SGGEVX(1) LAPACK driver routine (version 3.2) SGGEVX(1)NAME
SGGEVX - computes for a pair of N-by-N real nonsymmetric matrices (A,B)
SYNOPSIS
SUBROUTINE SGGEVX( 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
REAL ABNRM, BBNRM
LOGICAL BWORK( * )
INTEGER IWORK( * )
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
BETA( * ), LSCALE( * ), RCONDE( * ), RCONDV( * ),
RSCALE( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )
PURPOSE
SGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
the generalized eigenvalues, and optionally, the left and/or right gen‐
eralized 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 eigen‐
values (RCONDE), and reciprocal condition numbers for the right eigen‐
vectors (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 singu‐
lar. It is usually represented as the pair (alpha,beta), as there is a
reasonable interpretation for beta=0, and even for both being zero.
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 balanc‐
ing. 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) REAL 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" ver‐
sions of the input A and B.
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) REAL 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" ver‐
sions of the input A and B.
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
ALPHAR (output) REAL array, dimension (N)
ALPHAI (output) REAL array, dimension (N) BETA (output)
REAL array, dimension (N) On exit, (ALPHAR(j) +
ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigen‐
values. 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. How‐
ever, 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) 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 eigen‐
values. 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) 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 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 eigen‐
values 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.
ILO (output) INTEGER
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) REAL 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 inter‐
changed 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) REAL 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) REAL
The one-norm of the balanced matrix A.
BBNRM (output) REAL
The one-norm of the balanced matrix B.
RCONDE (output) REAL array, dimension (N)
If SENSE = 'E' or 'B', the reciprocal condition numbers of the
eigenvalues, stored in consecutive elements of the array. For
a complex conjugate pair of eigenvalues two consecutive ele‐
ments 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 j-th eigenpair. If SENSE = 'N' or 'V', RCONDE is not ref‐
erenced.
RCONDV (output) REAL array, dimension (N)
If SENSE = 'V' or 'B', the estimated reciprocal condition num‐
bers of the 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 = 'N' or 'E', RCONDV is not referenced.
WORK (workspace/output) REAL 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(1,2*N). If BAL‐
ANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V', LWORK >=
max(1,6*N). If SENSE = 'E', LWORK >= max(1,10*N). If SENSE =
'V' or 'B', LWORK >= 2*N*N+8*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 itera‐
tion failed in SHGEQZ.
=N+2: error return from STGEVC.
FURTHER DETAILS
Balancing a matrix pair (A,B) includes, first, permuting rows and col‐
umns 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 eigenval‐
ue lambda is
chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I) An
approximate error bound for the angle between the i-th computed eigen‐
vector 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.
LAPACK driver routine (version 3November 2008 SGGEVX(1)
