       Re: Re: Re: Numerical Problem

```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 = {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
]->
]->

```

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