NAME¶

DGEEV - computes for an N-by-N real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors

SYNOPSIS¶

SUBROUTINE DGEEV(

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

CHARACTER JOBVL, JOBVR INTEGER INFO, LDA, LDVL, LDVR, LWORK, N DOUBLE PRECISION A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), WI( * ), WORK( * ), WR( * )

PURPOSE¶

DGEEV 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.

ARGUMENTS¶

JOBVL (input) CHARACTER*1

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

JOBVR (input) CHARACTER*1

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

N (input) INTEGER

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

A (input/output) DOUBLE PRECISION array, dimension (LDA,N)

On entry, the N-by-N matrix A. On exit, A has been overwritten.

LDA (input) INTEGER

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

WR (output) DOUBLE PRECISION array, dimension (N)

WI (output) DOUBLE PRECISION 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 (output) DOUBLE PRECISION 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 (input) INTEGER

The leading dimension of the array VL. LDVL >= 1; if JOBVL = 'V', LDVL >= N.

VR (output) DOUBLE PRECISION 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 (input) INTEGER

The leading dimension of the array VR. LDVR >= 1; if JOBVR = 'V', LDVR >= N.

WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))

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

LWORK (input) 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 (output) 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.