.TH "tgevc" 3 "Tue Jan 28 2025 00:54:31" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
tgevc \- tgevc: eigvec of pair of matrices
.SH SYNOPSIS
.br
.PP
.SS "Functions"
.in +1c
.ti -1c
.RI "subroutine \fBctgevc\fP (side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, rwork, info)"
.br
.RI "\fBCTGEVC\fP "
.ti -1c
.RI "subroutine \fBdtgevc\fP (side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, info)"
.br
.RI "\fBDTGEVC\fP "
.ti -1c
.RI "subroutine \fBstgevc\fP (side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, info)"
.br
.RI "\fBSTGEVC\fP "
.ti -1c
.RI "subroutine \fBztgevc\fP (side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, rwork, info)"
.br
.RI "\fBZTGEVC\fP "
.in -1c
.SH "Detailed Description"
.PP
.SH "Function Documentation"
.PP
.SS "subroutine ctgevc (character side, character howmny, logical, dimension( * ) select, integer n, complex, dimension( lds, * ) s, integer lds, complex, dimension( ldp, * ) p, integer ldp, complex, dimension( ldvl, * ) vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)"
.PP
\fBCTGEVC\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
CTGEVC computes some or all of the right and/or left eigenvectors of
a pair of complex matrices (S,P), where S and P are upper triangular\&.
Matrix pairs of this type are produced by the generalized Schur
factorization of a complex matrix pair (A,B):
A = Q*S*Z**H, B = Q*P*Z**H
as computed by CGGHRD + CHGEQZ\&.
The right eigenvector x and the left eigenvector y of (S,P)
corresponding to an eigenvalue w are defined by:
S*x = w*P*x, (y**H)*S = w*(y**H)*P,
where y**H denotes the conjugate transpose of y\&.
The eigenvalues are not input to this routine, but are computed
directly from the diagonal elements of S and P\&.
This routine returns the matrices X and/or Y of right and left
eigenvectors of (S,P), or the products Z*X and/or Q*Y,
where Z and Q are input matrices\&.
If Q and Z are the unitary factors from the generalized Schur
factorization of a matrix pair (A,B), then Z*X and Q*Y
are the matrices of right and left eigenvectors of (A,B)\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fISIDE\fP
.PP
.nf
SIDE is CHARACTER*1
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors\&.
.fi
.PP
.br
\fIHOWMNY\fP
.PP
.nf
HOWMNY is CHARACTER*1
= 'A': compute all right and/or left eigenvectors;
= 'B': compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= 'S': compute selected right and/or left eigenvectors,
specified by the logical array SELECT\&.
.fi
.PP
.br
\fISELECT\fP
.PP
.nf
SELECT is LOGICAL array, dimension (N)
If HOWMNY='S', SELECT specifies the eigenvectors to be
computed\&. The eigenvector corresponding to the j-th
eigenvalue is computed if SELECT(j) = \&.TRUE\&.\&.
Not referenced if HOWMNY = 'A' or 'B'\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
The order of the matrices S and P\&. N >= 0\&.
.fi
.PP
.br
\fIS\fP
.PP
.nf
S is COMPLEX array, dimension (LDS,N)
The upper triangular matrix S from a generalized Schur
factorization, as computed by CHGEQZ\&.
.fi
.PP
.br
\fILDS\fP
.PP
.nf
LDS is INTEGER
The leading dimension of array S\&. LDS >= max(1,N)\&.
.fi
.PP
.br
\fIP\fP
.PP
.nf
P is COMPLEX array, dimension (LDP,N)
The upper triangular matrix P from a generalized Schur
factorization, as computed by CHGEQZ\&. P must have real
diagonal elements\&.
.fi
.PP
.br
\fILDP\fP
.PP
.nf
LDP is INTEGER
The leading dimension of array P\&. LDP >= max(1,N)\&.
.fi
.PP
.br
\fIVL\fP
.PP
.nf
VL is COMPLEX array, dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
contain an N-by-N matrix Q (usually the unitary matrix Q
of left Schur vectors returned by CHGEQZ)\&.
On exit, if SIDE = 'L' or 'B', VL contains:
if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
if HOWMNY = 'B', the matrix Q*Y;
if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
SELECT, stored consecutively in the columns of
VL, in the same order as their eigenvalues\&.
Not referenced if SIDE = 'R'\&.
.fi
.PP
.br
\fILDVL\fP
.PP
.nf
LDVL is INTEGER
The leading dimension of array VL\&. LDVL >= 1, and if
SIDE = 'L' or 'l' or 'B' or 'b', LDVL >= N\&.
.fi
.PP
.br
\fIVR\fP
.PP
.nf
VR is COMPLEX array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
contain an N-by-N matrix Z (usually the unitary matrix Z
of right Schur vectors returned by CHGEQZ)\&.
On exit, if SIDE = 'R' or 'B', VR contains:
if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
if HOWMNY = 'B', the matrix Z*X;
if HOWMNY = 'S', the right eigenvectors of (S,P) specified by
SELECT, stored consecutively in the columns of
VR, in the same order as their eigenvalues\&.
Not referenced if SIDE = 'L'\&.
.fi
.PP
.br
\fILDVR\fP
.PP
.nf
LDVR is INTEGER
The leading dimension of the array VR\&. LDVR >= 1, and if
SIDE = 'R' or 'B', LDVR >= N\&.
.fi
.PP
.br
\fIMM\fP
.PP
.nf
MM is INTEGER
The number of columns in the arrays VL and/or VR\&. MM >= M\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors\&. If HOWMNY = 'A' or 'B', M
is set to N\&. Each selected eigenvector occupies one column\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is COMPLEX array, dimension (2*N)
.fi
.PP
.br
\fIRWORK\fP
.PP
.nf
RWORK is REAL array, dimension (2*N)
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
INFO is INTEGER
= 0: successful exit\&.
< 0: if INFO = -i, the i-th argument had an illegal value\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
.SS "subroutine dtgevc (character side, character howmny, logical, dimension( * ) select, integer n, double precision, dimension( lds, * ) s, integer lds, double precision, dimension( ldp, * ) p, integer ldp, double precision, dimension( ldvl, * ) vl, integer ldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, double precision, dimension( * ) work, integer info)"
.PP
\fBDTGEVC\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
DTGEVC computes some or all of the right and/or left eigenvectors of
a pair of real matrices (S,P), where S is a quasi-triangular matrix
and P is upper triangular\&. Matrix pairs of this type are produced by
the generalized Schur factorization of a matrix pair (A,B):
A = Q*S*Z**T, B = Q*P*Z**T
as computed by DGGHRD + DHGEQZ\&.
The right eigenvector x and the left eigenvector y of (S,P)
corresponding to an eigenvalue w are defined by:
S*x = w*P*x, (y**H)*S = w*(y**H)*P,
where y**H denotes the conjugate transpose of y\&.
The eigenvalues are not input to this routine, but are computed
directly from the diagonal blocks of S and P\&.
This routine returns the matrices X and/or Y of right and left
eigenvectors of (S,P), or the products Z*X and/or Q*Y,
where Z and Q are input matrices\&.
If Q and Z are the orthogonal factors from the generalized Schur
factorization of a matrix pair (A,B), then Z*X and Q*Y
are the matrices of right and left eigenvectors of (A,B)\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fISIDE\fP
.PP
.nf
SIDE is CHARACTER*1
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors\&.
.fi
.PP
.br
\fIHOWMNY\fP
.PP
.nf
HOWMNY is CHARACTER*1
= 'A': compute all right and/or left eigenvectors;
= 'B': compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= 'S': compute selected right and/or left eigenvectors,
specified by the logical array SELECT\&.
.fi
.PP
.br
\fISELECT\fP
.PP
.nf
SELECT is LOGICAL array, dimension (N)
If HOWMNY='S', SELECT specifies the eigenvectors to be
computed\&. If w(j) is a real eigenvalue, the corresponding
real eigenvector is computed if SELECT(j) is \&.TRUE\&.\&.
If w(j) and w(j+1) are the real and imaginary parts of a
complex eigenvalue, the corresponding complex eigenvector
is computed if either SELECT(j) or SELECT(j+1) is \&.TRUE\&.,
and on exit SELECT(j) is set to \&.TRUE\&. and SELECT(j+1) is
set to \&.FALSE\&.\&.
Not referenced if HOWMNY = 'A' or 'B'\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
The order of the matrices S and P\&. N >= 0\&.
.fi
.PP
.br
\fIS\fP
.PP
.nf
S is DOUBLE PRECISION array, dimension (LDS,N)
The upper quasi-triangular matrix S from a generalized Schur
factorization, as computed by DHGEQZ\&.
.fi
.PP
.br
\fILDS\fP
.PP
.nf
LDS is INTEGER
The leading dimension of array S\&. LDS >= max(1,N)\&.
.fi
.PP
.br
\fIP\fP
.PP
.nf
P is DOUBLE PRECISION array, dimension (LDP,N)
The upper triangular matrix P from a generalized Schur
factorization, as computed by DHGEQZ\&.
2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
of S must be in positive diagonal form\&.
.fi
.PP
.br
\fILDP\fP
.PP
.nf
LDP is INTEGER
The leading dimension of array P\&. LDP >= max(1,N)\&.
.fi
.PP
.br
\fIVL\fP
.PP
.nf
VL is DOUBLE PRECISION array, dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of left Schur vectors returned by DHGEQZ)\&.
On exit, if SIDE = 'L' or 'B', VL contains:
if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
if HOWMNY = 'B', the matrix Q*Y;
if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
SELECT, stored consecutively in the columns of
VL, in the same order as their eigenvalues\&.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part, and the second the imaginary part\&.
Not referenced if SIDE = 'R'\&.
.fi
.PP
.br
\fILDVL\fP
.PP
.nf
LDVL is INTEGER
The leading dimension of array VL\&. LDVL >= 1, and if
SIDE = 'L' or 'B', LDVL >= N\&.
.fi
.PP
.br
\fIVR\fP
.PP
.nf
VR is DOUBLE PRECISION array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
contain an N-by-N matrix Z (usually the orthogonal matrix Z
of right Schur vectors returned by DHGEQZ)\&.
On exit, if SIDE = 'R' or 'B', VR contains:
if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
if HOWMNY = 'B' or 'b', the matrix Z*X;
if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
specified by SELECT, stored consecutively in the
columns of VR, in the same order as their
eigenvalues\&.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part and the second the imaginary part\&.
Not referenced if SIDE = 'L'\&.
.fi
.PP
.br
\fILDVR\fP
.PP
.nf
LDVR is INTEGER
The leading dimension of the array VR\&. LDVR >= 1, and if
SIDE = 'R' or 'B', LDVR >= N\&.
.fi
.PP
.br
\fIMM\fP
.PP
.nf
MM is INTEGER
The number of columns in the arrays VL and/or VR\&. MM >= M\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors\&. If HOWMNY = 'A' or 'B', M
is set to N\&. Each selected real eigenvector occupies one
column and each selected complex eigenvector occupies two
columns\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is DOUBLE PRECISION array, dimension (6*N)
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
INFO is INTEGER
= 0: successful exit\&.
< 0: if INFO = -i, the i-th argument had an illegal value\&.
> 0: the 2-by-2 block (INFO:INFO+1) does not have a complex
eigenvalue\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
Allocation of workspace:
---------- -- ---------
WORK( j ) = 1-norm of j-th column of A, above the diagonal
WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
WORK( 2*N+1:3*N ) = real part of eigenvector
WORK( 3*N+1:4*N ) = imaginary part of eigenvector
WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector
Rowwise vs\&. columnwise solution methods:
------- -- ---------- -------- -------
Finding a generalized eigenvector consists basically of solving the
singular triangular system
(A - w B) x = 0 (for right) or: (A - w B)**H y = 0 (for left)
Consider finding the i-th right eigenvector (assume all eigenvalues
are real)\&. The equation to be solved is:
n i
0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,\&. \&. \&.,1
k=j k=j
where C = (A - w B) (The components v(i+1:n) are 0\&.)
The 'rowwise' method is:
(1) v(i) := 1
for j = i-1,\&. \&. \&.,1:
i
(2) compute s = - sum C(j,k) v(k) and
k=j+1
(3) v(j) := s / C(j,j)
Step 2 is sometimes called the 'dot product' step, since it is an
inner product between the j-th row and the portion of the eigenvector
that has been computed so far\&.
The 'columnwise' method consists basically in doing the sums
for all the rows in parallel\&. As each v(j) is computed, the
contribution of v(j) times the j-th column of C is added to the
partial sums\&. Since FORTRAN arrays are stored columnwise, this has
the advantage that at each step, the elements of C that are accessed
are adjacent to one another, whereas with the rowwise method, the
elements accessed at a step are spaced LDS (and LDP) words apart\&.
When finding left eigenvectors, the matrix in question is the
transpose of the one in storage, so the rowwise method then
actually accesses columns of A and B at each step, and so is the
preferred method\&.
.fi
.PP
.RE
.PP
.SS "subroutine stgevc (character side, character howmny, logical, dimension( * ) select, integer n, real, dimension( lds, * ) s, integer lds, real, dimension( ldp, * ) p, integer ldp, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, real, dimension( * ) work, integer info)"
.PP
\fBSTGEVC\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
STGEVC computes some or all of the right and/or left eigenvectors of
a pair of real matrices (S,P), where S is a quasi-triangular matrix
and P is upper triangular\&. Matrix pairs of this type are produced by
the generalized Schur factorization of a matrix pair (A,B):
A = Q*S*Z**T, B = Q*P*Z**T
as computed by SGGHRD + SHGEQZ\&.
The right eigenvector x and the left eigenvector y of (S,P)
corresponding to an eigenvalue w are defined by:
S*x = w*P*x, (y**H)*S = w*(y**H)*P,
where y**H denotes the conjugate transpose of y\&.
The eigenvalues are not input to this routine, but are computed
directly from the diagonal blocks of S and P\&.
This routine returns the matrices X and/or Y of right and left
eigenvectors of (S,P), or the products Z*X and/or Q*Y,
where Z and Q are input matrices\&.
If Q and Z are the orthogonal factors from the generalized Schur
factorization of a matrix pair (A,B), then Z*X and Q*Y
are the matrices of right and left eigenvectors of (A,B)\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fISIDE\fP
.PP
.nf
SIDE is CHARACTER*1
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors\&.
.fi
.PP
.br
\fIHOWMNY\fP
.PP
.nf
HOWMNY is CHARACTER*1
= 'A': compute all right and/or left eigenvectors;
= 'B': compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= 'S': compute selected right and/or left eigenvectors,
specified by the logical array SELECT\&.
.fi
.PP
.br
\fISELECT\fP
.PP
.nf
SELECT is LOGICAL array, dimension (N)
If HOWMNY='S', SELECT specifies the eigenvectors to be
computed\&. If w(j) is a real eigenvalue, the corresponding
real eigenvector is computed if SELECT(j) is \&.TRUE\&.\&.
If w(j) and w(j+1) are the real and imaginary parts of a
complex eigenvalue, the corresponding complex eigenvector
is computed if either SELECT(j) or SELECT(j+1) is \&.TRUE\&.,
and on exit SELECT(j) is set to \&.TRUE\&. and SELECT(j+1) is
set to \&.FALSE\&.\&.
Not referenced if HOWMNY = 'A' or 'B'\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
The order of the matrices S and P\&. N >= 0\&.
.fi
.PP
.br
\fIS\fP
.PP
.nf
S is REAL array, dimension (LDS,N)
The upper quasi-triangular matrix S from a generalized Schur
factorization, as computed by SHGEQZ\&.
.fi
.PP
.br
\fILDS\fP
.PP
.nf
LDS is INTEGER
The leading dimension of array S\&. LDS >= max(1,N)\&.
.fi
.PP
.br
\fIP\fP
.PP
.nf
P is REAL array, dimension (LDP,N)
The upper triangular matrix P from a generalized Schur
factorization, as computed by SHGEQZ\&.
2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
of S must be in positive diagonal form\&.
.fi
.PP
.br
\fILDP\fP
.PP
.nf
LDP is INTEGER
The leading dimension of array P\&. LDP >= max(1,N)\&.
.fi
.PP
.br
\fIVL\fP
.PP
.nf
VL is REAL array, dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of left Schur vectors returned by SHGEQZ)\&.
On exit, if SIDE = 'L' or 'B', VL contains:
if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
if HOWMNY = 'B', the matrix Q*Y;
if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
SELECT, stored consecutively in the columns of
VL, in the same order as their eigenvalues\&.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part, and the second the imaginary part\&.
Not referenced if SIDE = 'R'\&.
.fi
.PP
.br
\fILDVL\fP
.PP
.nf
LDVL is INTEGER
The leading dimension of array VL\&. LDVL >= 1, and if
SIDE = 'L' or 'B', LDVL >= N\&.
.fi
.PP
.br
\fIVR\fP
.PP
.nf
VR is REAL array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
contain an N-by-N matrix Z (usually the orthogonal matrix Z
of right Schur vectors returned by SHGEQZ)\&.
On exit, if SIDE = 'R' or 'B', VR contains:
if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
if HOWMNY = 'B' or 'b', the matrix Z*X;
if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
specified by SELECT, stored consecutively in the
columns of VR, in the same order as their
eigenvalues\&.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part and the second the imaginary part\&.
Not referenced if SIDE = 'L'\&.
.fi
.PP
.br
\fILDVR\fP
.PP
.nf
LDVR is INTEGER
The leading dimension of the array VR\&. LDVR >= 1, and if
SIDE = 'R' or 'B', LDVR >= N\&.
.fi
.PP
.br
\fIMM\fP
.PP
.nf
MM is INTEGER
The number of columns in the arrays VL and/or VR\&. MM >= M\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors\&. If HOWMNY = 'A' or 'B', M
is set to N\&. Each selected real eigenvector occupies one
column and each selected complex eigenvector occupies two
columns\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is REAL array, dimension (6*N)
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
INFO is INTEGER
= 0: successful exit\&.
< 0: if INFO = -i, the i-th argument had an illegal value\&.
> 0: the 2-by-2 block (INFO:INFO+1) does not have a complex
eigenvalue\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
\fBFurther Details:\fP
.RS 4
.PP
.nf
Allocation of workspace:
---------- -- ---------
WORK( j ) = 1-norm of j-th column of A, above the diagonal
WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
WORK( 2*N+1:3*N ) = real part of eigenvector
WORK( 3*N+1:4*N ) = imaginary part of eigenvector
WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector
Rowwise vs\&. columnwise solution methods:
------- -- ---------- -------- -------
Finding a generalized eigenvector consists basically of solving the
singular triangular system
(A - w B) x = 0 (for right) or: (A - w B)**H y = 0 (for left)
Consider finding the i-th right eigenvector (assume all eigenvalues
are real)\&. The equation to be solved is:
n i
0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,\&. \&. \&.,1
k=j k=j
where C = (A - w B) (The components v(i+1:n) are 0\&.)
The 'rowwise' method is:
(1) v(i) := 1
for j = i-1,\&. \&. \&.,1:
i
(2) compute s = - sum C(j,k) v(k) and
k=j+1
(3) v(j) := s / C(j,j)
Step 2 is sometimes called the 'dot product' step, since it is an
inner product between the j-th row and the portion of the eigenvector
that has been computed so far\&.
The 'columnwise' method consists basically in doing the sums
for all the rows in parallel\&. As each v(j) is computed, the
contribution of v(j) times the j-th column of C is added to the
partial sums\&. Since FORTRAN arrays are stored columnwise, this has
the advantage that at each step, the elements of C that are accessed
are adjacent to one another, whereas with the rowwise method, the
elements accessed at a step are spaced LDS (and LDP) words apart\&.
When finding left eigenvectors, the matrix in question is the
transpose of the one in storage, so the rowwise method then
actually accesses columns of A and B at each step, and so is the
preferred method\&.
.fi
.PP
.RE
.PP
.SS "subroutine ztgevc (character side, character howmny, logical, dimension( * ) select, integer n, complex*16, dimension( lds, * ) s, integer lds, complex*16, dimension( ldp, * ) p, integer ldp, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)"
.PP
\fBZTGEVC\fP
.PP
\fBPurpose:\fP
.RS 4
.PP
.nf
ZTGEVC computes some or all of the right and/or left eigenvectors of
a pair of complex matrices (S,P), where S and P are upper triangular\&.
Matrix pairs of this type are produced by the generalized Schur
factorization of a complex matrix pair (A,B):
A = Q*S*Z**H, B = Q*P*Z**H
as computed by ZGGHRD + ZHGEQZ\&.
The right eigenvector x and the left eigenvector y of (S,P)
corresponding to an eigenvalue w are defined by:
S*x = w*P*x, (y**H)*S = w*(y**H)*P,
where y**H denotes the conjugate transpose of y\&.
The eigenvalues are not input to this routine, but are computed
directly from the diagonal elements of S and P\&.
This routine returns the matrices X and/or Y of right and left
eigenvectors of (S,P), or the products Z*X and/or Q*Y,
where Z and Q are input matrices\&.
If Q and Z are the unitary factors from the generalized Schur
factorization of a matrix pair (A,B), then Z*X and Q*Y
are the matrices of right and left eigenvectors of (A,B)\&.
.fi
.PP
.RE
.PP
\fBParameters\fP
.RS 4
\fISIDE\fP
.PP
.nf
SIDE is CHARACTER*1
= 'R': compute right eigenvectors only;
= 'L': compute left eigenvectors only;
= 'B': compute both right and left eigenvectors\&.
.fi
.PP
.br
\fIHOWMNY\fP
.PP
.nf
HOWMNY is CHARACTER*1
= 'A': compute all right and/or left eigenvectors;
= 'B': compute all right and/or left eigenvectors,
backtransformed by the matrices in VR and/or VL;
= 'S': compute selected right and/or left eigenvectors,
specified by the logical array SELECT\&.
.fi
.PP
.br
\fISELECT\fP
.PP
.nf
SELECT is LOGICAL array, dimension (N)
If HOWMNY='S', SELECT specifies the eigenvectors to be
computed\&. The eigenvector corresponding to the j-th
eigenvalue is computed if SELECT(j) = \&.TRUE\&.\&.
Not referenced if HOWMNY = 'A' or 'B'\&.
.fi
.PP
.br
\fIN\fP
.PP
.nf
N is INTEGER
The order of the matrices S and P\&. N >= 0\&.
.fi
.PP
.br
\fIS\fP
.PP
.nf
S is COMPLEX*16 array, dimension (LDS,N)
The upper triangular matrix S from a generalized Schur
factorization, as computed by ZHGEQZ\&.
.fi
.PP
.br
\fILDS\fP
.PP
.nf
LDS is INTEGER
The leading dimension of array S\&. LDS >= max(1,N)\&.
.fi
.PP
.br
\fIP\fP
.PP
.nf
P is COMPLEX*16 array, dimension (LDP,N)
The upper triangular matrix P from a generalized Schur
factorization, as computed by ZHGEQZ\&. P must have real
diagonal elements\&.
.fi
.PP
.br
\fILDP\fP
.PP
.nf
LDP is INTEGER
The leading dimension of array P\&. LDP >= max(1,N)\&.
.fi
.PP
.br
\fIVL\fP
.PP
.nf
VL is COMPLEX*16 array, dimension (LDVL,MM)
On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
contain an N-by-N matrix Q (usually the unitary matrix Q
of left Schur vectors returned by ZHGEQZ)\&.
On exit, if SIDE = 'L' or 'B', VL contains:
if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
if HOWMNY = 'B', the matrix Q*Y;
if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
SELECT, stored consecutively in the columns of
VL, in the same order as their eigenvalues\&.
Not referenced if SIDE = 'R'\&.
.fi
.PP
.br
\fILDVL\fP
.PP
.nf
LDVL is INTEGER
The leading dimension of array VL\&. LDVL >= 1, and if
SIDE = 'L' or 'l' or 'B' or 'b', LDVL >= N\&.
.fi
.PP
.br
\fIVR\fP
.PP
.nf
VR is COMPLEX*16 array, dimension (LDVR,MM)
On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
contain an N-by-N matrix Z (usually the unitary matrix Z
of right Schur vectors returned by ZHGEQZ)\&.
On exit, if SIDE = 'R' or 'B', VR contains:
if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
if HOWMNY = 'B', the matrix Z*X;
if HOWMNY = 'S', the right eigenvectors of (S,P) specified by
SELECT, stored consecutively in the columns of
VR, in the same order as their eigenvalues\&.
Not referenced if SIDE = 'L'\&.
.fi
.PP
.br
\fILDVR\fP
.PP
.nf
LDVR is INTEGER
The leading dimension of the array VR\&. LDVR >= 1, and if
SIDE = 'R' or 'B', LDVR >= N\&.
.fi
.PP
.br
\fIMM\fP
.PP
.nf
MM is INTEGER
The number of columns in the arrays VL and/or VR\&. MM >= M\&.
.fi
.PP
.br
\fIM\fP
.PP
.nf
M is INTEGER
The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors\&. If HOWMNY = 'A' or 'B', M
is set to N\&. Each selected eigenvector occupies one column\&.
.fi
.PP
.br
\fIWORK\fP
.PP
.nf
WORK is COMPLEX*16 array, dimension (2*N)
.fi
.PP
.br
\fIRWORK\fP
.PP
.nf
RWORK is DOUBLE PRECISION array, dimension (2*N)
.fi
.PP
.br
\fIINFO\fP
.PP
.nf
INFO is INTEGER
= 0: successful exit\&.
< 0: if INFO = -i, the i-th argument had an illegal value\&.
.fi
.PP
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee
.PP
Univ\&. of California Berkeley
.PP
Univ\&. of Colorado Denver
.PP
NAG Ltd\&.
.RE
.PP
.SH "Author"
.PP
Generated automatically by Doxygen for LAPACK from the source code\&.