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Syntaxe et mise en oeuvre :
GNU/Linux |
CentOS 4.8 |
i386 |
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dgeevx(l) |
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DGEEVX - compute for an N-by-N real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors
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SUBROUTINE DGEEVX( |
BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV, WORK, LWORK, IWORK, INFO ) | ||
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CHARACTER |
BALANC, JOBVL, JOBVR, SENSE | ||
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INTEGER |
IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N | ||
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DOUBLE |
PRECISION ABNRM | ||
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INTEGER |
IWORK( * ) | ||
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DOUBLE |
PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ), SCALE( * ), VL( LDVL, * ), VR( LDVR, * ), WI( * ), WORK( * ), WR( * ) |
DGEEVX 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.
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) DOUBLE PRECISION 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) 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 will 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, 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) DOUBLE PRECISION 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) DOUBLE PRECISION
The one-norm of the balanced matrix (the maximum of the sum of absolute values of elements of any column).
RCONDE (output) DOUBLE PRECISION array, dimension (N)
RCONDE(j) is the reciprocal condition number of the j-th eigenvalue.
RCONDV (output) DOUBLE PRECISION array, dimension (N)
RCONDV(j) is the reciprocal condition number of the j-th right eigenvector.
WORK (workspace/output) DOUBLE PRECISION 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.
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.
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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dgeevx(l) | |