Numerical investigation of quasilinearization method in quantum mechanics

Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs

“…In a series of recent papers, [1,2] the possibility of applying a very powerful approximation technique called the quasilinearization method (QLM) to physical problems has been discussed. The QLM is designed to confront the nonlinear aspects of physical processes.…”

Quasilinearization method and WKB

“…Recently, however, it was shown [10] by one of the present authors (VBM) that a different proof of the convergence can be provided which allows to extend the applicability of the method to realistic forces defined on infinite intervals with possible singularities at certain points. This proof was generalized and elaborated in the subsequent works [11,12,13,14].…”

“…These equations, unlike the nonlinear Calogero equation [5] considered in references [10,12], contain not only quadratic nonlinear terms but various other forms of nonlinearity and not only the first, but also higher derivatives. It was shown that again just a small number of the QLM iterations yield fast convergent and uniformly excellent and stable numerical results.…”

“…If the initial QLM guess is properly chosen the wave function in all QLM iterations, unlike the WKB wave function, is free of unphysical turning point singularities. Since the first QLM iteration is given by an analytic expression [7,8,10,11,12,13], it allows one to analytically estimate the role of different parameters and the influence of their variation on different characteristics of a quantum system. The next iterates display very fast quadratic convergence so that accuracy of energies obtained after a few iterations is extremely high, reaching up to 20 significant figures for a sixth iterate as we show on the example of different widely used physical potentials.…”

On New High Order Iterative Schemes for Solving Initial Value Problems in Epidemiology

“…At first order, the hybrid iteration scheme reduces to quasilinearization method (QLM) which was originally developed in [1]. More recently Mandelzweig and his co-workers [8][9][10] have extended the application of the QLM to a wide variety of nonlinear BVPs and established that the method converges quadratically. In this work we demonstrate that the proposed hybrid iteration schemes are more accurate and converge faster than the QLM approach.…”