Approximate Jacobian Research Articles

Solution of nonlinear Poisson-type equations

We consider approximate and truncated Newton methods for solving nonlinear Poisson-type systems of equations with emphasis on vector computers. We have selected two model differential equations for our analysis. The first, after finite difference discretization, yields a nonlinear system with a symmetric positive definite Jacobian matrix, while the second yields a nonsymmetric Jacobian matrix which is not, in general, positive definite. In the case of a symmetric positive definite Jacobian matrix, the linearized Newton systems are solved by (preconditioned) conjugate gradient methods. In particular, we consider incomplete Cholesky factorization in conjunction with the red-black ordering scheme and symmetric diagonal scaling as preconditioners. In the case of a nonsymmetric Jacobian matrix, we construct symmetric positive definite approximations and analyze the effect of these approximations. We develop, and present theory for, a truncated approximate Newton method. This method uses an approximate Jacobian matrix in addition to truncation in the solution of the Newton system. Convergence properties and work reduction of this new method have been found to be very competitive.

Hybrid method for nonlinear least-square problems without calculating derivatives

This paper presents a no-derivative modification of the hybrid Gauss-Newton-BFGS method for nonlinear least-square problems suggested initially by Al-Baali and Fletcher and modified later by Fletcher and Xu. The modification is made in such a way that, in a Gauss-Newton step, the Broyden's rank-one updating formula is used to obtain an approximate Jacobian and, in a BFGS step, the Jacobian is estimated using difference formulas. A set of numerical comparisons among the new hybrid method, the Gauss-Newton-Broyden method, and the finite-difference BFGS method is made and shows that the new hybrid method combines the better features of the Gauss-Newton-Broyden method and the finite-difference BFGS method. This paper also extends to the least-square problem the finite-termination property of the Broyden method, proved for a nonsingular system of equations by Gay and for the full-rank rectangular system of equations by Gerber and Luk.

Voltage stability assessment through measurement of a reduced state vector

The use of a real-time measurement system for the monitoring and control of voltage stability in power systems calls for the investigation of the possibility of reducing the number of the measurements. A clustering algorithm based on graph theory and an arbitrary coherency function is presented, and its applicability is tested for two proposed coherency criteria. The objective of the measurements was the monitoring of the proximity of the system state to voltage instability. The proximity indicators selected are the minimum singular values of the system Jacobian matrix. In the case of the reduced number of measurements, the approximate Jacobian was determined by assuming that all the elements of the state vector from one cluster have the same values as the one representative measurement taken from that cluster. The proximity indicator was then calculated from such an approximated matrix and compared with the values obtained by simulation when the acquisition of the complete state vector was assumed. The results show that reduced measurements are adequate for predicting voltage instability.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A New Improved Algorithm for OLTC Transformers in the Fast Decoupled Load Flow

This paper describes a new improved version of the fast decoupled load flow that directly incorporates the on-load tap changing transformer ratios into the approximate Jacobian matrix so that the tap ratios are adjusted during solution in the same manner as the bus voltage magnitudes. This algorithm has been found to have superior convergence reliability to the normal fast decoupled load flow, particularly for severe contingency cases where the power system is close to voltage collapse. This algorithm has been found to be particularly suited to on-line applications where convergence reliability takes precedence over solution speed.

A Quasi-Newton Method Employing Direct Secant Updates of Matrix Factorizations

A quasi-Newton algorithm is presented which employs an initial factorization of the approximate Jacobian matrix. The method then updates the upper and lower triangular factors directly at each step. Iterates are generated using forward and backward substitution employing the updated factorizations. The method is shown to be locally and q-superlinearly convergent, and a numerical algorithm is presented.

Two classes of internally 𝑆-stable generalized Runge-Kutta processes which remain consistent with an inaccurate Jacobian

Generalized Runge-Kutta Processes for stiff systems of ordinary differential equations usually require an accurate evaluation of a Jacobian at every step. However, it is possible to derive processes which are Internally S-stable when an accurate Jacobian is used but still remain consistent and highly stable if an approximate Jacobian is used. It is shown that these processes require at least as many function evaluations as an explicit Runge-Kutta process of the same order, and second and third order processes are developed. A second class of Generalized Runge-Kutta is introduced which requires that the Jacobian be evaluated accurately less than once every step. A third order process of this class is developed, and all three methods contain an error estimator similar to those of Fehlberg or England.

Risoluzione di un sistema di equazioni non lineari con metodo numerico ai minimi quadrati

We introduce a numerical method for solving nonlinear algebraic equations related to linear least squares fitting with moving weights. The method considered introduces little overhead in the process. The approach is based on updating two matrices so that the product is an approximation of the inverse Jacobian. We may expect: 1. The tecnique performs good for stochastic systems 2. We never may yield an unbounded correction 3. We can use the algorithm for ill-conditioned, also completely singular problems.

Implementation of Implicit Formulas for the Solution of ODEs

Implicit formulas are quite popular for the solution of ODES. They seem to be necessary for the solution of stiff problems. Every code based on an implicit formula must deal with certain tasks studied in this paper: (i) A choice of basic variable has to be made. The literature is confusing as to what the possibilities are and the consequences of the choice. These matters are clarified. (ii) A test for convergence of the method for solving the implicit equations must be made. Ways of improving the reliability of this test are studied. (iii) Deciding when to form a new approximate Jacobian and/or iteration matrix is a crucial issue for the efficiency of a code. New insight which suggests rather specific actions will be developed. The paper closes with an interesting numerical example.

Recursive Collocation for the Numerical Solution of Stiff Ordinary Differential Equations

The exact solution of a given stiff system of nonlinear (homogeneous) ordinary differential equations on a given interval I is approximated, on each subinterval ${\sigma _k}$ corresponding to a partition ${\pi _N}$ of I, by a linear combination ${U_k}(x)$ of exponential functions. The function ${U_k}(x)$ will involve only the "significant" eigenvalues (in a sense to be made precise) of the approximate Jacobian for ${\sigma _k}$. The unknown vectors in ${U_k}(x)$ are computed recursively by requiring that ${U_k}(x)$ satisfy the given system at certain suitable points in ${\sigma _k}$ (collocation), with the additional condition that the collection of these functions $\{ {U_k}\}$ represent a continuous function on I satisfying the given initial conditions.

Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian

For solving large systems of nonlinear equations by quasi-Newton methods it may often be preferable to store an approximation to the Jacobian rather than an approximation to the inverse Jacobian. The main reason is that when the Jacobian is sparse and the locations of the zeroes are known, the updating procedure can be made more efficient for the approximate Jacobian than for the approximate inverse Jacobian.

Modification of a Quasi-Newton Method for Nonlinear Equations with a Sparse Jacobian

For solving large systems of nonlinear equations by quasi-Newton methods it may often be preferable to store an approximation to the Jacobian rather than an approximation to the inverse Jacobian. The main reason is that when the Jacobian is sparse and the locations of the zeroes are known, the updating procedure can be made more efficient for the approximate Jacobian than for the approximate inverse Jacobian.