       Re: erroneous result when adding reals

• To: mathgroup at smc.vnet.net
• Subject: [mg100912] Re: [mg100853] erroneous result when adding reals
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Thu, 18 Jun 2009 04:52:00 -0400 (EDT)
• References: <200906170151.VAA25825@smc.vnet.net>

```Vlad Seghete wrote:
> Hi all,
>
> I'm fairly new to Mathematica and today I ran into an issue that
> confuses me endlessly. It has to do with simple addition of real
> numbers. It seems like adding x (anything) to 0. (the real) produces a
> result different from x, within something close to machine precision.
> The problem becomes more serious when I do the addition in a Do loop,
> like below:
>
> Module[{tnew, tcur = 0., h = 1/100},
>   Clear[ts]; ts = {tcur};     (* ts is a list *)
>   Do[tcur = ts[[-1]];
>    tnew = tcur + h;          (* add h to the last element of the list
> *)
>    AppendTo[ts, tnew],   (* and then push it at the end of the list *)
>    {step, 1, 300}             (* repead 300 times *)
>    ]
> ];
> ListLinePlot[Table[h, {h, 0, 3, 1/100}] - ts, InterpolationOrder ->
> 0]
>
> The plot I get is NOT constant, and the error introduced through the
> "real addition" done in the Do loop is systematic and adds up to
> something relatively large. Notice that if I substitute 0. (the real)
> with 0 (the integer or rational), then the result is exactly like
> expected.
>
> Do any of you know why this happens and how I could avoid it, other
> than working with *only* rational numbers? Thank you!

It follows from the fact that 1/100 is not exactly representable as a
machine double.

You can get the same tables and plot from simpler code, as below.

h = 1/100;
t1 = h*Range;
elem = 0.;
t2 = Table[elem += h, {300}];

ListLinePlot[t1 - t2, InterpolationOrder -> 0]

What I do not fully grasp is why there seems to be a change in
direction, and rate, of the bias. I would have expected one line from
the plot, whereas I see two distinct segments.

If I change the range from 300 to 1800, I get yet a third segment,
taking the discrepancy back towards zero. Upping again, to 4800,
introduces a fourth segment. And it seems the "overshoots" grow. I
confess I find this behavior mysterious.

Daniel Lichtblau
Wolfram Research

```

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