SGEEV computes for an N-by-N real nonsymmetric matrix A, the


 eigenvalues and, optionally, the left and/or right eigenvectors.


 The right eigenvector v(j) of A satisfies


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


 where lambda(j) is its eigenvalue.


 The left eigenvector u(j) of A satisfies


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


 where u(j)**H denotes the conjugate-transpose of u(j).


 The computed eigenvectors are normalized to have Euclidean norm


 equal to 1 and largest component real.

JOBVL

          JOBVL is CHARACTER*1


          = 'N': left eigenvectors of A are not computed;


          = 'V': left eigenvectors of A are computed.

JOBVR

          JOBVR is CHARACTER*1


          = 'N': right eigenvectors of A are not computed;


          = 'V': right eigenvectors of A are computed.

N

          N is INTEGER


          The order of the matrix A. N >= 0.

A

          A is REAL array, dimension (LDA,N)


          On entry, the N-by-N matrix A.


          On exit, A has been overwritten.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,N).

WR

          WR is REAL array, dimension (N)

WI

          WI is REAL array, dimension (N)


          WR and WI contain the real and imaginary parts,


          respectively, of the computed eigenvalues.  Complex


          conjugate pairs of eigenvalues appear consecutively


          with the eigenvalue having the positive imaginary part


          first.

VL

          VL is REAL array, dimension (LDVL,N)


          If JOBVL = 'V', the left eigenvectors u(j) are stored one


          after another in the columns of VL, in the same order


          as their eigenvalues.


          If JOBVL = 'N', VL is not referenced.


          If the j-th eigenvalue is real, then u(j) = VL(:,j),


          the j-th column of VL.


          If the j-th and (j+1)-st eigenvalues form a complex


          conjugate pair, then u(j) = VL(:,j) + i*VL(:,j+1) and


          u(j+1) = VL(:,j) - i*VL(:,j+1).

LDVL

          LDVL is INTEGER


          The leading dimension of the array VL.  LDVL >= 1; if


          JOBVL = 'V', LDVL >= N.

VR

          VR is REAL array, dimension (LDVR,N)


          If JOBVR = 'V', the right eigenvectors v(j) are stored one


          after another in the columns of VR, in the same order


          as their eigenvalues.


          If JOBVR = 'N', VR is not referenced.


          If the j-th eigenvalue is real, then v(j) = VR(:,j),


          the j-th column of VR.


          If the j-th and (j+1)-st eigenvalues form a complex


          conjugate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and


          v(j+1) = VR(:,j) - i*VR(:,j+1).

LDVR

          LDVR is INTEGER


          The leading dimension of the array VR.  LDVR >= 1; if


          JOBVR = 'V', LDVR >= N.

WORK

          WORK is REAL array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.  LWORK >= max(1,3*N), and


          if JOBVL = 'V' or JOBVR = 'V', LWORK >= 4*N.  For good


          performance, LWORK must generally be larger.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value.


          > 0:  if INFO = i, the QR algorithm failed to compute all the


                eigenvalues, and no eigenvectors have been computed;


                elements i+1:N of WR and WI contain eigenvalues which


                have converged.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

September 2012