SGEGV(3F) SGEGV(3F)
SGEGV - compute for a pair of n-by-n real nonsymmetric matrices A and B,
the generalized eigenvalues (alphar +/- alphai*i, beta), and optionally,
the left and/or right generalized eigenvectors (VL and VR)
SUBROUTINE SGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA,
VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
BETA( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )
SGEGV computes for a pair of n-by-n real nonsymmetric matrices A and B,
the generalized eigenvalues (alphar +/- alphai*i, beta), and optionally,
the left and/or right generalized eigenvectors (VL and VR).
A generalized eigenvalue for a pair of matrices (A,B) is, roughly
speaking, a scalar w or a ratio alpha/beta = w, such that A - w*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.
A good beginning reference is the book, "Matrix Computations", by G.
Golub & C. van Loan (Johns Hopkins U. Press)
A right generalized eigenvector corresponding to a generalized eigenvalue
w for a pair of matrices (A,B) is a vector r such that (A - w B) r =
0 . A left generalized eigenvector is a vector l such that l**H * (A - w
B) = 0, where l**H is the
conjugate-transpose of l.
Note: this routine performs "full balancing" on A and B -- see "Further
Details", below.
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.
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 first of the pair of matrices whose generalized
eigenvalues and (optionally) generalized eigenvectors are to be
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SGEGV(3F) SGEGV(3F)
computed. On exit, the contents will have been destroyed. (For
a description of the contents of A on exit, see "Further
Details", below.)
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) REAL array, dimension (LDB, N)
On entry, the second of the pair of matrices whose generalized
eigenvalues and (optionally) generalized eigenvectors are to be
computed. On exit, the contents will have been destroyed. (For
a description of the contents of B on exit, see "Further
Details", below.)
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 eigenvalues. If ALPHAI(j) is
zero, then the j-th eigenvalue is real; if positive, then the jth
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) REAL array, dimension (LDVL,N)
If JOBVL = 'V', the left generalized eigenvectors. (See
"Purpose", above.) Real eigenvectors take one column, complex
take two columns, the first for the real part and the second for
the imaginary part. Complex eigenvectors correspond to an
eigenvalue with positive imaginary part. Each eigenvector will
be scaled so the largest component will have abs(real part) +
abs(imag. part) = 1, *except* that for eigenvalues with
alpha=beta=0, a zero vector will be returned as the corresponding
eigenvector. 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 JOBVL = 'V', the right generalized eigenvectors. (See
"Purpose", above.) Real eigenvectors take one column, complex
take two columns, the first for the real part and the second for
the imaginary part. Complex eigenvectors correspond to an
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SGEGV(3F) SGEGV(3F)
eigenvalue with positive imaginary part. Each eigenvector will
be scaled so the largest component will have abs(real part) +
abs(imag. part) = 1, *except* that for eigenvalues with
alpha=beta=0, a zero vector will be returned as the corresponding
eigenvector. Not referenced if JOBVR = 'N'.
LDVR (input) INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR =
'V', LDVR >= N.
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. LWORK >= max(1,8*N). For good
performance, LWORK must generally be larger. To compute the
optimal value of LWORK, call ILAENV to get blocksizes (for
SGEQRF, SORMQR, and SORGQR.) Then compute: NB -- MAX of the
blocksizes for SGEQRF, SORMQR, and SORGQR; The optimal LWORK is:
2*N + MAX( 6*N, N*(NB+1) ).
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: errors that usually indicate
LAPACK problems:
=N+1: error return from SGGBAL
=N+2: error return from SGEQRF
=N+3: error return from SORMQR
=N+4: error return from SORGQR
=N+5: error return from SGGHRD
=N+6: error return from SHGEQZ (other than failed iteration)
=N+7: error return from STGEVC
=N+8: error return from SGGBAK (computing VL)
=N+9: error return from SGGBAK (computing VR)
=N+10: error return from SLASCL (various calls)
FURTHER DETAILS
Balancing
---------
This driver calls SGGBAL to both permute and scale rows and columns of A
and B. The permutations PL and PR are chosen so that PL*A*PR and PL*B*R
will be upper triangular except for the diagonal blocks A(i:j,i:j) and
B(i:j,i:j), with i and j as close together as possible. The diagonal
scaling matrices DL and DR are chosen so that the pair DL*PL*A*PR*DR,
DL*PL*B*PR*DR have elements close to one (except for the elements that
start out zero.)
After the eigenvalues and eigenvectors of the balanced matrices have been
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SGEGV(3F) SGEGV(3F)
computed, SGGBAK transforms the eigenvectors back to what they would have
been (in perfect arithmetic) if they had not been balanced.
Contents of A and B on Exit
-------- -- - --- - -- ----
If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or both),
then on exit the arrays A and B will contain the real Schur form[*] of
the "balanced" versions of A and B. If no eigenvectors are computed,
then only the diagonal blocks will be correct.
[*] See SHGEQZ, SGEGS, or read the book "Matrix Computations",
by Golub & van Loan, pub. by Johns Hopkins U. Press.
SGEGV(3F) SGEGV(3F)
SGEGV - compute for a pair of n-by-n real nonsymmetric matrices A and B,
the generalized eigenvalues (alphar +/- alphai*i, beta), and optionally,
the left and/or right generalized eigenvectors (VL and VR)
SUBROUTINE SGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI, BETA,
VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
BETA( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )
SGEGV computes for a pair of n-by-n real nonsymmetric matrices A and B,
the generalized eigenvalues (alphar +/- alphai*i, beta), and optionally,
the left and/or right generalized eigenvectors (VL and VR).
A generalized eigenvalue for a pair of matrices (A,B) is, roughly
speaking, a scalar w or a ratio alpha/beta = w, such that A - w*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.
A good beginning reference is the book, "Matrix Computations", by G.
Golub & C. van Loan (Johns Hopkins U. Press)
A right generalized eigenvector corresponding to a generalized eigenvalue
w for a pair of matrices (A,B) is a vector r such that (A - w B) r =
0 . A left generalized eigenvector is a vector l such that l**H * (A - w
B) = 0, where l**H is the
conjugate-transpose of l.
Note: this routine performs "full balancing" on A and B -- see "Further
Details", below.
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.
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 first of the pair of matrices whose generalized
eigenvalues and (optionally) generalized eigenvectors are to be
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SGEGV(3F) SGEGV(3F)
computed. On exit, the contents will have been destroyed. (For
a description of the contents of A on exit, see "Further
Details", below.)
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
B (input/output) REAL array, dimension (LDB, N)
On entry, the second of the pair of matrices whose generalized
eigenvalues and (optionally) generalized eigenvectors are to be
computed. On exit, the contents will have been destroyed. (For
a description of the contents of B on exit, see "Further
Details", below.)
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 eigenvalues. If ALPHAI(j) is
zero, then the j-th eigenvalue is real; if positive, then the jth
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) REAL array, dimension (LDVL,N)
If JOBVL = 'V', the left generalized eigenvectors. (See
"Purpose", above.) Real eigenvectors take one column, complex
take two columns, the first for the real part and the second for
the imaginary part. Complex eigenvectors correspond to an
eigenvalue with positive imaginary part. Each eigenvector will
be scaled so the largest component will have abs(real part) +
abs(imag. part) = 1, *except* that for eigenvalues with
alpha=beta=0, a zero vector will be returned as the corresponding
eigenvector. 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 JOBVL = 'V', the right generalized eigenvectors. (See
"Purpose", above.) Real eigenvectors take one column, complex
take two columns, the first for the real part and the second for
the imaginary part. Complex eigenvectors correspond to an
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SGEGV(3F) SGEGV(3F)
eigenvalue with positive imaginary part. Each eigenvector will
be scaled so the largest component will have abs(real part) +
abs(imag. part) = 1, *except* that for eigenvalues with
alpha=beta=0, a zero vector will be returned as the corresponding
eigenvector. Not referenced if JOBVR = 'N'.
LDVR (input) INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR =
'V', LDVR >= N.
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. LWORK >= max(1,8*N). For good
performance, LWORK must generally be larger. To compute the
optimal value of LWORK, call ILAENV to get blocksizes (for
SGEQRF, SORMQR, and SORGQR.) Then compute: NB -- MAX of the
blocksizes for SGEQRF, SORMQR, and SORGQR; The optimal LWORK is:
2*N + MAX( 6*N, N*(NB+1) ).
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: errors that usually indicate
LAPACK problems:
=N+1: error return from SGGBAL
=N+2: error return from SGEQRF
=N+3: error return from SORMQR
=N+4: error return from SORGQR
=N+5: error return from SGGHRD
=N+6: error return from SHGEQZ (other than failed iteration)
=N+7: error return from STGEVC
=N+8: error return from SGGBAK (computing VL)
=N+9: error return from SGGBAK (computing VR)
=N+10: error return from SLASCL (various calls)
FURTHER DETAILS
Balancing
---------
This driver calls SGGBAL to both permute and scale rows and columns of A
and B. The permutations PL and PR are chosen so that PL*A*PR and PL*B*R
will be upper triangular except for the diagonal blocks A(i:j,i:j) and
B(i:j,i:j), with i and j as close together as possible. The diagonal
scaling matrices DL and DR are chosen so that the pair DL*PL*A*PR*DR,
DL*PL*B*PR*DR have elements close to one (except for the elements that
start out zero.)
After the eigenvalues and eigenvectors of the balanced matrices have been
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computed, SGGBAK transforms the eigenvectors back to what they would have
been (in perfect arithmetic) if they had not been balanced.
Contents of A and B on Exit
-------- -- - --- - -- ----
If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or both),
then on exit the arrays A and B will contain the real Schur form[*] of
the "balanced" versions of A and B. If no eigenvectors are computed,
then only the diagonal blocks will be correct.
[*] See SHGEQZ, SGEGS, or read the book "Matrix Computations",
by Golub & van Loan, pub. by Johns Hopkins U. Press.
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