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
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
|
.TH SGGES l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
SGGES - compute for a pair of N-by-N real nonsymmetric matrices (A,B),
.SH SYNOPSIS
.TP 18
SUBROUTINE SGGES(
JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR,
LDVSR, WORK, LWORK, BWORK, INFO )
.TP 18
.ti +4
CHARACTER
JOBVSL, JOBVSR, SORT
.TP 18
.ti +4
INTEGER
INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
.TP 18
.ti +4
LOGICAL
BWORK( * )
.TP 18
.ti +4
REAL
A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
VSR( LDVSR, * ), WORK( * )
.TP 18
.ti +4
LOGICAL
SELCTG
.TP 18
.ti +4
EXTERNAL
SELCTG
.SH PURPOSE
SGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, the generalized real Schur form (S,T),
optionally, the left and/or right matrices of Schur vectors (VSL and
VSR). This gives the generalized Schur factorization
.br
(A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
.br
Optionally, it also orders the eigenvalues so that a selected cluster
of eigenvalues appears in the leading diagonal blocks of the upper
quasi-triangular matrix S and the upper triangular matrix T.The
leading columns of VSL and VSR then form an orthonormal basis for the
corresponding left and right eigenspaces (deflating subspaces).
(If only the generalized eigenvalues are needed, use the driver
SGGEV instead, which is faster.)
.br
A generalized eigenvalue for a pair of matrices (A,B) is 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 or both being zero.
.br
A pair of matrices (S,T) is in generalized real Schur form if T is
upper triangular with non-negative diagonal and S is block upper
triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond
to real generalized eigenvalues, while 2-by-2 blocks of S will be
"standardized" by making the corresponding elements of T have the
form:
.br
[ a 0 ]
.br
[ 0 b ]
.br
and the pair of corresponding 2-by-2 blocks in S and T will have a
complex conjugate pair of generalized eigenvalues.
.br
.SH ARGUMENTS
.TP 8
JOBVSL (input) CHARACTER*1
= 'N': do not compute the left Schur vectors;
.br
= 'V': compute the left Schur vectors.
.TP 8
JOBVSR (input) CHARACTER*1
.br
= 'N': do not compute the right Schur vectors;
.br
= 'V': compute the right Schur vectors.
.TP 8
SORT (input) CHARACTER*1
Specifies whether or not to order the eigenvalues on the
diagonal of the generalized Schur form.
= 'N': Eigenvalues are not ordered;
.br
= 'S': Eigenvalues are ordered (see SELCTG);
.TP 8
SELCTG (input) LOGICAL FUNCTION of three REAL arguments
SELCTG must be declared EXTERNAL in the calling subroutine.
If SORT = 'N', SELCTG is not referenced.
If SORT = 'S', SELCTG is used to select eigenvalues to sort
to the top left of the Schur form.
An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
SELCTG(ALPHAR(j),ALPHAI(j),BETA(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 in the ill-conditioned case, a selected complex
eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
in this case.
.TP 8
N (input) INTEGER
The order of the matrices A, B, VSL, and VSR. N >= 0.
.TP 8
A (input/output) REAL array, dimension (LDA, N)
On entry, the first of the pair of matrices.
On exit, A has been overwritten by its generalized Schur
form S.
.TP 8
LDA (input) INTEGER
The leading dimension of A. LDA >= max(1,N).
.TP 8
B (input/output) REAL array, dimension (LDB, N)
On entry, the second of the pair of matrices.
On exit, B has been overwritten by its generalized Schur
form T.
.TP 8
LDB (input) INTEGER
The leading dimension of B. LDB >= max(1,N).
.TP 8
SDIM (output) INTEGER
If SORT = 'N', SDIM = 0.
If SORT = 'S', SDIM = number of eigenvalues (after sorting)
for which SELCTG is true. (Complex conjugate pairs for which
SELCTG is true for either eigenvalue count as 2.)
.TP 8
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. ALPHAR(j) + ALPHAI(j)*i,
and BETA(j),j=1,...,N are the diagonals of the complex Schur
form (S,T) that would result if the 2-by-2 diagonal blocks of
the real Schur form of (A,B) were further reduced to
triangular form using 2-by-2 complex unitary transformations.
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.
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).
.TP 8
VSL (output) REAL array, dimension (LDVSL,N)
If JOBVSL = 'V', VSL will contain the left Schur vectors.
Not referenced if JOBVSL = 'N'.
.TP 8
LDVSL (input) INTEGER
The leading dimension of the matrix VSL. LDVSL >=1, and
if JOBVSL = 'V', LDVSL >= N.
.TP 8
VSR (output) REAL array, dimension (LDVSR,N)
If JOBVSR = 'V', VSR will contain the right Schur vectors.
Not referenced if JOBVSR = 'N'.
.TP 8
LDVSR (input) INTEGER
The leading dimension of the matrix VSR. LDVSR >= 1, and
if JOBVSR = 'V', LDVSR >= N.
.TP 8
WORK (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
.TP 8
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 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.
.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
= 1,...,N:
The QZ iteration failed. (A,B) are not in Schur
form, 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.
.br
=N+2: after reordering, roundoff changed values of
some complex eigenvalues so that leading
eigenvalues in the Generalized Schur form no
longer satisfy SELCTG=.TRUE. This could also
be caused due to scaling.
=N+3: reordering failed in STGSEN.
|
