Residual function Newton-Raphson iteration

Fig. 4.2. Newton-Raphson iteration for solving two nonlinear equations containing the unknown variables x and y. Planes are drawn tangent to the residual functions R and R2 at an initial estimate (r, > (o)) to the value of the root. The improved guess (v(l y(l)) is the point at which the tangent planes intersect each other and the plane R = 0.

At each step in the Newton-Raphson iteration, we evaluate the residual functions and Jacobian matrix. We then calculate a correction vector as the solution to the matrix equation... 

To solve for the chemical system at t, we use Newton-Raphson iteration to minimize a set of residual functions, as discussed in Chapter 4. For a kinetic... 

The solution, performed at each step in the Newton-Raphson iteration, is accomplished by setting Eqn. 8.2 equal to Eqn. 8.3. We write a residual function for the surface potential... 

We employ the Newton-Raphson method to iterate toward a set of values for the unknown variables (nw, mi, mp)r for which the residual functions become vanishingly small. 

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