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man pages->IRIX man pages -> complib/dgeesx (3)
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### Contents

```
DGEESX(3F)							    DGEESX(3F)

```

### NAME[Toc][Back]

```     DGEESX - compute for an N-by-N real nonsymmetric matrix A,	the
eigenvalues, the real Schur form T, and, optionally, the matrix of	Schur
vectors Z
```

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	DGEESX(	JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, WR, WI,
VS, LDVS, RCONDE, RCONDV, WORK,	LWORK, IWORK, LIWORK,
BWORK, INFO )

CHARACTER	JOBVS, SENSE, SORT

INTEGER	INFO, LDA, LDVS, LIWORK, LWORK,	N, SDIM

DOUBLE		PRECISION RCONDE, RCONDV

LOGICAL	BWORK( * )

INTEGER	IWORK( * )

DOUBLE		PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( *
), WR( * )

LOGICAL	SELECT

EXTERNAL	SELECT
```

### PURPOSE[Toc][Back]

```     DGEESX computes for an N-by-N real	nonsymmetric matrix A, the
eigenvalues, the real Schur form T, and, optionally, the matrix of	Schur
vectors Z.	 This gives the	Schur factorization A =	Z*T*(Z**T).

Optionally, it also orders	the eigenvalues	on the diagonal	of the real
Schur form	so that	selected eigenvalues are at the	top left; computes a
reciprocal	condition number for the average of the	selected eigenvalues
(RCONDE); and computes a reciprocal condition number for the right
invariant subspace	corresponding to the selected eigenvalues (RCONDV).
The leading columns of Z form an orthonormal basis	for this invariant
subspace.

For further explanation of	the reciprocal condition numbers RCONDE	and
RCONDV, see Section 4.10 of the LAPACK Users' Guide (where	these
quantities	are called s and sep respectively).

A real matrix is in real Schur form if it is upper	quasi-triangular with
1-by-1 and	2-by-2 blocks. 2-by-2 blocks will be standardized in the form
[  a  b	]
[  c  a	]

where b*c < 0. The	eigenvalues of such a block are	a +- sqrt(bc).

Page 1

DGEESX(3F)							    DGEESX(3F)

```

### ARGUMENTS[Toc][Back]

```     JOBVS   (input) CHARACTER*1
= 'N': Schur vectors are not computed;
= 'V': Schur vectors are computed.

SORT    (input) CHARACTER*1
Specifies whether or not to order the eigenvalues on the diagonal
of	the Schur form.	 = 'N':	Eigenvalues are	not ordered;
= 'S': Eigenvalues	are ordered (see SELECT).

SELECT  (input) LOGICAL FUNCTION of two DOUBLE PRECISION arguments
SELECT must be declared EXTERNAL in the calling subroutine.  If
SORT = 'S', SELECT	is used	to select eigenvalues to sort to the
top left of the Schur form.  If SORT = 'N', SELECT	is not
referenced.  An eigenvalue	WR(j)+sqrt(-1)*WI(j) is	selected if
SELECT(WR(j),WI(j)) is true; i.e.,	if either one of a complex
conjugate pair of eigenvalues is selected,	then both are.	Note
that a selected complex eigenvalue	may no longer satisfy
SELECT(WR(j),WI(j)) = .TRUE. after	ordering, since	ordering may
change the	value of complex eigenvalues (especially if the
eigenvalue	is ill-conditioned); in	this case INFO may be set to
N+3 (see INFO below).

SENSE   (input) CHARACTER*1
Determines	which reciprocal condition numbers are computed.  =
'N': None are computed;
= 'E': Computed for average of selected eigenvalues only;
= 'V': Computed for selected right	invariant subspace only;
= 'B': Computed for both.	If SENSE = 'E',	'V' or 'B', SORT must
equal 'S'.

N	     (input) INTEGER
The order of the matrix A.	N >= 0.

A	     (input/output) DOUBLE PRECISION array, dimension (LDA, N)
On	entry, the N-by-N matrix A.  On	exit, A	is overwritten by its
real Schur	form T.

LDA     (input) INTEGER
The leading dimension of the array	A.  LDA	>= max(1,N).

SDIM    (output) INTEGER
If	SORT = 'N', SDIM = 0.  If SORT = 'S', SDIM = number of
eigenvalues (after	sorting) for which SELECT is true. (Complex
conjugate pairs for which SELECT is true for either eigenvalue
count as 2.)

WR	     (output) DOUBLE PRECISION array, dimension	(N)
WI	     (output) DOUBLE PRECISION array, dimension	(N) WR and WI
contain the real and imaginary parts, respectively, of the
computed eigenvalues, in the same order that they appear on the
diagonal of the output Schur form T.  Complex conjugate pairs of

Page 2

DGEESX(3F)							    DGEESX(3F)

eigenvalues appear	consecutively with the eigenvalue having the
positive imaginary	part first.

VS	     (output) DOUBLE PRECISION array, dimension	(LDVS,N)
If	JOBVS =	'V', VS	contains the orthogonal	matrix Z of Schur
vectors.  If JOBVS	= 'N', VS is not referenced.

LDVS    (input) INTEGER
The leading dimension of the array	VS.  LDVS >= 1,	and if JOBVS =
'V', LDVS >= N.

RCONDE  (output) DOUBLE PRECISION
If	SENSE =	'E' or 'B', RCONDE contains the	reciprocal condition
number for	the average of the selected eigenvalues.  Not
referenced	if SENSE = 'N' or 'V'.

RCONDV  (output) DOUBLE PRECISION
If	SENSE =	'V' or 'B', RCONDV contains the	reciprocal condition
number for	the selected right invariant subspace.	Not referenced
if	SENSE =	'N' or 'E'.

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(1,3*N).  Also, if
SENSE = 'E' or 'V'	or 'B',	LWORK >= N+2*SDIM*(N-SDIM), where SDIM
is	the number of selected eigenvalues computed by this routine.
Note that N+2*SDIM*(N-SDIM) <= N+N*N/2.  For good performance,
LWORK must	generally be larger.

IWORK   (workspace) INTEGER array,	dimension (LIWORK)
Not referenced if SENSE = 'N' or 'E'.

LIWORK  (input) INTEGER
The dimension of the array	IWORK.	LIWORK >= 1; if	SENSE =	'V' or
'B', LIWORK >= SDIM*(N-SDIM).

BWORK   (workspace) LOGICAL array,	dimension (N)
Not referenced if SORT = 'N'.

INFO    (output) INTEGER
= 0: successful exit
< 0: if INFO = -i,	the i-th argument had an illegal value.
> 0: if INFO = i, and i is
<=	N: the QR algorithm failed to compute all the
eigenvalues; elements 1:ILO-1 and i+1:N of	WR and WI contain
those eigenvalues which have converged; if	JOBVS =	'V', VS
contains the transformation which reduces A to its	partially
converged Schur form.  = N+1: the eigenvalues could not be
reordered because some eigenvalues	were too close to separate
(the problem is very ill-conditioned); = N+2: after reordering,

Page 3

DGEESX(3F)							    DGEESX(3F)

roundoff changed values of	some complex eigenvalues so that
leading eigenvalues in the	Schur form no longer satisfy
SELECT=.TRUE.  This could also be caused by underflow due to
scaling.
DGEESX(3F)							    DGEESX(3F)

```

### NAME[Toc][Back]

```     DGEESX - compute for an N-by-N real nonsymmetric matrix A,	the
eigenvalues, the real Schur form T, and, optionally, the matrix of	Schur
vectors Z
```

### SYNOPSIS[Toc][Back]

```     SUBROUTINE	DGEESX(	JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, WR, WI,
VS, LDVS, RCONDE, RCONDV, WORK,	LWORK, IWORK, LIWORK,
BWORK, INFO )

CHARACTER	JOBVS, SENSE, SORT

INTEGER	INFO, LDA, LDVS, LIWORK, LWORK,	N, SDIM

DOUBLE		PRECISION RCONDE, RCONDV

LOGICAL	BWORK( * )

INTEGER	IWORK( * )

DOUBLE		PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( *
), WR( * )

LOGICAL	SELECT

EXTERNAL	SELECT
```

### PURPOSE[Toc][Back]

```     DGEESX computes for an N-by-N real	nonsymmetric matrix A, the
eigenvalues, the real Schur form T, and, optionally, the matrix of	Schur
vectors Z.	 This gives the	Schur factorization A =	Z*T*(Z**T).

Optionally, it also orders	the eigenvalues	on the diagonal	of the real
Schur form	so that	selected eigenvalues are at the	top left; computes a
reciprocal	condition number for the average of the	selected eigenvalues
(RCONDE); and computes a reciprocal condition number for the right
invariant subspace	corresponding to the selected eigenvalues (RCONDV).
The leading columns of Z form an orthonormal basis	for this invariant
subspace.

For further explanation of	the reciprocal condition numbers RCONDE	and
RCONDV, see Section 4.10 of the LAPACK Users' Guide (where	these
quantities	are called s and sep respectively).

A real matrix is in real Schur form if it is upper	quasi-triangular with
1-by-1 and	2-by-2 blocks. 2-by-2 blocks will be standardized in the form
[  a  b	]
[  c  a	]

where b*c < 0. The	eigenvalues of such a block are	a +- sqrt(bc).

Page 1

DGEESX(3F)							    DGEESX(3F)

```

### ARGUMENTS[Toc][Back]

```     JOBVS   (input) CHARACTER*1
= 'N': Schur vectors are not computed;
= 'V': Schur vectors are computed.

SORT    (input) CHARACTER*1
Specifies whether or not to order the eigenvalues on the diagonal
of	the Schur form.	 = 'N':	Eigenvalues are	not ordered;
= 'S': Eigenvalues	are ordered (see SELECT).

SELECT  (input) LOGICAL FUNCTION of two DOUBLE PRECISION arguments
SELECT must be declared EXTERNAL in the calling subroutine.  If
SORT = 'S', SELECT	is used	to select eigenvalues to sort to the
top left of the Schur form.  If SORT = 'N', SELECT	is not
referenced.  An eigenvalue	WR(j)+sqrt(-1)*WI(j) is	selected if
SELECT(WR(j),WI(j)) is true; i.e.,	if either one of a complex
conjugate pair of eigenvalues is selected,	then both are.	Note
that a selected complex eigenvalue	may no longer satisfy
SELECT(WR(j),WI(j)) = .TRUE. after	ordering, since	ordering may
change the	value of complex eigenvalues (especially if the
eigenvalue	is ill-conditioned); in	this case INFO may be set to
N+3 (see INFO below).

SENSE   (input) CHARACTER*1
Determines	which reciprocal condition numbers are computed.  =
'N': None are computed;
= 'E': Computed for average of selected eigenvalues only;
= 'V': Computed for selected right	invariant subspace only;
= 'B': Computed for both.	If SENSE = 'E',	'V' or 'B', SORT must
equal 'S'.

N	     (input) INTEGER
The order of the matrix A.	N >= 0.

A	     (input/output) DOUBLE PRECISION array, dimension (LDA, N)
On	entry, the N-by-N matrix A.  On	exit, A	is overwritten by its
real Schur	form T.

LDA     (input) INTEGER
The leading dimension of the array	A.  LDA	>= max(1,N).

SDIM    (output) INTEGER
If	SORT = 'N', SDIM = 0.  If SORT = 'S', SDIM = number of
eigenvalues (after	sorting) for which SELECT is true. (Complex
conjugate pairs for which SELECT is true for either eigenvalue
count as 2.)

WR	     (output) DOUBLE PRECISION array, dimension	(N)
WI	     (output) DOUBLE PRECISION array, dimension	(N) WR and WI
contain the real and imaginary parts, respectively, of the
computed eigenvalues, in the same order that they appear on the
diagonal of the output Schur form T.  Complex conjugate pairs of

Page 2

DGEESX(3F)							    DGEESX(3F)

eigenvalues appear	consecutively with the eigenvalue having the
positive imaginary	part first.

VS	     (output) DOUBLE PRECISION array, dimension	(LDVS,N)
If	JOBVS =	'V', VS	contains the orthogonal	matrix Z of Schur
vectors.  If JOBVS	= 'N', VS is not referenced.

LDVS    (input) INTEGER
The leading dimension of the array	VS.  LDVS >= 1,	and if JOBVS =
'V', LDVS >= N.

RCONDE  (output) DOUBLE PRECISION
If	SENSE =	'E' or 'B', RCONDE contains the	reciprocal condition
number for	the average of the selected eigenvalues.  Not
referenced	if SENSE = 'N' or 'V'.

RCONDV  (output) DOUBLE PRECISION
If	SENSE =	'V' or 'B', RCONDV contains the	reciprocal condition
number for	the selected right invariant subspace.	Not referenced
if	SENSE =	'N' or 'E'.

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(1,3*N).  Also, if
SENSE = 'E' or 'V'	or 'B',	LWORK >= N+2*SDIM*(N-SDIM), where SDIM
is	the number of selected eigenvalues computed by this routine.
Note that N+2*SDIM*(N-SDIM) <= N+N*N/2.  For good performance,
LWORK must	generally be larger.

IWORK   (workspace) INTEGER array,	dimension (LIWORK)
Not referenced if SENSE = 'N' or 'E'.

LIWORK  (input) INTEGER
The dimension of the array	IWORK.	LIWORK >= 1; if	SENSE =	'V' or
'B', LIWORK >= SDIM*(N-SDIM).

BWORK   (workspace) LOGICAL array,	dimension (N)
Not referenced if SORT = 'N'.

INFO    (output) INTEGER
= 0: successful exit
< 0: if INFO = -i,	the i-th argument had an illegal value.
> 0: if INFO = i, and i is
<=	N: the QR algorithm failed to compute all the
eigenvalues; elements 1:ILO-1 and i+1:N of	WR and WI contain
those eigenvalues which have converged; if	JOBVS =	'V', VS
contains the transformation which reduces A to its	partially
converged Schur form.  = N+1: the eigenvalues could not be
reordered because some eigenvalues	were too close to separate
(the problem is very ill-conditioned); = N+2: after reordering,

Page 3

DGEESX(3F)							    DGEESX(3F)

roundoff changed values of	some complex eigenvalues so that
leading eigenvalues in the	Schur form no longer satisfy
SELECT=.TRUE.  This could also be caused by underflow due to
scaling.

PPPPaaaaggggeeee 4444```
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