1SGEESX(1) LAPACK driver routine (version 3.2) SGEESX(1)
2
6 SGEESX - computes for an N-by-N real nonsymmetric matrix A, the eigen‐
7 values, the real Schur form T, and, optionally, the matrix of Schur
8 vectors Z
9
11 SUBROUTINE SGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, WR, WI,
12 VS, LDVS, RCONDE, RCONDV, WORK, LWORK, IWORK,
13 LIWORK, BWORK, INFO )
14 CHARACTER JOBVS, SENSE, SORT
16 INTEGER INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM
18 REAL RCONDE, RCONDV
20 LOGICAL BWORK( * )
22 INTEGER IWORK( * )
24 REAL A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), WR(
26 * )
27 LOGICAL SELECT
29 EXTERNAL SELECT
31
33 SGEESX computes for an N-by-N real nonsymmetric matrix A, the eigenval‐
34 ues, the real Schur form T, and, optionally, the matrix of Schur vec‐
35 tors Z. This gives the Schur factorization A = Z*T*(Z**T). Option‐
36 ally, it also orders the eigenvalues on the diagonal of the real Schur
37 form so that selected eigenvalues are at the top left; computes a
38 reciprocal condition number for the average of the selected eigenvalues
39 (RCONDE); and computes a reciprocal condition number for the right
40 invariant subspace corresponding to the selected eigenvalues (RCONDV).
41 The leading columns of Z form an orthonormal basis for this invariant
42 subspace.
43 For further explanation of the reciprocal condition numbers RCONDE and
44 RCONDV, see Section 4.10 of the LAPACK Users' Guide (where these quan‐
45 tities are called s and sep respectively).
46 A real matrix is in real Schur form if it is upper quasi-triangular
47 with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in
48 the form
49 [ a b ]
50 [ c a ]
51 where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
52
54 JOBVS (input) CHARACTER*1
55 = 'N': Schur vectors are not computed;
56 = 'V': Schur vectors are computed.
57 SORT (input) CHARACTER*1
59 Specifies whether or not to order the eigenvalues on the diago‐
60 nal of the Schur form. = 'N': Eigenvalues are not ordered;
61 = 'S': Eigenvalues are ordered (see SELECT).
62 SELECT (external procedure) LOGICAL FUNCTION of two REAL arguments
64 SELECT must be declared EXTERNAL in the calling subroutine. If
65 SORT = 'S', SELECT is used to select eigenvalues to sort to the
66 top left of the Schur form. If SORT = 'N', SELECT is not ref‐
67 erenced. An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
68 SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
69 conjugate pair of eigenvalues is selected, then both are. Note
70 that a selected complex eigenvalue may no longer satisfy
71 SELECT(WR(j),WI(j)) = .TRUE. after ordering, since ordering may
72 change the value of complex eigenvalues (especially if the ei‐
73 genvalue is ill-conditioned); in this case INFO may be set to
74 N+3 (see INFO below).
75 SENSE (input) CHARACTER*1
77 Determines which reciprocal condition numbers are computed. =
78 'N': None are computed;
79 = 'E': Computed for average of selected eigenvalues only;
80 = 'V': Computed for selected right invariant subspace only;
81 = 'B': Computed for both. If SENSE = 'E', 'V' or 'B', SORT
82 must equal 'S'.
83 N (input) INTEGER
85 The order of the matrix A. N >= 0.
86 A (input/output) REAL array, dimension (LDA, N)
88 On entry, the N-by-N matrix A. On exit, A is overwritten by
89 its real Schur form T.
90 LDA (input) INTEGER
92 The leading dimension of the array A. LDA >= max(1,N).
93 SDIM (output) INTEGER
95 If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of ei‐
96 genvalues (after sorting) for which SELECT is true. (Complex
97 conjugate pairs for which SELECT is true for either eigenvalue
98 count as 2.)
99 WR (output) REAL array, dimension (N)
101 WI (output) REAL array, dimension (N) WR and WI contain
102 the real and imaginary parts, respectively, of the computed ei‐
103 genvalues, in the same order that they appear on the diagonal
104 of the output Schur form T. Complex conjugate pairs of eigen‐
105 values appear consecutively with the eigenvalue having the pos‐
106 itive imaginary part first.
107 VS (output) REAL array, dimension (LDVS,N)
109 If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
110 vectors. If JOBVS = 'N', VS is not referenced.
111 LDVS (input) INTEGER
113 The leading dimension of the array VS. LDVS >= 1, and if JOBVS
114 = 'V', LDVS >= N.
115 RCONDE (output) REAL
117 If SENSE = 'E' or 'B', RCONDE contains the reciprocal condition
118 number for the average of the selected eigenvalues. Not refer‐
119 enced if SENSE = 'N' or 'V'.
120 RCONDV (output) REAL
122 If SENSE = 'V' or 'B', RCONDV contains the reciprocal condition
123 number for the selected right invariant subspace. Not refer‐
124 enced if SENSE = 'N' or 'E'.
125 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
127 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
128 LWORK (input) INTEGER
130 The dimension of the array WORK. LWORK >= max(1,3*N). Also,
131 if SENSE = 'E' or 'V' or 'B', LWORK >= N+2*SDIM*(N-SDIM), where
132 SDIM is the number of selected eigenvalues computed by this
133 routine. Note that N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also
134 that an error is only returned if LWORK < max(1,3*N), but if
135 SENSE = 'E' or 'V' or 'B' this may not be large enough. For
136 good performance, LWORK must generally be larger. If LWORK =
137 -1, then a workspace query is assumed; the routine only calcu‐
138 lates upper bounds on the optimal sizes of the arrays WORK and
139 IWORK, returns these values as the first entries of the WORK
140 and IWORK arrays, and no error messages related to LWORK or
141 LIWORK are issued by XERBLA.
142 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
144 On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
145 LIWORK (input) INTEGER
147 The dimension of the array IWORK. LIWORK >= 1; if SENSE = 'V'
148 or 'B', LIWORK >= SDIM*(N-SDIM). Note that SDIM*(N-SDIM) <=
149 N*N/4. Note also that an error is only returned if LIWORK < 1,
150 but if SENSE = 'V' or 'B' this may not be large enough. If
151 LIWORK = -1, then a workspace query is assumed; the routine
152 only calculates upper bounds on the optimal sizes of the arrays
153 WORK and IWORK, returns these values as the first entries of
154 the WORK and IWORK arrays, and no error messages related to
155 LWORK or LIWORK are issued by XERBLA.
156 BWORK (workspace) LOGICAL array, dimension (N)
158 Not referenced if SORT = 'N'.
159 INFO (output) INTEGER
161 = 0: successful exit
162 < 0: if INFO = -i, the i-th argument had an illegal value.
163 > 0: if INFO = i, and i is
164 <= N: the QR algorithm failed to compute all the
165 eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI contain
166 those eigenvalues which have converged; if JOBVS = 'V', VS con‐
167 tains the transformation which reduces A to its partially con‐
168 verged Schur form. = N+1: the eigenvalues could not be
169 reordered because some eigenvalues were too close to separate
170 (the problem is very ill-conditioned); = N+2: after reordering,
171 roundoff changed values of some complex eigenvalues so that
172 leading eigenvalues in the Schur form no longer satisfy
173 SELECT=.TRUE. This could also be caused by underflow due to
174 scaling.
175 LAPACK driver routine (version 3.N2o)vember 2008 SGEESX(1)