Newton-Raphson algorithm Jacobian

Algorithmic Details for NLP Methods All the above NLP methods incorporate concepts from the Newton-Raphson method for equation solving. Essential features of these methods are that they rovide (1) accurate derivative information to solve for the KKT con-itions, (2) stabilization strategies to promote convergence of the Newton-like method from poor starting points, and (3) regularization of the Jacobian matrix in Newton s method (the so-called KKT matrix) if it becomes singular or ill-conditioned. 

C.M. Bethke (52) has shown that significant numerical advantages in such calculations can be realized by switching into the basis set a mineral species that is in partial equilibrium with the aqueous phase. This avoids expansion of the size of the Jacobian matrix and reduces computation time. A method based on this concept is being developed for use in the 3270 version of EQ3/6. The concept appears to show promise for improvement of the "optimizer" algorithm as well as the Newton-Raphson one. 

Broyden s algorithm consists of successively updating of the jacobian matrix of the Newton-Raphson equations by use of the correction matrix xCy7, that is,... 

After the Broyden correction for the independent variables has been computed, Broyden proposed that the inverse of the jacobian matrix of the Newton-Raphson equations be updated by use of Householder s formula. Herein lies the difficulty with Broyden s method. For Newton-Raphson formulations such as the Almost Band Algorithm for problems involving highly nonideal solutions, the corresponding jacobian matrices are exceedingly sparse, and the inverse of a sparse matrix is not necessarily sparse. The sparse characteristic of these jacobian matrices makes the application of Broyden s method (wherein the inverse of the jacobian matrix is updated by use of Householder s formula) impractical. 

Equations (15-58), (15-59), and (15-60) are solved simultaneously by the Newton-Raphson iterative method in which successive sets of the output variables are produced until the values of the M, E, and H functions are driven to within some tolerance of zero. During the iterations, nonzero values of the functions are called discrepancies or errors. Let the functions and output variables be grouped by stage in order from top to bottom. As will be shown, this is done to produce a block tridiagonal structure for the Jacobian matrix of partial derivatives so that the Thomas algorithm can be applied. Let... 

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