This paper deals with the quadratic inverse eigenvalue problem (QIEP), which here is proposed to be solved by exploiting the properties of the smallest singular value of a matrix, within the context of a Newton method and Newton-like methods. Under the assumptions that prescribed eigenvalues are distinct, QIEP has a solution $c$ and the Jacobian matrix $J_g(c)$ is nonsingular, the paper shows that the convergence is locally quadratic. Remarkably, the algorithms are described in detail. Significant numerical results are reported to enlighten the effectiveness of the proposed methods.