How to use dsyev routine to calculate eigenvalues?

问题

i am trying to write code to calculate eign vector and eign values for a symmetric matrix. I understand how to calculate evalues using pen & paper but i am slightly confused with the api!. I am a beginner so i may be wrong in interpreting the api parameters.

int main() {
        char jobz='V',uplo='U';
        int lda=3,n=3,info=8,lwork=9;
//      lapack_int lda=3,n=3,info=8;

        int i;
        double w[3],work[3];

        double a[9] = {
        3,2,4,
        2,0,2,
        4,2,3
        };




        info=LAPACKE_dsyev(LAPACK_ROW_MAJOR,jobz,uplo,  n  ,a,  lda  , w);    
        //dsyev_( &jobz,&uplo,&n, a, &lda, w,work , &lwork, &info );
        if( info > 0 ) {
                printf( "The algorithm failed to compute eigenvalues.\n" );
                exit( 1 );
        }
        for(i=0;i<3;i++)
        {
            printf("%f\n",w[i]);
         } 
        for(i=0;i<9;i++)
        {
                printf("%f\n",a[i]);
        }

        exit( 0 );
}

output: -1.000000 -1.000000 8.000000

0.617945 1.999713 -0.016938 0.010468 0.033876 0.999857 1.381966 0.618034 0.000000

whereas i expected k=-1: [1,-2,0] ,[4,2,-5] and k=8: [2,1,2] somewhere in the output!

am i using api incorrectly or am i reading the output incorrectly? also please suggest how do i do the same task with fortran api ? as with fortran i am unable to get proper eign values !. i.e. eign values i get with fortran: -0.134742 0.050742 0.523036

eign vectors: 0.617945 1.999713 -0.016938 0.010468 0.033876 0.999857 1.381966 0.618034 0.000000

回答1:

As suggested by @francis in the comment, the program works if you modify work[3] to work[9]. The obtained result is

Eigenvalues: w[0],w[1],w[2] => -1.000000 -1.000000 8.000000

1st eigenvector: a[0], a[1], a[2] => -0.494101 -0.472019 0.730111
2nd eigenvector: a[3], a[4], a[5] => -0.558050  0.816142 0.149979
3rd eigenvector: a[6], a[7], a[8] =>  0.666667  0.333333 0.666667

For comparison, let's diagonalize the same matrix with different programs. For example, Python/Numpy gives the result

>>> import numpy as np
>>> a = np.array([[3,2,4], [2,0,2], [4,2,3]], dtype=np.float )
>>> np.linalg.eig( a )
    (array([-1.,  8., -1.]),
     array([[-0.74535599,  0.66666667, -0.09414024],
            [ 0.2981424 ,  0.33333333, -0.84960833],
            [ 0.59628479,  0.66666667,  0.5189444 ]]))

while Julia gives

julia> a = Float64[ 3 2 4 ; 2 0 2 ; 4 2 3 ]
3x3 Array{Float64,2}:
 3.0  2.0  4.0
 2.0  0.0  2.0
 4.0  2.0  3.0

julia> eig( a )
([-0.9999999999999996,-0.9999999999999947,8.0],
3x3 Array{Float64,2}:
  0.447214  -0.596285  -0.666667
 -0.894427  -0.298142  -0.333333
  0.0        0.745356  -0.666667)

In both cases, the first three numbers are eigenvalues {-1,-1,8} and the following matrix are the corresponding eigenvectors (in their columns). You will see that all the programs give different results for eigenvectors. Because there are two degenerate eigenvalues (-1), any linear combination of the corresponding eigenvectors is also an eigenvector with the same eivenvalue, so the result is not unique. We can confirm that the degenerate eigenvectors obtained from C are related to those of Julia by a 2x2 orthogonal transformation (or a "rotation" by 101.6 degrees).

Interestingly, your expected eigenvectors [-1,2,0] and [4,2,-5] correspond exactly to the eigenvectors obtained from Julia (after normalization), but this is probably accidental and one cannot expect such agreement of degenerate eigenvectors.

来源：https://stackoverflow.com/questions/32327760/how-to-use-dsyev-routine-to-calculate-eigenvalues

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