SGGEVX(3) LAPACK routine of NEC Numeric Library Collection SGGEVX(3)
NAME
SGGEVX
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)
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
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.
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)
BALANC is 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)
JOBVL is CHARACTER*1
= 'N': do not compute the left generalized eigenvectors;
= 'V': compute the left generalized eigenvectors.
JOBVR (input)
JOBVR is CHARACTER*1
= 'N': do not compute the right generalized eigenvectors;
= 'V': compute the right generalized eigenvectors.
SENSE (input)
SENSE is 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)
N is INTEGER
The order of the matrices A, B, VL, and VR. N >= 0.
A (input/output)
A is 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" versions of the input A and B.
LDA (input)
LDA is INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output)
B is 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" versions of the input A and B.
LDB (input)
LDB is INTEGER
The leading dimension of B. LDB >= max(1,N).
ALPHAR (output)
ALPHAR is REAL array, dimension (N)
ALPHAI (output)
ALPHAI is REAL array, dimension (N)
BETA (output)
BETA is REAL 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)
VL is 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 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)
LDVL is INTEGER
The leading dimension of the matrix VL. LDVL >= 1, and
if JOBVL = 'V', LDVL >= N.
VR (output)
VR is 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 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)
LDVR is INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and
if JOBVR = 'V', LDVR >= N.
ILO (output)
ILO is INTEGER
IHI (output)
IHI is 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)
LSCALE is 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 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)
RSCALE is 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)
ABNRM is REAL
The one-norm of the balanced matrix A.
BBNRM (output)
BBNRM is REAL
The one-norm of the balanced matrix B.
RCONDE (output)
RCONDE is 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
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 j-th eigenpair.
If SENSE = 'N' or 'V', RCONDE is not referenced.
RCONDV (output)
RCONDV is REAL array, dimension (N)
If SENSE = 'V' or 'B', the estimated reciprocal condition
numbers 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 (output)
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,2*N).
If BALANC = '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 (output)
IWORK is INTEGER array, dimension (N+6)
If SENSE = 'E', IWORK is not referenced.
BWORK (output)
BWORK is LOGICAL array, dimension (N)
If SENSE = 'N', BWORK is not referenced.
INFO (output)
INFO is 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 SHGEQZ.
=N+2: error return from STGEVC.
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.
LAPACK routine 31 October 2017 SGGEVX(3)