On Saturday 28 April 2007 20:17, Henry Baker wrote:
I wasn't aware of the ill-conditioned nature of this problem. References?
I'm far from an expert in the field, so here's a not-terribly-highbrow reference, from Forman Acton's charming book "Numerical methods that (usually) work". This is from a brief rant in the middle entitled "What not to compute", under the heading "Exponential fitting". | One of the perennial problems that plagues, among others, | the analyzers of isotope decay is the fitting of data | by a series of exponential functions. How much of A and | how much of B, decaying at known rates a and b, are in | the sample whose activity was sampled several times | in the historic past? This question is quite tractable. [...] | Unfortunately there is a companion problem that looks | only slightly more complicated -- until you try it! | We again have {t_i,y_i} readings from a radioactive sample, | but the decaying materials are not known, hence the decay rates | a and b must also be fitted. [...] For it is well known | that an exponential equation of this type in which all four | parameters are to be fitted is extremely ill conditioned. | That is, there are many combinations of {a,b,A,B} that will | fit most exact data quite well indeed (will you believe four | significant figures?) and when experimental noise is thrown | into the pot, the entire operation becomes hopeless. -- g