proper supremum norm implementation

*To*: mathgroup at smc.vnet.net*Subject*: [mg108947] proper supremum norm implementation*From*: Benjamin Hell <hell at exoneon.de>*Date*: Thu, 8 Apr 2010 07:59:53 -0400 (EDT)

Hi, I am trying to calculate an approximation to the supremum norm of the difference of two functions on the interval [a,b]. The first one is a linear spline (lets call it spline, too) connecting data points pretty much lying on the graph of the second function (lets call it optf), which in my case is a piecewise linear function. So both functions are not really complicated, though quite some data points are involved. My current approach is to use NMaximize to get an approximation to the supremum norm. The whole procedure looks like: spline = Interpolation[data,InterpolationOrder->1]; f[t_?NumericQ]:=Norm[spline[t]-optf[t],Infinity]; result = NMaximize[{f[t], a <= t, t <= b}, t]; For a small number of data points this works pretty well, but more than 60 points deliver quite bad results. I know that the results are bad because I know a point c in [a,b] at which way higher values for Norm[spline[t]-optf[t],Infinity] appear than NMaximize delivers. Playing with AccuracyGoal and MaxIterations did not really help. I also tried using FindMaximum with t=c as starting value, but I didn't even get a result from that algorithm (only Null and warnings on convergence). Does anybody know a better way to implement the supremum norm in this case? Thanks in advance.