SGEEVX(3F) SGEEVX(3F)
SGEEVX - compute for an N-by-N real nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors
SUBROUTINE SGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL,
LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE,
RCONDV, WORK, LWORK, IWORK, INFO )
CHARACTER BALANC, JOBVL, JOBVR, SENSE
INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
REAL ABNRM
INTEGER IWORK( * )
REAL A( LDA, * ), RCONDE( * ), RCONDV( * ), SCALE( * ), VL(
LDVL, * ), VR( LDVR, * ), WI( * ), WORK( * ), WR( * )
SGEEVX computes for an N-by-N real nonsymmetric matrix A, the eigenvalues
and, optionally, the left and/or right eigenvectors.
Optionally also, it computes a balancing transformation to improve the
conditioning of the eigenvalues and eigenvectors (ILO, IHI, SCALE, and
ABNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and
reciprocal condition numbers for the right
eigenvectors (RCONDV).
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.
Balancing a matrix means permuting the rows and columns to make it more
nearly upper triangular, and applying a diagonal similarity
transformation D * A * D**(-1), where D is a diagonal matrix, to make its
rows and columns closer in norm and the condition numbers of its
eigenvalues and eigenvectors smaller. The computed reciprocal condition
numbers correspond to the balanced matrix. Permuting rows and columns
will not change the condition numbers (in exact arithmetic) but diagonal
scaling will. For further explanation of balancing, see section 4.10.2
of the LAPACK Users' Guide.
Page 1
SGEEVX(3F) SGEEVX(3F)
BALANC (input) CHARACTER*1
Indicates how the input matrix should be diagonally scaled and/or
permuted to improve the conditioning of its eigenvalues. = 'N':
Do not diagonally scale or permute;
= 'P': Perform permutations to make the matrix more nearly upper
triangular. Do not diagonally scale; = 'S': Diagonally scale the
matrix, i.e. replace A by D*A*D**(-1), where D is a diagonal
matrix chosen to make the rows and columns of A more equal in
norm. Do not permute; = 'B': Both diagonally scale and permute A.
Computed reciprocal condition numbers will be for the matrix
after balancing and/or permuting. Permuting does not change
condition numbers (in exact arithmetic), but balancing does.
JOBVL (input) CHARACTER*1
= 'N': left eigenvectors of A are not computed;
= 'V': left eigenvectors of A are computed. If SENSE = 'E' or
'B', JOBVL must = 'V'.
JOBVR (input) CHARACTER*1
= 'N': right eigenvectors of A are not computed;
= 'V': right eigenvectors of A are computed. If SENSE = 'E' or
'B', JOBVR must = 'V'.
SENSE (input) CHARACTER*1
Determines which reciprocal condition numbers are computed. =
'N': None are computed;
= 'E': Computed for eigenvalues only;
= 'V': Computed for right eigenvectors only;
= 'B': Computed for eigenvalues and right eigenvectors.
If SENSE = 'E' or 'B', both left and right eigenvectors must also
be computed (JOBVL = 'V' and JOBVR = 'V').
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.
If JOBVL = 'V' or JOBVR = 'V', A contains the real Schur form of
the balanced version of the input matrix A.
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 will appear
consecutively with the eigenvalue having the positive imaginary
part first.
Page 2
SGEEVX(3F) SGEEVX(3F)
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, and if JOBVR =
'V', LDVR >= N.
ILO,IHI (output) INTEGER ILO and IHI are integer values
determined when A was balanced. The balanced A(i,j) = 0 if I > J
and J = 1,...,ILO-1 or I = IHI+1,...,N.
SCALE (output) REAL array, dimension (N)
Details of the permutations and scaling factors applied when
balancing A. If P(j) is the index of the row and column
interchanged with row and column j, and D(j) is the scaling
factor applied to row and column j, then SCALE(J) = P(J), for
J = 1,...,ILO-1 = D(J), for J = ILO,...,IHI = P(J) for J =
IHI+1,...,N. The order in which the interchanges are made is N
to IHI+1, then 1 to ILO-1.
ABNRM (output) REAL
The one-norm of the balanced matrix (the maximum of the sum of
absolute values of elements of any column).
RCONDE (output) REAL array, dimension (N)
RCONDE(j) is the reciprocal condition number of the j-th
eigenvalue.
RCONDV (output) REAL array, dimension (N)
RCONDV(j) is the reciprocal condition number of the j-th right
eigenvector.
Page 3
SGEEVX(3F) SGEEVX(3F)
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. If SENSE = 'N' or 'E', LWORK
>= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V', LWORK >= 3*N.
If SENSE = 'V' or 'B', LWORK >= N*(N+6). For good performance,
LWORK must generally be larger.
IWORK (workspace) INTEGER array, dimension (2*N-2)
If SENSE = 'N' or 'E', not referenced.
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 or condition numbers have been
computed; elements 1:ILO-1 and i+1:N of WR and WI contain
eigenvalues which have converged.
SGEEVX(3F) SGEEVX(3F)
SGEEVX - compute for an N-by-N real nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors
SUBROUTINE SGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL,
LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE,
RCONDV, WORK, LWORK, IWORK, INFO )
CHARACTER BALANC, JOBVL, JOBVR, SENSE
INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
REAL ABNRM
INTEGER IWORK( * )
REAL A( LDA, * ), RCONDE( * ), RCONDV( * ), SCALE( * ), VL(
LDVL, * ), VR( LDVR, * ), WI( * ), WORK( * ), WR( * )
SGEEVX computes for an N-by-N real nonsymmetric matrix A, the eigenvalues
and, optionally, the left and/or right eigenvectors.
Optionally also, it computes a balancing transformation to improve the
conditioning of the eigenvalues and eigenvectors (ILO, IHI, SCALE, and
ABNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and
reciprocal condition numbers for the right
eigenvectors (RCONDV).
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.
Balancing a matrix means permuting the rows and columns to make it more
nearly upper triangular, and applying a diagonal similarity
transformation D * A * D**(-1), where D is a diagonal matrix, to make its
rows and columns closer in norm and the condition numbers of its
eigenvalues and eigenvectors smaller. The computed reciprocal condition
numbers correspond to the balanced matrix. Permuting rows and columns
will not change the condition numbers (in exact arithmetic) but diagonal
scaling will. For further explanation of balancing, see section 4.10.2
of the LAPACK Users' Guide.
Page 1
SGEEVX(3F) SGEEVX(3F)
BALANC (input) CHARACTER*1
Indicates how the input matrix should be diagonally scaled and/or
permuted to improve the conditioning of its eigenvalues. = 'N':
Do not diagonally scale or permute;
= 'P': Perform permutations to make the matrix more nearly upper
triangular. Do not diagonally scale; = 'S': Diagonally scale the
matrix, i.e. replace A by D*A*D**(-1), where D is a diagonal
matrix chosen to make the rows and columns of A more equal in
norm. Do not permute; = 'B': Both diagonally scale and permute A.
Computed reciprocal condition numbers will be for the matrix
after balancing and/or permuting. Permuting does not change
condition numbers (in exact arithmetic), but balancing does.
JOBVL (input) CHARACTER*1
= 'N': left eigenvectors of A are not computed;
= 'V': left eigenvectors of A are computed. If SENSE = 'E' or
'B', JOBVL must = 'V'.
JOBVR (input) CHARACTER*1
= 'N': right eigenvectors of A are not computed;
= 'V': right eigenvectors of A are computed. If SENSE = 'E' or
'B', JOBVR must = 'V'.
SENSE (input) CHARACTER*1
Determines which reciprocal condition numbers are computed. =
'N': None are computed;
= 'E': Computed for eigenvalues only;
= 'V': Computed for right eigenvectors only;
= 'B': Computed for eigenvalues and right eigenvectors.
If SENSE = 'E' or 'B', both left and right eigenvectors must also
be computed (JOBVL = 'V' and JOBVR = 'V').
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.
If JOBVL = 'V' or JOBVR = 'V', A contains the real Schur form of
the balanced version of the input matrix A.
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 will appear
consecutively with the eigenvalue having the positive imaginary
part first.
Page 2
SGEEVX(3F) SGEEVX(3F)
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, and if JOBVR =
'V', LDVR >= N.
ILO,IHI (output) INTEGER ILO and IHI are integer values
determined when A was balanced. The balanced A(i,j) = 0 if I > J
and J = 1,...,ILO-1 or I = IHI+1,...,N.
SCALE (output) REAL array, dimension (N)
Details of the permutations and scaling factors applied when
balancing A. If P(j) is the index of the row and column
interchanged with row and column j, and D(j) is the scaling
factor applied to row and column j, then SCALE(J) = P(J), for
J = 1,...,ILO-1 = D(J), for J = ILO,...,IHI = P(J) for J =
IHI+1,...,N. The order in which the interchanges are made is N
to IHI+1, then 1 to ILO-1.
ABNRM (output) REAL
The one-norm of the balanced matrix (the maximum of the sum of
absolute values of elements of any column).
RCONDE (output) REAL array, dimension (N)
RCONDE(j) is the reciprocal condition number of the j-th
eigenvalue.
RCONDV (output) REAL array, dimension (N)
RCONDV(j) is the reciprocal condition number of the j-th right
eigenvector.
Page 3
SGEEVX(3F) SGEEVX(3F)
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. If SENSE = 'N' or 'E', LWORK
>= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V', LWORK >= 3*N.
If SENSE = 'V' or 'B', LWORK >= N*(N+6). For good performance,
LWORK must generally be larger.
IWORK (workspace) INTEGER array, dimension (2*N-2)
If SENSE = 'N' or 'E', not referenced.
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 or condition numbers have been
computed; elements 1:ILO-1 and i+1:N of WR and WI contain
eigenvalues which have converged.
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