Newton-Raphson method tolerance

The residues (Nb-kNp at each node) are reduced to zero (a small positive number fixed by specifying an error tolerance at input) iteratively by computing corrections to current values of the unknowns using the Newton-Raphson method (14). Elements of the Jacobian matrix required by this method are computed from analytical expressions. The system of equations to be solved for the corrections has block tridiagonal form and is solved by use of a published software routine (1.5b... 

For the application of the Newton-Raphson method detailed above we must define the functions (as shown in (6.9)), their derivatives (i.e., the Jacobian matrix) and the vector of initial estimates that corresponds to the bulk concentrations of the different species for fc = 0 and, subsequently, to the concentrations of the previous timestep fc — 1. We also have to provide the tolerance criterion together with a maximmn number of iterations such that a convergence error is obtained if the accuracy required is not reached after a reasonable munber of cycles. Different criteria can be imposed in a multidimensional problem ... 

The Newton-Raphson iteration method is terminated when either the absolute values of the residuals and the differences between successive iterates are less than a specified tolerance, or when the specified maximum number of iterations is... 

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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