DGGEVX(3S)DGGEVX(3S)NAME
DGGEVX - compute for a pair of N-by-N real nonsymmetric matrices (A,B)
SYNOPSIS
SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, IHI,
LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK,
LWORK, IWORK, BWORK, INFO )
CHARACTER BALANC, JOBVL, JOBVR, SENSE
INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
DOUBLE PRECISION ABNRM, BBNRM
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B(
LDB, * ), BETA( * ), LSCALE( * ), RCONDE( * ), RCONDV(
* ), RSCALE( * ), VL( LDVL, * ), VR( LDVR, * ), WORK(
* )
IMPLEMENTATION
These routines are part of the SCSL Scientific Library and can be loaded
using either the -lscs or the -lscs_mp option. The -lscs_mp option
directs the linker to use the multi-processor version of the library.
When linking to SCSL with -lscs or -lscs_mp, the default integer size is
4 bytes (32 bits). Another version of SCSL is available in which integers
are 8 bytes (64 bits). This version allows the user access to larger
memory sizes and helps when porting legacy Cray codes. It can be loaded
by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
only one of the two versions; 4-byte integer and 8-byte integer library
calls cannot be mixed.
PURPOSE
DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B) the
generalized eigenvalues, and optionally, the left and/or right
generalized eigenvectors.
Optionally also, it computes a balancing transformation to improve the
conditioning of the eigenvalues and eigenvectors (ILO, IHI, LSCALE,
RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for the
eigenvalues (RCONDE), and reciprocal condition numbers for the right
eigenvectors (RCONDV).
A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda
or a ratio alpha/beta = lambda, such that A - lambda*B is singular. It is
usually represented as the pair (alpha,beta), as there is a reasonable
interpretation for beta=0, and even for both being zero.
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The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of
(A,B) satisfies
A * v(j) = lambda(j) * B * v(j) .
The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of
(A,B) satisfies
u(j)**H * A = lambda(j) * u(j)**H * B.
where u(j)**H is the conjugate-transpose of u(j).
ARGUMENTS
BALANC (input) CHARACTER*1
Specifies the balance option to be performed. = 'N': do not
diagonally scale or permute;
= 'P': permute only;
= 'S': scale only;
= 'B': both permute and scale. Computed reciprocal condition
numbers will be for the matrices after permuting and/or
balancing. Permuting does not change condition numbers (in exact
arithmetic), but balancing does.
JOBVL (input) CHARACTER*1
= 'N': do not compute the left generalized eigenvectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input) CHARACTER*1
= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors.
SENSE (input) CHARACTER*1
Determines which reciprocal condition numbers are computed. =
'N': none are computed;
= 'E': computed for eigenvalues only;
= 'V': computed for eigenvectors only;
= 'B': computed for eigenvalues and eigenvectors.
N (input) INTEGER
The order of the matrices A, B, VL, and VR. N >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
On entry, the matrix A in the pair (A,B). On exit, A has been
overwritten. If JOBVL='V' or JOBVR='V' or both, then A contains
the first part of the real Schur form of the "balanced" versions
of the input A and B.
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
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B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
On entry, the matrix B in the pair (A,B). On exit, B has been
overwritten. If JOBVL='V' or JOBVR='V' or both, then B contains
the second part of the real Schur form of the "balanced" versions
of the input A and B.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
ALPHAR (output) DOUBLE PRECISION array, dimension (N)
ALPHAI (output) DOUBLE PRECISION array, dimension (N) BETA
(output) DOUBLE PRECISION array, dimension (N) On exit,
(ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the
generalized eigenvalues. If ALPHAI(j) is zero, then the j-th
eigenvalue is real; if positive, then the j-th and (j+1)-st
eigenvalues are a complex conjugate pair, with ALPHAI(j+1)
negative.
Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may
easily over- or underflow, and BETA(j) may even be zero. Thus,
the user should avoid naively computing the ratio ALPHA/BETA.
However, ALPHAR and ALPHAI will be always less than and usually
comparable with norm(A) in magnitude, and BETA always less than
and usually comparable with norm(B).
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 the j-th eigenvalue is real, then u(j) = VL(:,j),
the j-th column of VL. If the j-th and (j+1)-th 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). Each eigenvector will be scaled so
the largest component have abs(real part) + abs(imag. part) = 1.
Not referenced if JOBVL = 'N'.
LDVL (input) INTEGER
The leading dimension of the matrix VL. LDVL >= 1, and 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 the j-th eigenvalue is real, then v(j) = VR(:,j),
the j-th column of VR. If the j-th and (j+1)-th 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). Each eigenvector will be scaled so
the largest component have abs(real part) + abs(imag. part) = 1.
Not referenced if JOBVR = 'N'.
LDVR (input) INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR =
'V', LDVR >= N.
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ILO,IHI (output) INTEGER ILO and IHI are integer values such that
on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j = 1,...,ILO-1 or
i = IHI+1,...,N. If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
LSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied to the
left side of A and B. If PL(j) is the index of the row
interchanged with row j, and DL(j) is the scaling factor applied
to row j, then LSCALE(j) = PL(j) for j = 1,...,ILO-1 = DL(j)
for j = ILO,...,IHI = PL(j) for j = IHI+1,...,N. The order in
which the interchanges are made is N to IHI+1, then 1 to ILO-1.
RSCALE (output) DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied to the
right side of A and B. If PR(j) is the index of the column
interchanged with column j, and DR(j) is the scaling factor
applied to column j, then RSCALE(j) = PR(j) for j = 1,...,ILO-1
= DR(j) for j = ILO,...,IHI = PR(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 A.
BBNRM (output) DOUBLE PRECISION
The one-norm of the balanced matrix B.
RCONDE (output) DOUBLE PRECISION array, dimension (N)
If SENSE = 'E' or 'B', the reciprocal condition numbers of the
selected eigenvalues, stored in consecutive elements of the
array. For a complex conjugate pair of eigenvalues two
consecutive elements of RCONDE are set to the same value. Thus
RCONDE(j), RCONDV(j), and the j-th columns of VL and VR all
correspond to the same eigenpair (but not in general the j-th
eigenpair, unless all eigenpairs are selected). If SENSE = 'V',
RCONDE is not referenced.
RCONDV (output) DOUBLE PRECISION array, dimension (N)
If SENSE = 'V' or 'B', the estimated reciprocal condition numbers
of the selected eigenvectors, stored in consecutive elements of
the array. For a complex eigenvector two consecutive elements of
RCONDV are set to the same value. If the eigenvalues cannot be
reordered to compute RCONDV(j), RCONDV(j) is set to 0; this can
only occur when the true value would be very small anyway. If
SENSE = 'E', RCONDV is not referenced.
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. LWORK >= max(1,6*N). If SENSE =
'E', LWORK >= 12*N. If SENSE = 'V' or 'B', LWORK >=
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2*N*N+12*N+16.
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 (N+6)
If SENSE = 'E', IWORK is not referenced.
BWORK (workspace) LOGICAL array, dimension (N)
If SENSE = 'N', BWORK is not referenced.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
= 1,...,N: The QZ iteration failed. No eigenvectors have been
calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) should be
correct for j=INFO+1,...,N. > N: =N+1: other than QZ iteration
failed in DHGEQZ.
=N+2: error return from DTGEVC.
FURTHER DETAILS
Balancing a matrix pair (A,B) includes, first, permuting rows and columns
to isolate eigenvalues, second, applying diagonal similarity
transformation to the rows and columns to make the rows and columns as
close in norm as possible. 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 4.11.1.2 of
LAPACK Users' Guide.
An approximate error bound on the chordal distance between the i-th
computed generalized eigenvalue w and the corresponding exact eigenvalue
lambda is
chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
An approximate error bound for the angle between the i-th computed
eigenvector VL(i) or VR(i) is given by
EPS * norm(ABNRM, BBNRM) / DIF(i).
For further explanation of the reciprocal condition numbers RCONDE and
RCONDV, see section 4.11 of LAPACK User's Guide.
SEE ALSOINTRO_LAPACK(3S), INTRO_SCSL(3S)
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This man page is available only online.
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