LAPACK gives me the wrong eigenvalues

Question

I use DSYEV and DSYEVD from the LAPACK library to search for eigenvalues ​​and eigenvectors (compilation syntax: gfortran -llapack). However, for a particular matrix, I find the wrong eigenvalues ​​( -0.44,0.35,0.88). Something went wrong?

It is easy to see that the matrix has a zero determinant, therefore, at least one eigenvalue must be equal to zero.

Here is my code (hopefully it is not too big):

    Program Real_Eigenvec
    implicit none

    integer, parameter:: n=3
    integer:: i,j, flag
    real*8:: A(n,n),X(n,n) 
    real*8:: lambda(n)
    real*8, parameter:: p=0.5d0/dsqrt(2.d0), q=1.d0-1.d0/dsqrt(2.d0)


    Print*,'Enter flag: 0 for DSYEV, 1 for DSYEVD'
    Read*, flag


    A= transpose(reshape((/ 0.d0, 1.d0, 0.d0, p, q, p, 0.5d0, 0.0d0, 0.5d0 /), shape(A)))


    print*,'Dimension of the matrix, n=',int(sqrt(float(size(A))))

    Print*,'A matrix in full form:'
    Do i=1,n
      print 100, (A(i,j),j=1,n)
    End Do

    call Eigen(A,lambda,X,n,flag)

    !    Print the eigenvalues and eigenvectors.

    PRINT 200
    DO i  = 1, n
      PRINT 201, lambda(i), (X(i,j), j=1,n)
    END DO

100 FORMAT (1X,10(:2X,F10.2))
200 FORMAT (/1X, 'Eigenvalue', 16X, 'Eigenvector^T')
201 FORMAT (1X, F10.2,4X,6(:f10.2))

    End Program Real_Eigenvec



!!!      SUBROUTINES -----------------------------------------


    Subroutine Eigen(A,lambda,X,n,flag)
    implicit none

    integer:: i,n,flag
    real*8:: A(n,n),Ap(n,n),X(n,n)
    real*8:: lambda(n)
    real*8, allocatable  :: work(:)
        integer, allocatable :: iwork(:)
    integer:: lwork,liwork,info

    print*,'n in Eiegen routine=',n

    lwork=3*n-1      ! DSYEV for flag=0
    if (flag==1) then ! DSYEVD for flag=1
      lwork=1+6*n+2*n**2
    end if 

    liwork=3+5*n


    allocate(work(lwork))
    allocate(iwork(liwork))

    Ap=A

    if (flag==0) then
      CALL DSYEV ('v', 'l', n, Ap, n, lambda, work, lwork, info)
    else
      CALL  DSYEVD ('V', 'U', n, Ap, n, lambda, work, &
                &   lwork, iwork, liwork, info)
      ! For doumentation visit: http://www.netlib.org/lapack/explore-html/d1/da2/dsyevd_8f.html
    end if

    X=Ap

    print*,'info=',info

    deallocate(work)
    deallocate(iwork)

    End Subroutine Eigen

Answer 1

As indicated in the lapack documentation DSYEV can be used for symmetric matrices.

DSYEV , , A.

A

Dimension of the matrix, n=           3
A matrix in full form:
     0.00        1.00        0.00
     0.35        0.29        0.35
     0.50        0.00        0.50

DGEEV,

DGEEV N--N A, , , / .

, WR WL. , , VL VR .

A * v(j) = lambda(j) * v(j)
u(j)**H * A = lambda(j) * u(j)**H

:

DGEEV(JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)

LWORK = 4*N
CALL DGEEV( 'N', 'V', n, A, n, wr, wl, Ap, n, Ap, n, work, lwork, info )

, ,

real*8:: A(n,n),VL(n,n),VR(n,n)
real*8:: wr(n),wl(n)
lwork = 4*N
allocate(work(lwork))
CALL DGEEV( 'V', 'V', n, A, n, wr, wl, VL, n, VR, n, work, lwork, info )

. , (1.00, -0.21, 0.00).