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A nonlinear multiresolution scheme within Harten's framework [12], [13] is presented, based on a new nonlinear, centered piecewise polynomial interpolation technique. Analytical properties of the resulting subdivision scheme, such as convergence, smoothness, and stability, are studied. The stability and the compression properties of the associated multiresolution transform are demonstrated on several numerical experiments on images.

The purpose of this article is to present results that amount to a description of the conjugacy classes of two fourth-order root-finding iterative methods, namely King's family of iterative methods and Jarratt's iterative method, for complex polynomials of degrees two, three and four. For degrees two and three, a full description of the conjugacy classes is accomplished, in each case, by a one-parameter family of polynomials. This is done in such a way that, when one applies one of these two root-finding iterative methods to the elements of these parametrized families, a family of iterative methods is obtained, and its dynamics represents, up to conjugacy, the dynamics of the corresponding iterative root-finding method applied to any complex polynomial having the same degree. For degree four, partial results analogous to the ones just described are presented.

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The dynamics of a classical third-order Newton-type iterative method is studied when it is applied to degrees two and three polynomials. The method is free of second derivatives which is the main limitation of the classical third-order iterative schemes for systems. Moreover, each iteration consists only in two steps of Newton's method having the same derivative. With these two properties the scheme becomes a real alternative to the classical Newton method. Affine conjugacy class of the method when is applied to a differentiable function is given. Chaotic dynamics have been investigated in several examples. Applying the root-finding method to a family of degree three polynomials, we have find a bifurcation diagram as those that appear in the bifurcation of the logistic map in the interval.

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This paper is devoted to the convergence and stability analysis of a class of nonlinear subdivision schemes and associated multiresolution transforms. As soon as a nonlinear scheme can be written as a specific perturbation of a linear and convergent subdivision scheme, we show that if some contractivity properties are satisfied, then stability and convergence can be achieved. This approach is applied to various schemes, which give different new results. More precisely, we study uncentered Lagrange interpolatory linear schemes, WENO scheme (Liu et al., J Comput Phys 115:200-212, 1994), PPH and Power-P schemes (Amat and Liandrat, Appl Comput Harmon Anal 18(2): 198-206, 2005; Serna and Marquina, J Comput Phys 194:632-658, 2004) and a nonlinear scheme using local spherical coordinates (Aspert et al., Comput Aided Geom Des 20:165-187, 2003). Finally, a stability proof is given for the multiresolution transform associated to a nonlinear scheme of Marinov et al. (2005).

This letter is devoted to the stability of the so-called piecewise polynomial harmonic (PPH) multiresolution transform that belongs to the class of data dependent nonlinear multiresolution algorithms. The presentation of the PPH multiresolution as some specific perturbation of a linear multiresolution allows to establish a two step contraction property that leads first to a convergence result and finally to the stability.