1DGEESX(1) LAPACK driver routine (version 3.2) DGEESX(1)
2
6 DGEESX - 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 DGEESX( 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 DOUBLE PRECISION RCONDE, RCONDV
20 LOGICAL BWORK( * )
22 INTEGER IWORK( * )
24 DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK(
26 * ), WR( * )
27 LOGICAL SELECT
29 EXTERNAL SELECT
31
33 DGEESX 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 DOUBLE PRECISION
64 arguments
65 SELECT must be declared EXTERNAL in the calling subroutine. If
66 SORT = 'S', SELECT is used to select eigenvalues to sort to the
67 top left of the Schur form. If SORT = 'N', SELECT is not ref‐
68 erenced. An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
69 SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
70 conjugate pair of eigenvalues is selected, then both are. Note
71 that a selected complex eigenvalue may no longer satisfy
72 SELECT(WR(j),WI(j)) = .TRUE. after ordering, since ordering may
73 change the value of complex eigenvalues (especially if the ei‐
74 genvalue is ill-conditioned); in this case INFO may be set to
75 N+3 (see INFO below).
76 SENSE (input) CHARACTER*1
78 Determines which reciprocal condition numbers are computed. =
79 'N': None are computed;
80 = 'E': Computed for average of selected eigenvalues only;
81 = 'V': Computed for selected right invariant subspace only;
82 = 'B': Computed for both. If SENSE = 'E', 'V' or 'B', SORT
83 must equal 'S'.
84 N (input) INTEGER
86 The order of the matrix A. N >= 0.
87 A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
89 On entry, the N-by-N matrix A. On exit, A is overwritten by
90 its real Schur form T.
91 LDA (input) INTEGER
93 The leading dimension of the array A. LDA >= max(1,N).
94 SDIM (output) INTEGER
96 If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of ei‐
97 genvalues (after sorting) for which SELECT is true. (Complex
98 conjugate pairs for which SELECT is true for either eigenvalue
99 count as 2.)
100 WR (output) DOUBLE PRECISION array, dimension (N)
102 WI (output) DOUBLE PRECISION array, dimension (N) WR and
103 WI contain the real and imaginary parts, respectively, of the
104 computed eigenvalues, in the same order that they appear on the
105 diagonal of the output Schur form T. Complex conjugate pairs
106 of eigenvalues appear consecutively with the eigenvalue having
107 the positive imaginary part first.
108 VS (output) DOUBLE PRECISION array, dimension (LDVS,N)
110 If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
111 vectors. If JOBVS = 'N', VS is not referenced.
112 LDVS (input) INTEGER
114 The leading dimension of the array VS. LDVS >= 1, and if JOBVS
115 = 'V', LDVS >= N.
116 RCONDE (output) DOUBLE PRECISION
118 If SENSE = 'E' or 'B', RCONDE contains the reciprocal condition
119 number for the average of the selected eigenvalues. Not refer‐
120 enced if SENSE = 'N' or 'V'.
121 RCONDV (output) DOUBLE PRECISION
123 If SENSE = 'V' or 'B', RCONDV contains the reciprocal condition
124 number for the selected right invariant subspace. Not refer‐
125 enced if SENSE = 'N' or 'E'.
126 WORK (workspace/output) DOUBLE PRECISION array, dimension
128 (MAX(1,LWORK))
129 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
130 LWORK (input) INTEGER
132 The dimension of the array WORK. LWORK >= max(1,3*N). Also,
133 if SENSE = 'E' or 'V' or 'B', LWORK >= N+2*SDIM*(N-SDIM), where
134 SDIM is the number of selected eigenvalues computed by this
135 routine. Note that N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also
136 that an error is only returned if LWORK < max(1,3*N), but if
137 SENSE = 'E' or 'V' or 'B' this may not be large enough. For
138 good performance, LWORK must generally be larger. If LWORK =
139 -1, then a workspace query is assumed; the routine only calcu‐
140 lates upper bounds on the optimal sizes of the arrays WORK and
141 IWORK, returns these values as the first entries of the WORK
142 and IWORK arrays, and no error messages related to LWORK or
143 LIWORK are issued by XERBLA.
144 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
146 On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
147 LIWORK (input) INTEGER
149 The dimension of the array IWORK. LIWORK >= 1; if SENSE = 'V'
150 or 'B', LIWORK >= SDIM*(N-SDIM). Note that SDIM*(N-SDIM) <=
151 N*N/4. Note also that an error is only returned if LIWORK < 1,
152 but if SENSE = 'V' or 'B' this may not be large enough. If
153 LIWORK = -1, then a workspace query is assumed; the routine
154 only calculates upper bounds on the optimal sizes of the arrays
155 WORK and IWORK, returns these values as the first entries of
156 the WORK and IWORK arrays, and no error messages related to
157 LWORK or LIWORK are issued by XERBLA.
158 BWORK (workspace) LOGICAL array, dimension (N)
160 Not referenced if SORT = 'N'.
161 INFO (output) INTEGER
163 = 0: successful exit
164 < 0: if INFO = -i, the i-th argument had an illegal value.
165 > 0: if INFO = i, and i is
166 <= N: the QR algorithm failed to compute all the
167 eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI contain
168 those eigenvalues which have converged; if JOBVS = 'V', VS con‐
169 tains the transformation which reduces A to its partially con‐
170 verged Schur form. = N+1: the eigenvalues could not be
171 reordered because some eigenvalues were too close to separate
172 (the problem is very ill-conditioned); = N+2: after reordering,
173 roundoff changed values of some complex eigenvalues so that
174 leading eigenvalues in the Schur form no longer satisfy
175 SELECT=.TRUE. This could also be caused by underflow due to
176 scaling.
177 LAPACK driver routine (version 3.N2o)vember 2008 DGEESX(1)