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

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   <command>
 
   <command>
     <title>Numerical.EigenValuesAndAllVectors</title>
+
     <title>Num.EigenValuesAndAllVectors</title>
     <short_description>Eigenvalues and left and right eigenvectors of a matrix</short_description>
+
     <short_description>Computes eigenvalues and left and right eigenvectors of a matrix.</short_description>
 
<syntax>
 
<syntax>
$numerical.EigenValues(A:Matrix):List
+
Num.EigenValuesAndAllVectors(A:MAT):[B:MAT, C:MAT, D:MAT, E:MAT , F:MAT]
 
</syntax>
 
</syntax>
 
     <description>
 
     <description>
This function returns a List of three matrices, containing numerical approximation to A's eigenvalues and right and left eigenvectors.  
+
<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 rectangular!
+
<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!
+
This function returns a list of five matrices, containing numerical approximation of the eigenvalues of the matrix <tt>A</tt> and right and left eigenvectors.  
The output contains first of a matrix B, where the number of rows contains one of A's eigenvalues. The first column contains the eigenvalue's real part, the second the imaginary.
+
 
The second element of the list is a matrix of the size of A, containing A's left hand eigenvectors, while the third element in the list is a matrix containing the right hand eigenvectors.
+
<itemize>
 +
<item>@param <em>A</em> A quadratic matrix with rational entries.</item>
 +
<item>@return The output is a list of five matrices <tt>[B:MAT, C:MAT, D:MAT, E:MAT, F:MAT]</tt>. The first matrix <tt>B</tt> contains the complex eigenvalues of the matrix <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 the eigenvalue. The matrices <tt>C</tt> and <tt>D</tt> represent the right eigenvectors of <tt>A</tt>, i.e. the <tt>j</tt>-th column of <tt>C</tt> contains the real part of the right eigenvector corresponding to eigenvalue <tt>j</tt> and the <tt>j</tt>-th column of D contains the imaginary part of the same right eigenvector corresponding to eigenvalue <tt>j</tt>. The matrices <tt>E</tt> and <tt>F</tt> store the left eigenvectors analogue to <tt>C</tt> and <tt>D</tt>.</item>
 +
</itemize>
 +
 
 +
 
 
<example>
 
<example>
 
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.EigenValuesAndAllVectors(A);
+
Dec(Num.EigenValuesAndAllVectors(A),3);
-- CoCoAServer: computing Cpu Time = 0.0031
+
-- CoCoAServer: computing Cpu Time = 0.016
 
-------------------------------
 
-------------------------------
 
[Mat([
 
[Mat([
   [2038617447977453/70368744177664, 1593056728295919/4503599627370496, 0, 1717983664400761/562949953421312],
+
   [<quotes>28.970</quotes>, <quotes>-13.677</quotes>, <quotes>0.353</quotes>, <quotes>0.353</quotes>],
   [-3850002255576293/281474976710656, 1593056728295919/4503599627370496, 0, -1717983664400761/562949953421312]
+
   [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>3.051</quotes>, <quotes>-3.051</quotes>]
 
]), Mat([
 
]), Mat([
   [-4846625556027553/9007199254740992, -675715895173401/1125899906842624, 6285574018989927/36028797018963968, -7024364631742823/18014398509481984],
+
   [<quotes>0.538</quotes>, <quotes>-0.600</quotes>, <quotes>0.389</quotes>, <quotes>0.389</quotes>],
   [-5611119929071853/18014398509481984, -8025389267782659/36028797018963968, -630161806301403/4503599627370496, 7963794620848619/18014398509481984],
+
   [<quotes>0.311</quotes>, <quotes>-0.222</quotes>, <quotes>-0.442</quotes>, <quotes>-0.442</quotes>],
   [-3851121972702563/9007199254740992, 6293666352540409/36028797018963968, -2394868378529203/9007199254740992, -1824257157284653/36028797018963968],
+
   [<quotes>0.427</quotes>, <quotes>0.174</quotes>, <quotes>0.050</quotes>, <quotes>0.050</quotes>],
   [-5910799605047357/9007199254740992, 6738448111784605/9007199254740992, 6552680769135833/9007199254740992, 0]
+
   [<quotes>0.656</quotes>, <quotes>0.748</quotes>, <quotes>0</quotes>, <quotes>0</quotes>]
 
]), Mat([
 
]), Mat([
   [-7110239176083849/18014398509481984, -5241040126502889/9007199254740992, -569232410323621/18014398509481984, 4695168387448581/18014398509481984],
+
   [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>-0.174</quotes>, <quotes>0.174</quotes>],
   [-7846388397589843/18014398509481984, -3981313256671163/9007199254740992, -2719422585742633/9007199254740992, -4930385173711605/9007199254740992],
+
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0.139</quotes>, <quotes>-0.139</quotes>],
   [-3437594604471165/4503599627370496, 2800381393796867/4503599627370496, 6128985174171139/9007199254740992, 0],
+
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0.265</quotes>, <quotes>-0.265</quotes>],
   [-1207381852306067/4503599627370496, 634514467740541/2251799813685248, -2469130937097749/9007199254740992, 6644460631770309/144115188075855872]
+
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>-0.727</quotes>, <quotes>0.727</quotes>]
 +
]), Mat([
 +
   [<quotes>0.394</quotes>, <quotes>-0.581</quotes>, <quotes>0.260</quotes>, <quotes>0.260</quotes>],
 +
  [<quotes>0.435</quotes>, <quotes>-0.442</quotes>, <quotes>-0.547</quotes>, <quotes>-0.547</quotes>],
 +
  [<quotes>0.763</quotes>, <quotes>0.621</quotes>, <quotes>0</quotes>, <quotes>0</quotes>],
 +
  [<quotes>0.268</quotes>, <quotes>0.281</quotes>, <quotes>0.046</quotes>, <quotes>0.046</quotes>]
 +
]), Mat([
 +
   [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>-0.031</quotes>, <quotes>0.031</quotes>],
 +
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>-0.301</quotes>, <quotes>0.301</quotes>],
 +
   [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0.680</quotes>, <quotes>-0.680</quotes>],
 +
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>-0.274</quotes>, <quotes>0.274</quotes>]
 
])]
 
])]
 
-------------------------------
 
-------------------------------
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     <seealso>
 
     <seealso>
 
       <see>Introduction to CoCoAServer</see>
 
       <see>Introduction to CoCoAServer</see>
       <see>Numerical.QR</see>
+
       <see>Num.QR</see>
       <see>Numerical.SVD</see>
+
       <see>Num.SVD</see>
       <see>Numerical.EigenValues</see>
+
       <see>Num.EigenValues</see>
       <see>Numerical.EigenValuesAndVectors</see>
+
       <see>Num.EigenValuesAndVectors</see>
 
     </seealso>
 
     </seealso>
     <wiki-category>Package_Numerical</wiki-category>
+
    <types>
 +
      <type>apcocoaserver</type>
 +
      <type>matrix</type>
 +
    </types>
 +
    <key>EigenValuesAndAllVectors</key>
 +
    <key>Num.EigenValuesAndAllVectors</key>
 +
    <key>numerical.eigenvaluesandallvectors</key>
 +
     <wiki-category>Package_numerical</wiki-category>
 
   </command>
 
   </command>

Revision as of 07:48, 8 July 2009

Num.EigenValuesAndAllVectors

Computes eigenvalues and left and right eigenvectors of a matrix.

Syntax

Num.EigenValuesAndAllVectors(A:MAT):[B:MAT, C:MAT, D:MAT, E:MAT , F: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 returns a list of five matrices, containing numerical approximation of the eigenvalues of the matrix A and right and left eigenvectors.

  • @param A A quadratic matrix with rational entries.

  • @return The output is a list of five matrices [B:MAT, C:MAT, D:MAT, E:MAT, F:MAT]. The first matrix B contains the complex eigenvalues of the matrix 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 the eigenvalue. The matrices C and D represent the right eigenvectors of A, i.e. the j-th column of C contains the real part of the right eigenvector corresponding to eigenvalue j and the j-th column of D contains the imaginary part of the same right eigenvector corresponding to eigenvalue j. The matrices E and F store the left eigenvectors analogue to C and D.


Example

A:=Mat([[1,2,7,18],[2,4,9,12],[23,8,9,10],[7,5,3,2]]); 
Dec(Num.EigenValuesAndAllVectors(A),3);
-- CoCoAServer: computing Cpu Time = 0.016
-------------------------------
[Mat([
  [<quotes>28.970</quotes>, <quotes>-13.677</quotes>, <quotes>0.353</quotes>, <quotes>0.353</quotes>],
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>3.051</quotes>, <quotes>-3.051</quotes>]
]), Mat([
  [<quotes>0.538</quotes>, <quotes>-0.600</quotes>, <quotes>0.389</quotes>, <quotes>0.389</quotes>],
  [<quotes>0.311</quotes>, <quotes>-0.222</quotes>, <quotes>-0.442</quotes>, <quotes>-0.442</quotes>],
  [<quotes>0.427</quotes>, <quotes>0.174</quotes>, <quotes>0.050</quotes>, <quotes>0.050</quotes>],
  [<quotes>0.656</quotes>, <quotes>0.748</quotes>, <quotes>0</quotes>, <quotes>0</quotes>]
]), Mat([
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>-0.174</quotes>, <quotes>0.174</quotes>],
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0.139</quotes>, <quotes>-0.139</quotes>],
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0.265</quotes>, <quotes>-0.265</quotes>],
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>-0.727</quotes>, <quotes>0.727</quotes>]
]), Mat([
  [<quotes>0.394</quotes>, <quotes>-0.581</quotes>, <quotes>0.260</quotes>, <quotes>0.260</quotes>],
  [<quotes>0.435</quotes>, <quotes>-0.442</quotes>, <quotes>-0.547</quotes>, <quotes>-0.547</quotes>],
  [<quotes>0.763</quotes>, <quotes>0.621</quotes>, <quotes>0</quotes>, <quotes>0</quotes>],
  [<quotes>0.268</quotes>, <quotes>0.281</quotes>, <quotes>0.046</quotes>, <quotes>0.046</quotes>]
]), Mat([
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>-0.031</quotes>, <quotes>0.031</quotes>],
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>-0.301</quotes>, <quotes>0.301</quotes>],
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>0.680</quotes>, <quotes>-0.680</quotes>],
  [<quotes>0</quotes>, <quotes>0</quotes>, <quotes>-0.274</quotes>, <quotes>0.274</quotes>]
])]
-------------------------------

See also

Introduction to CoCoAServer

Num.QR

Num.SVD

Num.EigenValues

Num.EigenValuesAndVectors