There are some issues related to divide and conquer algorithm. Sorting with Permutation Matrices [22] If d1 < d2 ¡¤¡¤¡¤ < dn then the sequence of eigenvalues obtained will be ¦Ë1 < ¦Ë2 < ¡¤¡¤¡¤ < ¦Ën. we may not come across a diagonal matrix with sorted diagonal elements after rank one modification. we need to apply permutation to sort them in ascending order using permutation matrix. The vertical blue lines represent the diagonal elements after the rank one modification and eigendecomposition of smaller matrices. the linear approximation would be horizontal and the next approximation would be a large negative number which is not useful. This happens particularly when the weights are small.The first scheme is named as approaching from left because the algorithm will produce a sequence of monotonically increasing approximations to desired root provided. the scheme will yield a sequence of approximations converging monotonically upwards to the desired root.