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Re: Mathematica 7 is now available

• To: mathgroup at smc.vnet.net
• Subject: [mg93740] Re: Mathematica 7 is now available
• From: "Nasser Abbasi" <nma at 12000.org>
• Date: Mon, 24 Nov 2008 04:15:31 -0500 (EST)
• References: <gg0qf8$c6$1@smc.vnet.net> <gg3c6m$k2t$1@smc.vnet.net> <gg8pih$k4r$1@smc.vnet.net>
• Reply-to: "Nasser Abbasi" <nma at 12000.org>

"Szabolcs" <szhorvat at gmail.com> wrote in message 
news:gg8pih$k4r$1 at smc.vnet.net...
> On Nov 21, 11:32 am, Michael Weyrauch <michael.weyra... at gmx.de> wrote:
>> The Tally[] problem is solved. This I checked with a prerelease version
>> of Mathematica 7 I got at this year's Mathematica users conference.
>>
>
> What about the other bug (the eigenvalue problem), linked from the
> same thread I mentioned?
>
> I copied the (wrong) results from Mathematica 6 here:
>
>
> In[1]:= mat = {{-6, 0, -Sqrt[3], 0, 0, Sqrt[3], 0, 0, 0, 0, 0, 0, 0,
>    0}, {0, -6, 0, -Sqrt[3], 0, 0, Sqrt[3], 0, 0, 0, 0, 0, 0,
>    0}, {-Sqrt[3], 0, -4, 2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4),
>    2 Sqrt[2/3], 0, 0, Sqrt[3], 0, 0, 0, 0, 0, 0}, {0, -Sqrt[3],
>    2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), -4/3, -(2 Sqrt[2])/3, 0, 0, 0,
>     Sqrt[3], 0, 0, 0, 0, 0}, {0, 0, 2 Sqrt[2/3], -(2 Sqrt[2])/3, 7/3,
>     0, 0, 0, 0, Sqrt[3], 0, 0, 0, 0}, {Sqrt[3], 0, 0, 0, 0, -4, 0,
>    2 (-1/(4 Sqrt[3]) + Sqrt[3]/4), 0, 0, 2 Sqrt[2/3], 0, 0, 0}, {0,
>    Sqrt[3], 0, 0, 0, 0, -4, 0, 2 (-1/(4 Sqrt[3]) + Sqrt[3]/4), 0, 0,
>    2 Sqrt[2/3], 0, 0}, {0, 0, Sqrt[3], 0, 0,
>    2 (-1/(4 Sqrt[3]) + Sqrt[3]/4), 0, -14/3,
>    2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), 2 Sqrt[2/3], (2 Sqrt[2])/3, 0,
>     0, 0}, {0, 0, 0, Sqrt[3], 0, 0, 2 (-1/(4 Sqrt[3]) + Sqrt[3]/4),
>    2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), -2, -(2 Sqrt[2])/3,
>    0, (2 Sqrt[2])/3, 0, 0}, {0, 0, 0, 0, Sqrt[3], 0, 0,
>    2 Sqrt[2/3], -(2 Sqrt[2])/3, -7/3, 0, 0,
>    2 (1/(3 Sqrt[2]) + (2 Sqrt[2])/3), Sqrt[10/3]}, {0, 0, 0, 0, 0,
>    2 Sqrt[2/3], 0, (2 Sqrt[2])/3, 0, 0, -16/3,
>    2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), 2 Sqrt[2/3], 0}, {0, 0, 0, 0,
>    0, 0, 2 Sqrt[2/3], 0, (2 Sqrt[2])/3, 0,
>    2 (-1/(4 Sqrt[3]) + (3 Sqrt[3])/4), -8/3, -(2 Sqrt[2])/3, 0}, {0,
>    0, 0, 0, 0, 0, 0, 0, 0, 2 (1/(3 Sqrt[2]) + (2 Sqrt[2])/3),
>    2 Sqrt[2/3], -(2 Sqrt[2])/3, 1/2,
>    2 (-Sqrt[5/3]/16 - Sqrt[15]/16)}, {0, 0, 0, 0, 0, 0, 0, 0, 0,
>    Sqrt[10/3], 0, 0, 2 (-Sqrt[5/3]/16 - Sqrt[15]/16), 7/2}};
>
> In[2]:= mat === Conjugate@Transpose[mat]
> Out[2]= True
>
> (mat is Hermitian so we expect real eigenvalues.)
>
> In[3]:= N@Eigenvalues[mat]
>
> Out[3]= {-9.41358 + 0.88758 I, -9.41358 - 0.88758 I, -7.37965 +
>  2.32729 I, -7.37965 - 2.32729 I, -4.46655 + 2.59738 I, -4.46655 -
>  2.59738 I, 4.36971, 3.21081, -2.32456 + 2.10914 I, -2.32456 -
>  2.10914 I, 2.04366+ 0.552265 I,
> 2.04366- 0.552265 I, -0.249588 + 1.29034 I, -0.249588 - 1.29034 I}
>
> In[4]:= Eigenvalues[N[mat]]
>
> Out[4]= {-9.09122, -7.41855, -7.41855, -7.2915, 4.33734, -4., -4., \
> 3.2915, -3.24612, -2.38787, -2.38787, 1.80642, 1.80642,
> 9.21707*10^-16}
>

It seems to be fixed in M7:

In[36]:= $Version
Out[36]= 7.0 for Microsoft Windows (32-bit) (November 10, 2008)

In[32]:= mat = {{-6, 0, -Sqrt[3], 0, 0, Sqrt[3], 0,
     0, 0, 0, 0, 0, 0, 0}, {0, -6, 0,
     -Sqrt[3], 0, 0, Sqrt[3], 0, 0, 0, 0,
     0, 0, 0}, {-Sqrt[3], 0, -4,
     2*(-(4*Sqrt[3])^(-1) + (3*Sqrt[3])/
        4), 2*Sqrt[2/3], 0, 0, Sqrt[3], 0,
     0, 0, 0, 0, 0}, {0, -Sqrt[3],
     2*(-(4*Sqrt[3])^(-1) + (3*Sqrt[3])/
        4), -4/3, -(2*Sqrt[2])/3, 0, 0, 0,
     Sqrt[3], 0, 0, 0, 0, 0},
    {0, 0, 2*Sqrt[2/3], -(2*Sqrt[2])/3,
     7/3, 0, 0, 0, 0, Sqrt[3], 0, 0, 0,
     0}, {Sqrt[3], 0, 0, 0, 0, -4, 0,
     2*(-(4*Sqrt[3])^(-1) + Sqrt[3]/4), 0,
     0, 2*Sqrt[2/3], 0, 0, 0},
    {0, Sqrt[3], 0, 0, 0, 0, -4, 0,
     2*(-(4*Sqrt[3])^(-1) + Sqrt[3]/4), 0,
     0, 2*Sqrt[2/3], 0, 0},
    {0, 0, Sqrt[3], 0, 0,
     2*(-(4*Sqrt[3])^(-1) + Sqrt[3]/4), 0,
     -14/3, 2*(-(4*Sqrt[3])^(-1) +
       (3*Sqrt[3])/4), 2*Sqrt[2/3],
     (2*Sqrt[2])/3, 0, 0, 0},
    {0, 0, 0, Sqrt[3], 0, 0,
     2*(-(4*Sqrt[3])^(-1) + Sqrt[3]/4),
     2*(-(4*Sqrt[3])^(-1) + (3*Sqrt[3])/
        4), -2, -(2*Sqrt[2])/3, 0,
     (2*Sqrt[2])/3, 0, 0}, {0, 0, 0, 0,
     Sqrt[3], 0, 0, 2*Sqrt[2/3],
     -(2*Sqrt[2])/3, -7/3, 0, 0,
     2*(1/(3*Sqrt[2]) + (2*Sqrt[2])/3),
     Sqrt[10/3]}, {0, 0, 0, 0, 0,
     2*Sqrt[2/3], 0, (2*Sqrt[2])/3, 0, 0,
     -16/3, 2*(-(4*Sqrt[3])^(-1) +
       (3*Sqrt[3])/4), 2*Sqrt[2/3], 0},
    {0, 0, 0, 0, 0, 0, 2*Sqrt[2/3], 0,
     (2*Sqrt[2])/3, 0,
     2*(-(4*Sqrt[3])^(-1) + (3*Sqrt[3])/
        4), -8/3, -(2*Sqrt[2])/3, 0},
    {0, 0, 0, 0, 0, 0, 0, 0, 0,
     2*(1/(3*Sqrt[2]) + (2*Sqrt[2])/3),
     2*Sqrt[2/3], -(2*Sqrt[2])/3, 1/2,
     2*(-Sqrt[5/3]/16 - Sqrt[15]/16)},
    {0, 0, 0, 0, 0, 0, 0, 0, 0,
     Sqrt[10/3], 0, 0, 2*(-Sqrt[5/3]/16 -
       Sqrt[15]/16), 7/2}};

In[33]:= mat === Conjugate[Transpose[mat]]
Out[33]= True

In[34]:= N[Eigenvalues[mat]]
Out[34]= {-9.091215416949623, -7.4185507188738455,
  -7.4185507188738455, -7.291502622129181,
  4.337337307188519, -4., -4.,
  3.2915026221291814, -3.2461218902388955,
  -2.387873132949261, -2.387873132949261,
  1.8064238518231066, 1.8064238518231066,
  0.}

In[35]:= Eigenvalues[N[mat]]
Out[35]= {-9.091215416949622, -7.4185507188738455,
  -7.418550718873844, -7.291502622129181,
  4.337337307188519, -4.000000000000002,
  -3.999999999999999, 3.2915026221291814,
  -3.246121890238896, -2.387873132949261,
  -2.3878731329492604, 1.8064238518231066,
  1.8064238518231046,
  -2.8189256280805394*^-16}

Nasser