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Re: The Integrator (Wolfram)
*To*: mathgroup at smc.vnet.net
*Subject*: [mg26898] Re: The Integrator (Wolfram)
*From*: "Paul Lutus" <nospam at nosite.com>
*Date*: Fri, 26 Jan 2001 23:29:44 -0500 (EST)
*References*: <94r83n$ih0@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
"Phil Freedenberg" <pfreedenberg at hotmail.com> wrote in message
news:94r83n$ih0 at smc.vnet.net...
>
> Hello
> Are you aware that if you integrate Sqrt[1 + a/x^(0.5)] you get zero but
> the integral of Sqrt[1 + a/x^(1/2)] looks correct? Is this a bug?
When Mathematica is presented with a floating-point constant, its behavior
changes. Unless precautions are taken(*), it will use what are called
"machine-precision numbers," numbers whose accuracy depends on the
platform's own floating-point resources.
This remark applies only to floating-point values, not to integers or
undefined variable names or variables to which integers have been assigned.
To experiment with this difference, write the equation in two ways:
f1[x_] := Sqrt[1 + a/x^(0.5)]
f2[x_] := Sqrt[1 + a/x^(1/2)]
Note the difference when integers or unassigned variable names are applied
to the two functions.
f1[5] =
Sqrt[1 + 0.447214 a]
f2[5] =
a
Sqrt[1 + -------]
Sqrt[5]
This difference is discussed at length in the Mathematica documentation
("3.1.4 Numerical Precision").
* One precaution is to force a FP number to be larger than machine-precision
range. This increases resolution arbitrarily, but it doesn't prevent some of
the undesirable behavior in the above example. To force a FP number to be
taken as larger than it would otherwise be interpreted, enter it this way:
0.5`100 (* means 0.5 followed by 100 zeros *)
--
Paul Lutus
www.arachnoid.com
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