I realize that methods for solving equations that don't have an exact solution ( a solution in closed form) might be a little off-topic here. But, indirectly, it's a sundial question, because most likely there are sundial problems that result in equations without an exact solution.
For example, in the earlier discussion about finding azimuth, when h, dec, and Lat are known, there was some discussion of an iterative solution by Successive Substutions. I pointed out that the equation has an exact solution. But that discussion brought out the relevance of iterative equation solutions to the sundial topic. My question is about one particular relatively new version of Regula Falsi, described by some university mathematicians, in a paper. The page with the paper doesn't show a url, but here is the url of the google page a which their paper can be found and linked to.It's the link that says "An Improved Regula Falsi Method..." : https://www.google.com/#q=Naghipoor%2C+J.%2C+Ahmadian+S.A.%2C+and+Soheili%2C+A.R.%2C+%E2%80%9CAn+ImprovedRegula ------------ Newton's Method is deservedly popular, because it rarely fails to converge, and its convergence is usually very fast, especially if the initial guess is close. Of course it doesn't always converge, and sometimes only converges slowly. At the oppose extreme is Bisection, which _reliably_ converges at a predictable, respectable and useful (but not spectacular) rate. Regula Falsi is appealing, because it, too, always converges. And it has many improved versions that offer to make converge faster than Bisection except with the most deviously pathological equations. Maybe the sheer number of Regula Falsi improvements proposed is an indication of that methods promising-ness. Anyway, the paper referred to above is about one of those improved Regula Falsi methods. That's the subject of my question: The paper tells the proposed procecure, but it says nothing about the justification or motivation for the procedure. So, my question is, if anyone is interested in Regula Falsi improvements, can anyone tell me what is the justification or motivation for the procedure described in the paper that I mentioned above? (I emphasize that it isn't the only improved linear Regula Falsi version) Evidently the Regula Falsi versions usually requiring the fewest function-evaluations are the ones that replace Regula Falsi's linear interpolation with a curve--quadratic, inverse-quadratic, or exponential. But they require more work for each iteration. The paper that I referred to only compares its method to ordinary, un-improved Regula-Falsi, a method that probably isn't advocated anymore. But the authors seem to imply that their method compares well to other linear Regula Falsi methods. Michael Ossipoff
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