Is it possible to use the deflation algorithm to compute the eigenvalues of a large sparse matrix












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I am trying to compute the eigenvalues of a large sparse matrix (about 10% of the values are nonzero). The matrix is real valued, but since it is accumulated by a stochastic process it is not fully symmetric. I have used power iteration to compute the largest eigenvalue and the method worked fine. Unfortunately if I would apply the standard methods of deflation to compute the other eigenvalues I will spoil the sparsity.



I tried exploiting the fact that the eigenvectors of a real symmetric matrix form a basis. In this basis every vector can be expressed as



${bf x} = a_1{bf v}_1+a_2{bf v}_2 + dots + a_n{bf v}_n$



One can eliminate the eigenvalues one by one by subtracting the component along the corresponding eigenvector.



$ {bf x}_{n+1} = {bf A }{bf x}_{n} $

$ {bf x}_{n+1} = {bf x}_{n+1} - a_1{bf v}_1 $



Where $a_1 = frac{<{bf v}_1,{bf x}_{n+1}>}{lVert {bf x}_{n+1} rVert}$, and ${bf v}_1$ is the eigenvector corresponding to the largest eigenvalue as computed by the power iteration method.



The problem is that my matrix is not really symmetric. Therefore, I presume that my strategy is wrong. In the case of an asymmetric matrix, the eigenvalues can be complex, and some of the eigenvalues might have multiplicity bigger than one.



I looked at other possibilities, for instance



${bf A } - {bf x}_1{bf u}_1^T $



Where different choices of ${bf u}_1$ can be made e.g. left eigenvector of ${bf A }$ or ${bf u}_1=lambda_1{bf x}_1$ etc..
Unfortunately the resulting matrix is no longer sparse.



I will appreciate ideas how to apply the deflation algorithm to the spectrum of the problem outlined above.



Thank you in advance,



Alex






matrices eigenvalues-eigenvectors







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    $begingroup$


    I am trying to compute the eigenvalues of a large sparse matrix (about 10% of the values are nonzero). The matrix is real valued, but since it is accumulated by a stochastic process it is not fully symmetric. I have used power iteration to compute the largest eigenvalue and the method worked fine. Unfortunately if I would apply the standard methods of deflation to compute the other eigenvalues I will spoil the sparsity.



    I tried exploiting the fact that the eigenvectors of a real symmetric matrix form a basis. In this basis every vector can be expressed as



    ${bf x} = a_1{bf v}_1+a_2{bf v}_2 + dots + a_n{bf v}_n$



    One can eliminate the eigenvalues one by one by subtracting the component along the corresponding eigenvector.



    $ {bf x}_{n+1} = {bf A }{bf x}_{n} $

    $ {bf x}_{n+1} = {bf x}_{n+1} - a_1{bf v}_1 $



    Where $a_1 = frac{<{bf v}_1,{bf x}_{n+1}>}{lVert {bf x}_{n+1} rVert}$, and ${bf v}_1$ is the eigenvector corresponding to the largest eigenvalue as computed by the power iteration method.



    The problem is that my matrix is not really symmetric. Therefore, I presume that my strategy is wrong. In the case of an asymmetric matrix, the eigenvalues can be complex, and some of the eigenvalues might have multiplicity bigger than one.



    I looked at other possibilities, for instance



    ${bf A } - {bf x}_1{bf u}_1^T $



    Where different choices of ${bf u}_1$ can be made e.g. left eigenvector of ${bf A }$ or ${bf u}_1=lambda_1{bf x}_1$ etc..
    Unfortunately the resulting matrix is no longer sparse.



    I will appreciate ideas how to apply the deflation algorithm to the spectrum of the problem outlined above.



    Thank you in advance,



    Alex






    matrices eigenvalues-eigenvectors







    share|cite|improve this question











    $endgroup$















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      0








      0





      $begingroup$


      I am trying to compute the eigenvalues of a large sparse matrix (about 10% of the values are nonzero). The matrix is real valued, but since it is accumulated by a stochastic process it is not fully symmetric. I have used power iteration to compute the largest eigenvalue and the method worked fine. Unfortunately if I would apply the standard methods of deflation to compute the other eigenvalues I will spoil the sparsity.



      I tried exploiting the fact that the eigenvectors of a real symmetric matrix form a basis. In this basis every vector can be expressed as



      ${bf x} = a_1{bf v}_1+a_2{bf v}_2 + dots + a_n{bf v}_n$



      One can eliminate the eigenvalues one by one by subtracting the component along the corresponding eigenvector.



      $ {bf x}_{n+1} = {bf A }{bf x}_{n} $

      $ {bf x}_{n+1} = {bf x}_{n+1} - a_1{bf v}_1 $



      Where $a_1 = frac{<{bf v}_1,{bf x}_{n+1}>}{lVert {bf x}_{n+1} rVert}$, and ${bf v}_1$ is the eigenvector corresponding to the largest eigenvalue as computed by the power iteration method.



      The problem is that my matrix is not really symmetric. Therefore, I presume that my strategy is wrong. In the case of an asymmetric matrix, the eigenvalues can be complex, and some of the eigenvalues might have multiplicity bigger than one.



      I looked at other possibilities, for instance



      ${bf A } - {bf x}_1{bf u}_1^T $



      Where different choices of ${bf u}_1$ can be made e.g. left eigenvector of ${bf A }$ or ${bf u}_1=lambda_1{bf x}_1$ etc..
      Unfortunately the resulting matrix is no longer sparse.



      I will appreciate ideas how to apply the deflation algorithm to the spectrum of the problem outlined above.



      Thank you in advance,



      Alex






      matrices eigenvalues-eigenvectors







      share|cite|improve this question











      $endgroup$




      I am trying to compute the eigenvalues of a large sparse matrix (about 10% of the values are nonzero). The matrix is real valued, but since it is accumulated by a stochastic process it is not fully symmetric. I have used power iteration to compute the largest eigenvalue and the method worked fine. Unfortunately if I would apply the standard methods of deflation to compute the other eigenvalues I will spoil the sparsity.



      I tried exploiting the fact that the eigenvectors of a real symmetric matrix form a basis. In this basis every vector can be expressed as



      ${bf x} = a_1{bf v}_1+a_2{bf v}_2 + dots + a_n{bf v}_n$



      One can eliminate the eigenvalues one by one by subtracting the component along the corresponding eigenvector.



      $ {bf x}_{n+1} = {bf A }{bf x}_{n} $

      $ {bf x}_{n+1} = {bf x}_{n+1} - a_1{bf v}_1 $



      Where $a_1 = frac{<{bf v}_1,{bf x}_{n+1}>}{lVert {bf x}_{n+1} rVert}$, and ${bf v}_1$ is the eigenvector corresponding to the largest eigenvalue as computed by the power iteration method.



      The problem is that my matrix is not really symmetric. Therefore, I presume that my strategy is wrong. In the case of an asymmetric matrix, the eigenvalues can be complex, and some of the eigenvalues might have multiplicity bigger than one.



      I looked at other possibilities, for instance



      ${bf A } - {bf x}_1{bf u}_1^T $



      Where different choices of ${bf u}_1$ can be made e.g. left eigenvector of ${bf A }$ or ${bf u}_1=lambda_1{bf x}_1$ etc..
      Unfortunately the resulting matrix is no longer sparse.



      I will appreciate ideas how to apply the deflation algorithm to the spectrum of the problem outlined above.



      Thank you in advance,



      Alex






      matrices eigenvalues-eigenvectors




      matrices eigenvalues-eigenvectors






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      edited Aug 6 '14 at 9:55







      Alexander Cska

















      asked Jul 30 '14 at 15:46









      Alexander CskaAlexander Cska

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          wasWell the answer to the above question is yes. The trouble I was having is due to a coding bug. I tried the methodology on a 45000 x 45000 sparse matrix having about 11% non zero elements. The first eigenvalue coincided exactly whit the values given by eigs() from matlab when printed in double precision. The second eigenvalue coincided up to 9 digits after the decimal point. So it looks that although my matrix is not symmetric and positive definite, the above algorithm somehow worked.



          If you hove other ideas, write me a comment.






          share|cite|improve this answer









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            $begingroup$

            wasWell the answer to the above question is yes. The trouble I was having is due to a coding bug. I tried the methodology on a 45000 x 45000 sparse matrix having about 11% non zero elements. The first eigenvalue coincided exactly whit the values given by eigs() from matlab when printed in double precision. The second eigenvalue coincided up to 9 digits after the decimal point. So it looks that although my matrix is not symmetric and positive definite, the above algorithm somehow worked.



            If you hove other ideas, write me a comment.






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              wasWell the answer to the above question is yes. The trouble I was having is due to a coding bug. I tried the methodology on a 45000 x 45000 sparse matrix having about 11% non zero elements. The first eigenvalue coincided exactly whit the values given by eigs() from matlab when printed in double precision. The second eigenvalue coincided up to 9 digits after the decimal point. So it looks that although my matrix is not symmetric and positive definite, the above algorithm somehow worked.



              If you hove other ideas, write me a comment.






              share|cite|improve this answer









              $endgroup$
















                0












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                0





                $begingroup$

                wasWell the answer to the above question is yes. The trouble I was having is due to a coding bug. I tried the methodology on a 45000 x 45000 sparse matrix having about 11% non zero elements. The first eigenvalue coincided exactly whit the values given by eigs() from matlab when printed in double precision. The second eigenvalue coincided up to 9 digits after the decimal point. So it looks that although my matrix is not symmetric and positive definite, the above algorithm somehow worked.



                If you hove other ideas, write me a comment.






                share|cite|improve this answer









                $endgroup$



                wasWell the answer to the above question is yes. The trouble I was having is due to a coding bug. I tried the methodology on a 45000 x 45000 sparse matrix having about 11% non zero elements. The first eigenvalue coincided exactly whit the values given by eigs() from matlab when printed in double precision. The second eigenvalue coincided up to 9 digits after the decimal point. So it looks that although my matrix is not symmetric and positive definite, the above algorithm somehow worked.



                If you hove other ideas, write me a comment.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Aug 6 '14 at 11:35









                Alexander CskaAlexander Cska

                253112




                253112






























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