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RE: Re: Loss of precision when using Simplify
*To*: mathgroup at smc.vnet.net
*Subject*: [mg36995] RE: [mg36958] Re: [mg36925] Loss of precision when using Simplify
*From*: "DrBob" <drbob at bigfoot.com>
*Date*: Fri, 4 Oct 2002 05:01:45 -0400 (EDT)
*Reply-to*: <drbob at bigfoot.com>
*Sender*: owner-wri-mathgroup at wolfram.com
I get a somewhat different answer with the same function:
CW[n_] := Coefficient[Simplify[1.001`17 + Sum[(x[i] - 1.001`17)/2^i, {i,
\
n}]], x[1]]
Split[CW /@ Range[100]]
{{1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2},
{0.5`12.3085, 0.5`12.0074, 0.5`11.7064, 0.5`11.4054, 0.5`11.1043,
0.5`10.8033, 0.5`10.5023, 0.5`10.2012, 0.5`9.9002, 0.5`9.5992,
0.5`9.2982, 0.5`8.9971, 0.5`8.6961, 0.5`8.3951, 0.5`8.094, 0.5`7.793,
0.5`7.492, 0.5`7.1909, 0.5`6.8899, 0.5`6.5889, 0.5`6.2879, 0.5`5.9868,
0.5`5.6858, 0.5`5.3848, 0.5`5.0837, 0.5`4.7827, 0.5`4.4817, 0.5`4.1806,
0.5`3.8796, 0.5`3.5786, 0.5`3.2776, 0.5`2.9765, 0.5`2.6755, 0.5`2.3745,
0.5`2.3745, 0.5`1.7724, 0.5`1.7724, 0.5`1.1703, 0.5`1.1703, 0.5`0.5683,
0.5`0.2673, 0.5`0.2673},
{0.125`0.2673}, {1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2,
1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2,
1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2,
1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2}}
Notice the split was different than yours.
Bobby
-----Original Message-----
From: Andrzej Kozlowski [mailto:andrzej at platon.c.u-tokyo.ac.jp]
To: mathgroup at smc.vnet.net
Subject: [mg36995] [mg36958] Re: [mg36925] Loss of precision when using Simplify
Dear Carl
You have discovered what is perhaps a bug but maybe something even mor
einteresting . However, I think you stopped your investigation a little
prematurely. Here is a function that just computes the coefficient of
x[1] in your Simplified expression for various values of n:
CW[n_] := Coefficient[Simplify[1.001`17 + Sum[(x[i] - 1.001`17)/2^i,
{i, \
n}]], x[1]]
Now lets try CW for the first 100 values of n:
In[30]:=
Table[CW[n], {n, 100}]
Out[30]=
{1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2,
0.5`12.3085, 0.5`12.0074, 0.5`11.7064, 0.5`11.4054,
0.5`11.1043, 0.5`10.8033, 0.5`10.5023, 0.5`10.2012,
0.5`9.9002, 0.5`9.5992, 0.5`9.2982, 0.5`8.9971,
0.5`8.6961, 0.5`8.3951, 0.5`8.094, 0.5`7.793, 0.5`7.492,
0.5`7.1909, 0.5`6.8899, 0.5`6.5889, 0.5`6.2879,
0.5`5.9868, 0.5`5.6858, 0.5`5.3848, 0.5`5.0837,
0.5`4.7827, 0.5`4.4817, 0.5`4.1806, 0.5`3.8796,
0.5`3.5786, 0.5`3.2776, 0.5`2.9765, 0.5`2.6755,
0.5`2.3745, 0.5`2.3745, 0.5`1.7724, 0.5`1.7724,
0.5`1.1703, 0.5`1.1703, 0.5`0.5683, 0``0.1423, 0``0.1423,
0``0.1423, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2,
1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2,
1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2,
1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2,
1/2, 1/2, 1/2}
In[31]:= Map[Length,Split[#]]
Out[31]=
{12,43,45}
This is indeed curious.The problem seems to occur for values between 13
and 55 and then go away (for good???) At first I thought it maybe in
some fascinating way related to some properties of the integer n, but
now I am not sure. It certainly worth a careful examination. I hope
whoever discovers the cause of this will let us know.
Andrzej
Andrzej Kozlowski
Yokohama, Japan
http://www.mimuw.edu.pl/~akoz/
http://platon.c.u-tokyo.ac.jp/andrzej/
On Wednesday, October 2, 2002, at 04:32 PM, Carl K. Woll wrote:
> To Technical Support and the Mathematica User community,
>
> I'm writing to report what I consider to be a bug. First, I want to
> show a
> simplified example of the problem. Consider the following expression:
>
> expr=0.22 + x[0] + (3*(-0.16+ x[1]))/4 + (9*(0.546 + x[2]))/16;
>
> When simplified I expected to get some real number plus
> x[0]+3x[1]/4+9x[2]/16, but instead I get the following:
>
> Simplify[expr]
> 0.407125 + x[0] + 0.75 x[1] + 0.5625 x[2]
>
> As you can see, for some reason Mathematica converted the fractions
> 3/4 and
> 9/16 to real machine numbers. I consider this to be a bug.
>
> Now, for an example more representative of the situation that I've
been
> coming across.
>
> expr12 = 1.001`17 + Sum[(x[i] - 1.001`17)/2^i, {i, 12}];
> expr13 = 1.001`17 + Sum[(x[i] - 1.001`17)/2^i, {i, 13}];
> expr55 = 1.001`17 + Sum[(x[i] - 1.001`17)/2^i, {i, 55}];
>
> As you can see, I have replaced the real numbers by extended precision
> numbers. The simplified example above demonstrates that the problem
> exists
> when using machine numbers. Now, we'll see what happens when we use
> arbitrary precision numbers. First, let's simplify the expression with
> 12
> terms.
>
> Simplify[expr12]
> (1.0010000000000 + 2048 x[1] + 1024 x[2] + 512 x[3] + 256 x[4] + 128
> x[5] +
>
> 64 x[6] + 32 x[7] + 16 x[8] + 8 x[9] + 4 x[10] + 2 x[11] + x[12])
> / 4096
>
> As you can see, a sum with 12 terms upon simplification has
> coefficients
> which are still integers as they should be. However, increasing the
> number
> of terms to 13 yields
>
> Simplify[expr13]
> 0.0001221923828125 + 0.500000000000 x[1] + 0.250000000000 x[2] +
>
> 0.1250000000000 x[3] + 0.0625000000000 x[4] + 0.0312500000000 x[5] +
>
> 0.01562500000000 x[6] + 0.00781250000000 x[7] + 0.00390625000000
> x[8] +
>
> 0.001953125000000 x[9] + 0.000976562500000 x[10] + 0.000488281250000
> x[11]
> +
>
> 0.000244140625000 x[12] + 0.0001220703125000 x[13]
>
> Now, all of the coefficients are converted to real numbers,
> replicating the
> bug from the simplified example. Finally, let's see what happens when
> we
> have 55 terms. Rather than putting the resulting expression here, I
> will
> just leave it at the end of the post. The result though is somewhat
> surprising. Each of the coefficients of the x[i] are again real
> numbers, but
> now their precision is only 0! The proper result of course is the sum
> of
> some real number (with a precision close to 0 due to numerical
> cancellation)
> and an expression consisting of rational numbers multiplied by x[i].
> The
> loss of precision of the coefficients of the x[i] is precisely what
> occurred
> to me. Of course, in this simple example, simply using Expand instead
> of
> Simplify produces the expected result, and is my workaround. I hope
you
> agree with me that this is a bug, and one that Wolfram needs to
> correct.
>
> Carl Woll
> Physics Dept
> U of Washington
>
> Simplify[expr55]
> -16 -1 -1 -1
> 0. 10 + 0. x[1] + 0. x[2] + 0. 10 x[3] + 0. 10 x[4] + 0. 10
> x[5] +
>
> -2 -2 -2 -3 -3
> 0. 10 x[6] + 0. 10 x[7] + 0. 10 x[8] + 0. 10 x[9] + 0. 10
> x[10]
> +
>
> -3 -3 -4 -4
> -4
> 0. 10 x[11] + 0. 10 x[12] + 0. 10 x[13] + 0. 10 x[14] + 0.
10
> x[15] +
>
> -5 -5 -5 -6
> -6
> 0. 10 x[16] + 0. 10 x[17] + 0. 10 x[18] + 0. 10 x[19] + 0.
10
> x[20] +
>
> -6 -6 -7 -7
> -7
> 0. 10 x[21] + 0. 10 x[22] + 0. 10 x[23] + 0. 10 x[24] + 0.
10
> x[25] +
>
> -8 -8 -8 -9
> -9
> 0. 10 x[26] + 0. 10 x[27] + 0. 10 x[28] + 0. 10 x[29] + 0.
10
> x[30] +
>
> -9 -9 -10 -10
> 0. 10 x[31] + 0. 10 x[32] + 0. 10 x[33] + 0. 10 x[34] +
>
> -10 -11 -11 -11
> 0. 10 x[35] + 0. 10 x[36] + 0. 10 x[37] + 0. 10 x[38] +
>
> -12 -12 -12 -12
> 0. 10 x[39] + 0. 10 x[40] + 0. 10 x[41] + 0. 10 x[42] +
>
> -13 -13 -13 -14
> 0. 10 x[43] + 0. 10 x[44] + 0. 10 x[45] + 0. 10 x[46] +
>
> -14 -14 -15 -15
> 0. 10 x[47] + 0. 10 x[48] + 0. 10 x[49] + 0. 10 x[50] +
>
> -15 -15 -16 -16
> -
> 16
> 0. 10 x[51] + 0. 10 x[52] + 0. 10 x[53] + 0. 10 x[54] +
> 0. 10
> x[55]
>
>
>
>
>
>
>
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