1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
|
.TH SGEES l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
SGEES - 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
.SH SYNOPSIS
.TP 18
SUBROUTINE SGEES(
JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI,
VS, LDVS, WORK, LWORK, BWORK, INFO )
.TP 18
.ti +4
CHARACTER
JOBVS, SORT
.TP 18
.ti +4
INTEGER
INFO, LDA, LDVS, LWORK, N, SDIM
.TP 18
.ti +4
LOGICAL
BWORK( * )
.TP 18
.ti +4
REAL
A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ),
WR( * )
.TP 18
.ti +4
LOGICAL
SELECT
.TP 18
.ti +4
EXTERNAL
SELECT
.SH PURPOSE
SGEES 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.
The leading columns of Z then form an orthonormal basis for the
invariant subspace corresponding to the selected eigenvalues.
A 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
.br
[ a b ]
.br
[ c a ]
.br
where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
.SH ARGUMENTS
.TP 8
JOBVS (input) CHARACTER*1
= 'N': Schur vectors are not computed;
.br
= 'V': Schur vectors are computed.
.TP 8
SORT (input) CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the Schur form.
= 'N': Eigenvalues are not ordered;
.br
= 'S': Eigenvalues are ordered (see SELECT).
.TP 8
SELECT (input) LOGICAL FUNCTION of two REAL 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 complex
eigenvalues are selected.
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 is set to N+2 (see INFO below).
.TP 8
N (input) INTEGER
The order of the matrix A. N >= 0.
.TP 8
A (input/output) REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten by its real Schur form T.
.TP 8
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
.TP 8
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.)
.TP 8
WR (output) REAL array, dimension (N)
WI (output) REAL 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 eigenvalues will appear
consecutively with the eigenvalue having the positive
imaginary part first.
.TP 8
VS (output) REAL array, dimension (LDVS,N)
If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
vectors.
If JOBVS = 'N', VS is not referenced.
.TP 8
LDVS (input) INTEGER
The leading dimension of the array VS. LDVS >= 1; if
JOBVS = 'V', LDVS >= N.
.TP 8
WORK (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) contains the optimal LWORK.
.TP 8
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= max(1,3*N).
For good performance, LWORK must generally be larger.
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.
.TP 8
BWORK (workspace) LOGICAL array, dimension (N)
Not referenced if SORT = 'N'.
.TP 8
INFO (output) INTEGER
= 0: successful exit
.br
< 0: if INFO = -i, the i-th argument had an illegal value.
.br
> 0: if INFO = i, and i is
.br
<= N: the QR algorithm failed to compute all the
.br
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 matrix 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, 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.
|
