Jacobian matrices are used in a wide range of applications from solving the original DAEs to sensitivity analysis. Using Automatic Differentiation the necessary partial derivatives can be provided efficiently within a Modelica-Tool. This paper describes the corresponding implementation work within the OpenModelica Compiler (OMC) to create a symbolic derivative module. This new OMC-feature gener...

We develop general approximate Newton methods for solving Lipschitz continuous equations by replacing the iteration matrix with a consistently approximated Jacobian, thereby reducing the computation in the generalized Newton method. Locally superlinear convergence results are presented under moderate assumptions. To construct a consistently approximated Jacobian, we introduce two main methods: ...

We present an implicit immersed boundary method for the incompressible Navier–Stokes equations capable of handling three-dimensional membrane–fluid flow interactions. The goal of our approach is to greatly improve the time step by using the Jacobian-free Newton–Krylov method (JFNK) to advance the location of the elastic membrane implicitly. The most attractive feature of this Jacobian-free appr...

This paper provides a general proof of a relationship theorem between nonlinear analogue polynomial equations and the corresponding Jacobian matrix, presented recently by the present author. This theorem is also verified generally effective for all nonlinear polynomial algebraic system of equations. As two particular applications of this theorem, we gave a Newton formula without requiring the e...

A singularity handling method is proposed in this paper. It is done by introducing virtual redundant joints into the Jacobian matrix to maintain the rank of the Jacobian matrix when singularity occurs. These additional joints do not exist physically. Therefore, although mathematically stable, the manipulator still can not perform tasks in the degenerate direction(s). This method is comparativel...

A major problem in obtaining an efficient implementation of fully implicit RungeKutta (IRK) methods applied to systems of differential equations is to solve the underlying systems of nonlinear equations. Their solution is usually obtained by application of modified Newton iterations with an approximate Jacobian matrix. The systems of linear equations of the modified Newton method can actually b...

A major problem in obtaining an efficient implementation of fully implicit RungeKutta (IRK) methods applied to systems of differential equations is to solve the underlying systems of nonlinear equations. Their solution is usually obtained by application of modified Newton iterations with an approximate Jacobian matrix. The systems of linear equations of the modified Newton method can actually b...

We describe an algorithm for solving three-dimensional vector systems of parabolic partial diierential equations. Solutions are calculated using Galerkin's method with a high-order, piecewise polynomial hierarchical basis in space and the DAE code DASPK in time. A variant of DASPK that computes Jacobian vector products with a stored analytic Jacobian is compared with the default matrix-free app...

An extension to operational space (EXOS) is presented for the explicit representation of the null-space (NS) dynamics and its interaction with the operational-space dynamics. First, the EXOS Jacobian is formed by augmenting the Jacobian matrix with a minimum number of its NS vectors. Based on the EXOS Jacobian, free of algorithmic singularity, the kinematics, statics, and dynamics of a redundan...

Since metabolome data are derived from the underlying metabolic network, reverse engineering of such data to recover the network topology is of wide interest. Lyapunov equation puts a constraint to the link between data and network by coupling the covariance of data with the strength of interactions (Jacobian matrix). This equation, when expressed as a linear set of equations at steady state, c...