SGEEV(3F) SGEEV(3F)
SGEEV - compute for an N-by-N real nonsymmetric matrix A, the eigenvalues
and, optionally, the left and/or right eigenvectors
SUBROUTINE SGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR,
WORK, LWORK, INFO )
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDVL, LDVR, LWORK, N
REAL A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), WI( * ),
WORK( * ), WR( * )
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 (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) REAL 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) REAL array, dimension (N)
WI (output) REAL array, dimension (N) WR and WI contain the
real and imaginary parts, respectively, of the computed
eigenvalues. Complex conjugate pairs of eigenvalues appear
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SGEEV(3F) SGEEV(3F)
consecutively with the eigenvalue having the positive imaginary
part first.
VL (output) 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 (input) INTEGER
The leading dimension of the array VL. LDVL >= 1; if JOBVL =
'V', LDVL >= N.
VR (output) 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 (input) INTEGER
The leading dimension of the array VR. LDVR >= 1; if JOBVR =
'V', LDVR >= N.
WORK (workspace/output) REAL array, dimension (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.
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.
SGEEV(3F) SGEEV(3F)
SGEEV - compute for an N-by-N real nonsymmetric matrix A, the eigenvalues
and, optionally, the left and/or right eigenvectors
SUBROUTINE SGEEV( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR,
WORK, LWORK, INFO )
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDVL, LDVR, LWORK, N
REAL A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), WI( * ),
WORK( * ), WR( * )
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 (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) REAL 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) REAL array, dimension (N)
WI (output) REAL array, dimension (N) WR and WI contain the
real and imaginary parts, respectively, of the computed
eigenvalues. Complex conjugate pairs of eigenvalues appear
Page 1
SGEEV(3F) SGEEV(3F)
consecutively with the eigenvalue having the positive imaginary
part first.
VL (output) 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 (input) INTEGER
The leading dimension of the array VL. LDVL >= 1; if JOBVL =
'V', LDVL >= N.
VR (output) 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 (input) INTEGER
The leading dimension of the array VR. LDVR >= 1; if JOBVR =
'V', LDVR >= N.
WORK (workspace/output) REAL array, dimension (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.
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.
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