
    vt_minv_rhs = math_ops.matmul(v, minv_rhs, transpose_a=True)
    # C^{-1} V^T M^{-1} rhs
    cinv_vt_minv_rhs = linalg_ops.cholesky_solve(cchol, vt_minv_rhs)
    # V C^{-1} V^T M^{-1} rhs
    v_cinv_vt_minv_rhs = math_ops.matmul(v, cinv_vt_minv_rhs)
    # M^{-1} V C^{-1} V^T M^{-1} rhs
    minv_v_cinv_vt_minv_rhs = m.solve(v_cinv_vt_minv_rhs)

    # M^{-1} - M^{-1} V C^{-1} V^T M^{-1}
    return minv_rhs - minv_v_cinv_vt_minv_rhs