Difference between revisions of "ApCoCoA-1:Num.EigenValues"

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   <command>
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   {{Version|1}}
     <title>Numerical.EigenValues</title>
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<command>
     <short_description>eigenvalues of a matrix</short_description>
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     <title>Num.EigenValues</title>
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     <short_description>Computes the eigenvalues of a matrix.</short_description>
 
<syntax>
 
<syntax>
$numerical.EigenValues(A:Matrix):List
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Num.EigenValues(A:MAT):MAT
 
</syntax>
 
</syntax>
 
     <description>
 
     <description>
This function returns a matrix, containing numerical approximation to A's eigenvalues.  
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<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them.
Therefore the input matrix A has to be quadratic!
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<par/>
It is implemented in the ApCoCoA server, so you need a running server. It was not implemented in version 0.99.4 or previous. Also please keep in mind this method is based on blas/Lapack's eigenvalue solver and uses floating point arithmetic. This is not an exact, algebraic method!
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This function computes the approximate complex eigenvalues of the matrix <tt>A</tt>.
The output contains a matrix B. Each of the columns in B describe one of the eigenvalues of A. The first row of B contains the real part of the eigenvalues, the second row the imaginary ones.  
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 +
<itemize>
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<item>@param <em>A</em> A quadratic matrix with rational entries.</item>
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<item>@return The return value is a matrix with two rows. Each column of this matrix represents one approximate complex eigenvalue of <tt>A</tt>, i.e. the first entry of a column is the real part and the second entry of the same column is the imaginary part of one complex eigenvalue.</item>
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</itemize>
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<example>
 
<example>
 +
Use P::=QQ[x,y,z];
 +
 
A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]);
 
A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]);
Numerical.EigenValues(A);
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Dec(Num.EigenValues(A),3);
-- CoCoAServer: computing Cpu Time = 0.0049
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-- CoCoAServer: computing Cpu Time = 0.015
 
-------------------------------
 
-------------------------------
 
Mat([
 
Mat([
   [4077234895954899/140737488355328, -3850002255576291/281474976710656, 3186113456591853/9007199254740992, 3186113456591853/9007199254740992],
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   ["28.970", "-13.677", "0.353", "0.353"],
   [0, 0, 6871934657603045/2251799813685248, -6871934657603045/2251799813685248]
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   ["0", "0", "3.051", "-3.051"]
 
])
 
])
 
-------------------------------
 
-------------------------------
 +
 
</example>
 
</example>
 
     </description>
 
     </description>
 
     <seealso>
 
     <seealso>
       <see>Introduction to CoCoAServer</see>
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       <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see>
       <see>Numerical.QR</see>
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       <see>ApCoCoA-1:Num.QR|Num.QR</see>
       <see>Numerical.SVD</see>
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       <see>ApCoCoA-1:Num.SVD|Num.SVD</see>
       <see>Numerical.EigenValuesAndVectors</see>
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       <see>ApCoCoA-1:Num.EigenValuesAndVectors|Num.EigenValuesAndVectors</see>
       <see>Numerical.EigenValuesAndAllVectors</see>
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       <see>ApCoCoA-1:Num.EigenValuesAndAllVectors|Num.EigenValuesAndAllVectors</see>
 
     </seealso>
 
     </seealso>
 
     <types>
 
     <types>
       <type>cocoaserver</type>
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       <type>apcocoaserver</type>
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      <type>matrix</type>
 
     </types>
 
     </types>
     <key>heldt</key>
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     <key>Num.EigenValues</key>
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    <key>EigenValues</key>
 
     <key>numerical.eigenvalues</key>
 
     <key>numerical.eigenvalues</key>
     <wiki-category>Package_Numerical</wiki-category>
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     <wiki-category>ApCoCoA-1:Package_numerical</wiki-category>
 
   </command>
 
   </command>

Latest revision as of 13:47, 29 October 2020

This article is about a function from ApCoCoA-1.

Num.EigenValues

Computes the eigenvalues of a matrix.

Syntax

Num.EigenValues(A:MAT):MAT

Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

This function computes the approximate complex eigenvalues of the matrix A.

  • @param A A quadratic matrix with rational entries.

  • @return The return value is a matrix with two rows. Each column of this matrix represents one approximate complex eigenvalue of A, i.e. the first entry of a column is the real part and the second entry of the same column is the imaginary part of one complex eigenvalue.


Example

Use P::=QQ[x,y,z];

A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]);
Dec(Num.EigenValues(A),3);
-- CoCoAServer: computing Cpu Time = 0.015
-------------------------------
Mat([
  ["28.970", "-13.677", "0.353", "0.353"],
  ["0", "0", "3.051", "-3.051"]
])
-------------------------------

See also

Introduction to CoCoAServer

Num.QR

Num.SVD

Num.EigenValuesAndVectors

Num.EigenValuesAndAllVectors