FindFit and NormFunction

*To*: mathgroup at smc.vnet.net*Subject*: [mg64151] FindFit and NormFunction*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Thu, 2 Feb 2006 19:09:04 -0500 (EST)*Organization*: The University of Western Australia*Sender*: owner-wri-mathgroup at wolfram.com

I am having difficulty using the NormFunction option to FindFit. Let me give a concrete example. Abramowitz and Stegun Section 17.3.35 gives a nonlinear approximant to the complete elliptic integral. See www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&Page=592 This approximant to EllipticE[1-m] can be implemented as f[a_, b_, c_, d_][m_] = a m + b m^2 - (c m + d m^2) Log[m] + 1; After sampling EllipticE[1-m], data = N[Table[{m, EllipticE[1-m]}, {m, 10^-8, 1 - 10^-8, 10^-3}]]; using FindFit gives quite a decent approximant: FindFit[data, f[a, b, c, d][m], {a,b,c,d}, m] best[m_] =f[a, b, c, d][m] /. % The maximal absolute fractional error is ~6 x 10^-5 as seen from Plot[10^5 (1 - best[m]/EllipticE[1 - m]), {m, 0, 1}, PlotRange -> All] However, the Abramowitz and Stegun approximant has error ~4 x 10^-5. AS[m_] =f[0.4630151, 0.1077812, 0.2452727, 0.0412496][m]; Plot[10^5 (1 - AS[m]/EllipticE[1 - m]), {m, 0, 1}, PlotRange -> All] Essentially, one needs to fit with respect to the L-Infinity-Norm, see http://mathworld.wolfram.com/L-Infinity-Norm.html as the NormFunction in FindFit. However, when I try FindFit[data, f[a, b, c, d][m], {a,b,c,d}, m, NormFunction :> (Norm[#, Infinity]&)] I get a FindFit::"lmnl" message (which is reasonable and informative) and also a FindFit::"lstol" message: "The line search decreased the step size to within tolerance specified by AccuracyGoal and PrecisionGoal but was unable to find a sufficient decrease in the norm of the residual. You may need more than MachinePrecision digits of working precision to meet these tolerances." I can see (by monitoring the Norm of the residual) that the algorithm gets trapped a long way from the minimum -- but I don't see why more digits of working precision are required. As an aside, is there an easy way (a handle?) to monitor the norm of the residuals at each step? One way is to write a function to compute the residuals, r[a_, b_, c_, d_] = Norm[(f[a, b, c, d] /@ data[[All,1]]) - data[[All,2]], Infinity]; and then track them using StepMonitor or EvaluationMonitor Reap[FindFit[data, f[a, b, c, d][m], {a,b,c,d}, m, NormFunction :> (Norm[#,Infinity]&), EvaluationMonitor :> Sow[r[a,b,c,d]]]] but is there a _direct_ way of accessing the residuals which are computed by FindFit anyway? Cheers, Paul _______________________________________________________________________ Paul Abbott Phone: 61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) AUSTRALIA http://physics.uwa.edu.au/~paul _______________________________________________________________________ Paul Abbott Phone: 61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) AUSTRALIA http://physics.uwa.edu.au/~paul