Please look at the code sample given here. The matrix t.M has a precision of 19.657. When I set $MinPrecision to 20, I expected that the elements will change a little (in 20th digit) and eigenvalues may be slightly different. But notice that some of the eigenvalues now are different in the 4th decimal place. Is it possible to explain and fix this?
In[1]:= $Version
Out[1]= "10.0 for Mac OS X x86 (64-bit) (June 29, 2014)"
In[2]:= t = DiagonalMatrix[{1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1}];
M = {{0.7137918381815571041`20., 0, 0,
0, -0.0818845598913793050449093616880474190251622093580702678455`\
20. + 0``20.79396386163813 I, 0, 0,
0.2343688308366258291603440704025530543285644606954919531622`20. \
+ 0``20.337266027259396 I, \
-0.1781456332806043268557912392799948161104770751588753908233`20. +
0``21.549512304350156 I, 0,
0, -0.5098859155622219674745857688604570097577534403206352266676`\
20. + 0``21.092814469971415 I}, {0, 0.7137918381815571041`20., 0, 0,
0, -0.0818845598913793050449093616880474190251622093580702678455`\
20. + 0``20.79396386163813 I, \
-0.2343688308366258291603440704025530543285644606954919531622`20. +
0``20.337266027259396 I, 0,
0, -0.1781456332806043268557912392799948161104770751588753908233`\
20. + 0``21.549512304350156 I,
0.5098859155622219674745857688604570097577534403206352266676`20. \
+ 0``21.092814469971415 I, 0}, {0, 0, 0.7137918381815571041`20., 0, 0,
0.2343688308366258291603440704025530543285644606954919531622`20. \
+ 0``20.337266027259396 I, \
-0.0818845598913793050449093616880474190251622093580702678455`20. +
0``20.79396386163813 I, 0,
0, -0.5098859155622219674745857688604570097577534403206352266676`\
20. + 0``21.092814469971415 I, \
-0.1781456332806043268557912392799948161104770751588753908233`20. +
0``21.549512304350156 I, 0}, {0, 0, 0,
0.7137918381815571041`20., \
-0.2343688308366258291603440704025530543285644606954919531622`20. +
0``20.337266027259396 I, 0,
0, -0.0818845598913793050449093616880474190251622093580702678455`\
20. + 0``20.79396386163813 I,
0.5098859155622219674745857688604570097577534403206352266676`20. \
+ 0``21.092814469971415 I, 0,
0, -0.1781456332806043268557912392799948161104770751588753908233`\
20. + 0``21.549512304350156 I}, \
{-0.0818845598913793050449093616880474190251622093580702678455`20. +
0``20.79396386163813 I, 0,
0, -0.2343688308366258291603440704025530543285644606954919531622`\
20. + 0``20.337266027259396 I, 0.7137918381815571041`20., 0, 0,
0, -0.1060166164759502210167654308580758022516732692772395255612`\
20. + 0``20.835887101121145 I, 0, 0,
0.3034392623674499766124318272572987558005177815882278376081`20. \
+ 0``20.379189266742404 I}, {0, \
-0.0818845598913793050449093616880474190251622093580702678455`20. +
0``20.79396386163813 I,
0.2343688308366258291603440704025530543285644606954919531622`20. \
+ 0``20.337266027259396 I, 0, 0, 0.7137918381815571041`20., 0, 0,
0, -0.1060166164759502210167654308580758022516732692772395255612`\
20. + 0``20.835887101121145 I, \
-0.3034392623674499766124318272572987558005177815882278376081`20. +
0``20.379189266742404 I,
0}, {0, -0.\
2343688308366258291603440704025530543285644606954919531622`20. +
0``20.337266027259396 I, \
-0.0818845598913793050449093616880474190251622093580702678455`20. +
0``20.79396386163813 I, 0, 0, 0, 0.7137918381815571041`20., 0, 0,
0.3034392623674499766124318272572987558005177815882278376081`20. \
+ 0``20.379189266742404 I, \
-0.1060166164759502210167654308580758022516732692772395255612`20. +
0``20.835887101121145 I,
0}, {0.2343688308366258291603440704025530543285644606954919531622`\
20. + 0``20.337266027259396 I, 0,
0, -0.0818845598913793050449093616880474190251622093580702678455`\
20. + 0``20.79396386163813 I, 0, 0, 0,
0.7137918381815571041`20., \
-0.3034392623674499766124318272572987558005177815882278376081`20. +
0``20.379189266742404 I, 0,
0, -0.1060166164759502210167654308580758022516732692772395255612`\
20. + 0``20.835887101121145 I}, \
{-0.1781456332806043268557912392799948161104770751588753908233`20. +
0``21.549512304350156 I, 0, 0,
0.5098859155622219674745857688604570097577534403206352266676`20. \
+ 0``21.092814469971415 I, \
-0.1060166164759502210167654308580758022516732692772395255612`20. +
0``20.835887101121145 I, 0,
0, -0.3034392623674499766124318272572987558005177815882278376081`\
20. + 0``20.379189266742404 I, 0.7137918381815571041`20., 0, 0,
0}, {0, -0.\
1781456332806043268557912392799948161104770751588753908233`20. +
0``21.549512304350156 I, \
-0.5098859155622219674745857688604570097577534403206352266676`20. +
0``21.092814469971415 I, 0,
0, -0.1060166164759502210167654308580758022516732692772395255612`\
20. + 0``20.835887101121145 I,
0.3034392623674499766124318272572987558005177815882278376081`20. \
+ 0``20.379189266742404 I, 0, 0, 0.7137918381815571041`20., 0, 0}, {0,
0.5098859155622219674745857688604570097577534403206352266676`20. \
+ 0``21.092814469971415 I, \
-0.1781456332806043268557912392799948161104770751588753908233`20. +
0``21.549512304350156 I, 0,
0, -0.3034392623674499766124318272572987558005177815882278376081`\
20. + 0``20.379189266742404 I, \
-0.1060166164759502210167654308580758022516732692772395255612`20. +
0``20.835887101121145 I, 0, 0, 0, 0.7137918381815571041`20.,
0}, {-0.5098859155622219674745857688604570097577534403206352266676\
`20. + 0``21.092814469971415 I, 0,
0, -0.1781456332806043268557912392799948161104770751588753908233`\
20. + 0``21.549512304350156 I,
0.3034392623674499766124318272572987558005177815882278376081`20. \
+ 0``20.379189266742404 I, 0,
0, -0.1060166164759502210167654308580758022516732692772395255612`\
20. + 0``20.835887101121145 I, 0, 0, 0, 0.7137918381815571041`20.}};
In[4]:= Precision[t.M]
Out[4]= 19.657
In[5]:= Eigenvalues[t.M]
Out[5]= {-0.6434260521385389832, 0.6434260521385389832, \
-0.6434260521385389832, 0.6434260521385389832, 0.5495328432664610538, \
-0.5495328432664610537, 0.5495328432664610537, \
-0.5495328432664610537, -0.31288706030886822758, \
0.31288706030886822757, 0.31288706030886822757, \
-0.31288706030886822757}
In[6]:= $MinPrecision = 19;
Eigenvalues[t.M]
Out[7]= {-0.6434260521385389832, 0.6434260521385389832, \
-0.6434260521385389832, 0.6434260521385389832, 0.5495328432664610538, \
-0.5495328432664610537, 0.5495328432664610537, \
-0.5495328432664610537, -0.31288706030886822758, \
0.31288706030886822757, 0.31288706030886822757, \
-0.31288706030886822757}
In[8]:= $MinPrecision = 20;
Eigenvalues[t.M]
Out[9]= {0.64342605213853898318, -0.64342605213853898318, \
0.64339804735557256573, -0.64332543749200075174, \
0.54954065607255415410, -0.54953284326646105374, \
0.54953284326646105374, -0.54931890613227757447, \
-0.31288706030886822757, 0.31288706030886822757, \
-0.31288313900183744618, 0.31281340426449117539}