Second-order Tangent Solvers for Systems of Parameterized Nonlinear Equations

Algorithmic Differentiation of Numerical Methods: Second-Order Tangent and Adjoint Solvers for Systems of Parametrized Nonlinear Equations

Forward and reverse modes of algorithmic differentiation (AD) transform implementations of multivariate vector functions F : IR → IR as computer programs into tangent and adjoint code, respectively. The reapplication of the same ideas yields higher derivative code. In particular, second derivatives play an important role in nonlinear programming. Second-order methods based on Newton’s algorithm...

Algorithmic Differentiation of Numerical Methods: Tangent-Linear and Adjoint Solvers for Systems of Nonlinear Equations

We discuss software tool support for the Algorithmic Differentiation (also known as Automatic Differentiation; AD) of numerical simulation programs that contain calls to solvers for parameterized systems of n nonlinear equations. The local computational overhead as well as the additional memory requirement for the computation of directional derivatives or adjoints of the solution of the nonline...

THIRD-ORDER AND FOURTH-ORDER ITERATIVE METHODS FREE FROM SECOND DERIVATIVE FOR FINDING MULTIPLE ROOTS OF NONLINEAR EQUATIONS

In this paper, we present two new families of third-order and fourth-order methods for finding multiple roots of nonlinear equations. Each of them requires one evaluation of the function and two of its first derivative per iteration. Several numerical examples are given to illustrate the performance of the presented methods.    

Seventh-order iterative algorithm free from second derivative for solving algebraic nonlinear equations

Comparing Solvers for Large Systems of Nonlinear Algebraic Equations

The solution of systems of nonlinear algebraic equations is a problem which complexity grows as the number of equations increases. The number of engineering problems, which led to these systems, is increasing each year. In this paper we compare the performance of four advanced solvers for systems of nonlinear algebraic equations which have been collected from Internet repositories. We are parti...