Re: Re: Re: Numerical Problem
- To: mathgroup at smc.vnet.net
- Subject: [mg107170] Re: [mg107124] Re: [mg107063] Re: Numerical Problem
- From: "Tony Harker" <a.harker at ucl.ac.uk>
- Date: Thu, 4 Feb 2010 06:54:38 -0500 (EST)
- References: <hjulg2$sfk$1@smc.vnet.net> <201002020825.DAA08585@smc.vnet.net> <201002031111.GAA19569@smc.vnet.net> <4B69A3B8.70001@wolfram.com>
Dear Daniel, Yes, that's a nice demonstration. Tony ]-> -----Original Message----- ]-> From: Daniel Lichtblau [mailto:danl at wolfram.com] ]-> Sent: 03 February 2010 16:27 ]-> To: Tony Harker ]-> Cc: mathgroup at smc.vnet.net ]-> Subject: Re: [mg107124] Re: [mg107063] Re: Numerical Problem ]-> ]-> Tony Harker wrote: ]-> > Dear Steve, ]-> > ]-> > I agree with all you say: the result returned ]-> by Version 5 was ]-> > anomalous (and obscured the very point I was originally ]-> trying to make with ]-> > the example, taken from a course, about the different ]-> characteristics of ]-> > explicit, predictor-corrector, and implicit methods of ]-> the same order). I'm ]-> > happy with what version 7 is doing -- but I was surprised ]-> when I saw the ]-> > difference between versions 5 and 7. ]-> > ]-> > Tony ]-> ]-> To see what an anomaly your example was, try this ]-> variation. You will ]-> not get "nice" results on any platform. ]-> ]-> epcm[h_, y_List, f_List] := y + 1/2*h*(f.y + f.(y + h*f.y)) ]-> app[0] = {3., -1.}; ]-> mat = {{997, 2997}, {-999, -2999}}; ]-> ]-> Do[Print[InputForm[app[j] = epcm[1/10., app[j - 1], mat]]], ]-> {j, 1, 10}] ]-> ]-> Remains true if you use precision tricks (though raising ]-> precision will ]-> maintain reasonable results longer. ]-> ]-> The reason, as Steve alluded or maybe stated outright, is ]-> that we can ]-> represent exactly ratios of 2/1 in floating point ]-> approximations. And ]-> this is what happens to the vector components of your ]-> original example, ]-> when extended precision downsizes to fit into standard ]-> double precision ]-> reals. Cannot do this when the ratio is 3/1, and that's the ]-> case for the ]-> component magnitudes of the eigenvector corresponding to ]-> the smaller ]-> eigenvalue of the matrix above. ]-> ]-> Daniel Lichtblau ]-> Wolfram Research ]-> ]->
- References:
- Re: Numerical Problem
- From: schochet123 <schochet123@gmail.com>
- Re: Re: Numerical Problem
- From: "Tony Harker" <a.harker@ucl.ac.uk>
- Re: Numerical Problem