Iterative Component-Wise Solution of the Nonlinear Equations

3 Iterative Component-Wise Solution of the Nonlinear Equations 

An alternative method for solving (4.17) is based on solving for A component-wise [322]. To describe this algorithm let us enumerate the equations as 

This is scalar nonlinear equation for A Ai. We approximate its solution using a linear approximation at Q thus we have 

As is the case for Newton iteration, one may use the constraint residuals as an additional test. It is certainly unsafe to check only the latest difference of iterates (of a single component), and one should instead check that these are small for an entire cycle. The iteration described above is an example of a Gauss-Seidel-Newton iteration and it and many other iterative nonlinear solvers are discussed in [292], 

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