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I want to get (estimates of) the eigenvalues of a 10x10 matrix of floats:

fricas

(1) -> m := matrix([[random(1000)$Integer for i in 1..10] for j in 1..10]); sm := m + transpose(m); smf:Matrix Float := sm

$\label{eq1}\left[ \begin{array}{cccccccccc} {1720.0}&{857.0}&{911.0}&{1365.0}&{1144.0}&{1441.0}&{940.0}&{1 450.0}&{1184.0}&{1078.0} \ {857.0}&{1098.0}&{469.0}&{679.0}&{537.0}&{733.0}&{1088.0}&{89 4.0}&{1628.0}&{1526.0} \ {911.0}&{469.0}&{358.0}&{360.0}&{825.0}&{1102.0}&{666.0}&{108 2.0}&{775.0}&{1134.0} \ {1365.0}&{679.0}&{360.0}&{756.0}&{1070.0}&{245.0}&{1177.0}&{7 56.0}&{1719.0}&{647.0} \ {1144.0}&{537.0}&{825.0}&{1070.0}&{906.0}&{442.0}&{797.0}&{66 5.0}&{749.0}&{990.0} \ {1441.0}&{733.0}&{1102.0}&{245.0}&{442.0}&{100.0}&{1270.0}&{1 145.0}&{283.0}&{1204.0} \ {940.0}&{1088.0}&{666.0}&{1177.0}&{797.0}&{1270.0}&{1736.0}&{4 85.0}&{343.0}&{1312.0} \ {1450.0}&{894.0}&{1082.0}&{756.0}&{665.0}&{1145.0}&{485.0}&{1 168.0}&{1043.0}&{656.0} \ {1184.0}&{1628.0}&{775.0}&{1719.0}&{749.0}&{283.0}&{343.0}&{1 043.0}&{1040.0}&{1385.0} \ {1078.0}&{1526.0}&{1134.0}&{647.0}&{990.0}&{1204.0}&{1312.0}&{6 56.0}&{1385.0}&{178.0}$

(1)

Type: Matrix(Float)

The problem is: If I now call eigenvalues(smf) on the symmetric float matrix smf Axiom 3.0 Beta (February 2005) runs for a very long time (uncomment code if you want to try it):

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)set messages time on

Try this::

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)set output tex off

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)set output algebra on

eigen:=eigenvalues(sm)

   (2)
   [
       %B
     | 
           10          9              8                 7                    6
         %B   - 9060 %B  - 10221631 %B  + 53692462934 %B  + 32991488690592 %B
       + 
                                5                          4
         - 103347836372632698 %B  - 39496518001034749286 %B
       + 
                                   3                                2
         67792682723480658193502 %B  + 13238675729925514491568164 %B
       + 
         - 6411346841697177723823479672 %B - 777550587908816635397837762624
     ]

Type: List(Union(Fraction(Polynomial(Integer)),SuchThat?(Symbol,Polynomial(Integer))))

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Time: 0.04 (OT) = 0.04 sec
solve(rhs(eigen.1),10.0^(-15))

   (3)
   [%B = - 110.9548091175_7676212, %B = - 367.9868971146_480825,
    %B = - 1238.2551231278_173671, %B = - 1395.4394073487_777468,
    %B = - 1651.8389981628_01269, %B = 9515.9502244910_992075,
    %B = 1609.8532163864_54188, %B = 1401.1491995200_013201,
    %B = 980.4992467521_7715242, %B = 317.0233477218_8936052]

Type: List(Equation(Polynomial(Float)))

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Time: 0.01 sec

Thank you! This helps, but doesn't answer everything. Since interestingly:

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charpol := reduce(*, [ rhs(x) - lhs(x) for x in % ])

   (4)
       10                            9                           8
     %B   - 9060.0000000000_000009 %B  - 10221630.9999999999_9 %B
   + 
                              7                            6
     5_3692462934.000000007 %B  + 3299_1488690591.999971 %B
   + 
                                5                             4
     - 10334783_6372632698.02 %B  - 3949651800_1034749284.0 %B
   + 
                                    3                                  2
     0.6779268272_3480658214 E 23 %B  + 0.1323867572_9925514525 E 26 %B
   + 
     - 0.6411346841_6971777309 E 28 %B - 0.7775505879_0881663822 E 30

Type: Polynomial(Float)

fricas

Time: 0 sec

we cannot recover the characteristic polynomial from this solution.

To recover characteristic polynomial one needs good precision, which means very small second argument to solve.

To have use of precise solution we also need to increase precision of other floating point computations using digits. Unfortunately, this leads to ugly display, so we only show final result:

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digits(40)

   (5)  20

Type: PositiveInteger?

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Time: 0 sec
ev:= solve(rhs(eigen.1),1.0*10^(-35));

Type: List(Equation(Polynomial(Float)))

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Time: 0 sec
cp:= reduce(*, [rhs(x)-lhs(x) for x in ev])

   (7)
       10            9                                                  8
     %B   - 9060.0 %B  - 10221630.9999999999_9999999999_9999999999_98 %B
   + 
                                                    7
     5_3692462933.9999999999_9999999999_999999996 %B
   + 
                                                    6
     3299_1488690591.9999999999_9999999999_999998 %B
   + 
                                                      5
     - 10334783_6372632697.9999999999_9999999999_97 %B
   + 
                                                     4
     - 3949651800_1034749286.0000000000_0000000017 %B
   + 
                                                    3
     677_9268272348_0658193501.9999999999_9999964 %B
   + 
                                                   2
     132386_7572992551_4491568164.0000000000_002 %B
   + 
     - 64113468_4169717772_3823479671.9999999999_53 %B
   + 
     - 7775505879_0881663539_7837762624.0000000043

Type: Polynomial(Float)

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Time: 0 sec
rhs(eigen.1) - cp

   (8)
                   8               7               6               5
     - 0.2 E -31 %B  + 0.4 E -28 %B  + 0.2 E -25 %B  - 0.3 E -21 %B
   + 
                 4               3               2
     0.2 E -18 %B  + 0.4 E -15 %B  - 0.2 E -12 %B  - 0.5 E -10 %B + 0.4 E -8

Type: Polynomial(Float)

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Time: 0 sec

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)set output algebra off

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)set output tex on

For matrices of expressions one has to explicitly specify package:

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A:=matrix[[cos(x),-sin(x)],[sin(x),cos(x)]]

$\label{eq2}\left[ \begin{array}{cc} {\cos \left({x}\right)}& -{\sin \left({x}\right)} \ {\sin \left({x}\right)}&{\cos \left({x}\right)}$

(2)

Type: Matrix(Expression(Integer))

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Time: 0.01 (OT) = 0.01 sec
A(1,1)

$\label{eq3}\cos \left({x}\right)$

(3)

Type: Expression(Integer)

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Time: 0 sec
eigen:=eigenvalues(A)$InnerEigenPackage(EXPR(INT))

$\label{eq4}\left[{{{?}^{2}}-{2 \ {\cos \left({x}\right)}\ ?}+{{\sin \left({x}\right)}^{2}}+{{\cos \left({x}\right)}^{2}}}\right]$

(4)

Type: List(Union(Expression(Integer),SparseUnivariatePolynomial?(Expression(Integer))))

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Time: 0 sec

Unfortunatly, result is unsimplified, siplification is separate, for example

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map(simplify, eigen(1))

$\label{eq5}{{?}^{2}}-{2 \ {\cos \left({x}\right)}\ ?}+ 1$

(5)

Type: SparseUnivariatePolynomial?(Expression(Integer))

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Time: 0 sec

test --unknown, Fri, 24 Jun 2005 08:05:16 -0500 reply

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A:=matrix[[cos(x)-L,-sin(x)],[sin(x),cos(x)-L]]

$\label{eq6}\left[ \begin{array}{cc} {{\cos \left({x}\right)}- L}& -{\sin \left({x}\right)} \ {\sin \left({x}\right)}&{{\cos \left({x}\right)}- L}$

(6)

Type: Matrix(Expression(Integer))

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Time: 0 sec
A(1,1)*A(2,2)

$\label{eq7}{{\cos \left({x}\right)}^{2}}-{2 \ L \ {\cos \left({x}\right)}}+{{L}^{2}}$

(7)

Type: Expression(Integer)

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Time: 0 sec
A(2,1)*A(1,2)

$\label{eq8}-{{\sin \left({x}\right)}^{2}}$

(8)

Type: Expression(Integer)

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Time: 0 sec
A(1,1)*A(2,2)-A(2,1)*A(1,2)

$\label{eq9}{{\sin \left({x}\right)}^{2}}+{{\cos \left({x}\right)}^{2}}-{2 \ L \ {\cos \left({x}\right)}}+{{L}^{2}}$

(9)

Type: Expression(Integer)

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Time: 0 sec
B := solve(A(1,1)*A(2,2)-A(2,1)*A(1,2)=0,L)

$\label{eq10}\left[{L ={{{\sqrt{- 1}}\ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}}, \:{L ={-{{\sqrt{- 1}}\ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}}\right]$

(10)

Type: List(Equation(Expression(Integer)))

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Time: 0 sec
B(1)

$\label{eq11}L ={{{\sqrt{- 1}}\ {\sin \left({x}\right)}}+{\cos \left({x}\right)}}$

(11)

Type: Equation(Expression(Integer))

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Time: 0 sec

test --unknown, Fri, 24 Jun 2005 08:16:56 -0500 reply

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solve(x^2 - 2,x)

$\label{eq12}\left[{{{{x}^{2}}- 2}= 0}\right]$

(12)

Type: List(Equation(Fraction(Polynomial(Integer))))

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Time: 0 sec
sqrt(2)

$\label{eq13}\sqrt{2}$

(13)

Type: AlgebraicNumber?

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Time: 0 sec
solve(x^2=4,x)

$\label{eq14}\left[{x = 2}, \:{x = - 2}\right]$

(14)

Type: List(Equation(Fraction(Polynomial(Integer))))

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Time: 0 sec

... --unknown, Fri, 07 Oct 2005 16:14:35 -0500 reply

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P:=matrix[[a, b], [1.0 - a, 1.0 - b]]

$\label{eq15}\left[ \begin{array}{cc} a & b \ {-{{1.0}\ a}+{1.0}}&{-{{1.0}\ b}+{1.0}}$

(15)

Type: Matrix(Polynomial(Float))

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Time: 0 sec
eigenvectors(P)

   There are 1 exposed and 1 unexposed library operations named 
      eigenvectors having 1 argument(s) but none was determined to be 
      applicable. Use HyperDoc Browse, or issue
                          )display op eigenvectors
      to learn more about the available operations. Perhaps 
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named 
      eigenvectors with argument type(s) 
                          Matrix(Polynomial(Float))

      Perhaps you should use "@" to indicate the required return type, 
      or "$" to specify which version of the function you need.

Subject: Be Bold !!

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