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I am using MKL to solve a nonlinear problem first with an inversion and then eigenvector solution, interestingly if the problem is well formed the solution converges quickly, 3 to 10 steps. But if the problem becomes not well formed the iterations can take 9000 or so steps for even a minor change in the loading vector. I developed a problem that takes about 30 steps and the error reduces at an exponential rate - shown on the graph.

I was thinking the non-linear solver is estimating a set of points, if I measure the change in the set of points in the first few iterations - I wonder if i can use newton ralphson to get a better estimate of the final solution points - the problem is there are 360 degrees of freedom, but likely only a few are non-linear -

Any ideas would be welcome.

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It is difficult to know the exact cause of the slowdown without seeing the full problem. However, the symptoms you describe are characteristic of successive substitution or fixed point iteration algorithms. For chemical engineering models that I deal with, a method known as Dominant Eigenvalue Acceleration has been successful in speeding up the convergence in difficult cases. It can be applied every few iterations and has many parameters that can be adjusted. If your system is dense and has 360 degrees of freedom, Newton's method may have other problems in its application.

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