Nearest singular polynomials (Q1281843)

The following problem is considered: Given a monic polynomial \(f(x)\) \[ f= x^m+ \sum^m_{j=1} f_j\cdot x^{m-j},\quad f_j\in\mathbb{C}. \] Find a monic polynomial \(h(x)\) \[ h= (x- c)^k\cdot \Biggl(x^{m-k}+ \sum^{m- k}_{j= 1} \Phi_j\cdot x^{m- k-j}\Biggr),\quad c,\Phi_j\in \mathbb{C},\;k\geq 2 \] such that \(\| f- h\|^2\to\) minimum, where \(\| p(x)\|^2= \sum^k_{j= 0}| p_j|^2\) for any \(p(x)= \sum^k_{j= 0} p_j\cdot x^{k-j}\), \(p_j\in\mathbb{C}\). Some recursive relations between the polynomials determining the multiple zeros for consecutive \(k\)'s are presented. Two numerical examples are given.