1CGEESX(1) LAPACK driver routine (version 3.2) CGEESX(1)
2
6 CGEESX - computes for an N-by-N complex nonsymmetric matrix A, the ei‐
7 genvalues, the Schur form T, and, optionally, the matrix of Schur vec‐
8 tors Z
9
11 SUBROUTINE CGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, W, VS,
12 LDVS, RCONDE, RCONDV, WORK, LWORK, RWORK, BWORK,
13 INFO )
14 CHARACTER JOBVS, SENSE, SORT
16 INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
18 REAL RCONDE, RCONDV
20 LOGICAL BWORK( * )
22 REAL RWORK( * )
24 COMPLEX A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
26 LOGICAL SELECT
28 EXTERNAL SELECT
30
32 CGEESX computes for an N-by-N complex nonsymmetric matrix A, the eigen‐
33 values, the Schur form T, and, optionally, the matrix of Schur vectors
34 Z. This gives the Schur factorization A = Z*T*(Z**H). Optionally, it
35 also orders the eigenvalues on the diagonal of the Schur form so that
36 selected eigenvalues are at the top left; computes a reciprocal condi‐
37 tion number for the average of the selected eigenvalues (RCONDE); and
38 computes a reciprocal condition number for the right invariant subspace
39 corresponding to the selected eigenvalues (RCONDV). The leading col‐
40 umns of Z form an orthonormal basis for this invariant subspace.
41 For further explanation of the reciprocal condition numbers RCONDE and
42 RCONDV, see Section 4.10 of the LAPACK Users' Guide (where these quan‐
43 tities are called s and sep respectively).
44 A complex matrix is in Schur form if it is upper triangular.
45
47 JOBVS (input) CHARACTER*1
48 = 'N': Schur vectors are not computed;
49 = 'V': Schur vectors are computed.
50 SORT (input) CHARACTER*1
52 Specifies whether or not to order the eigenvalues on the diago‐
53 nal of the Schur form. = 'N': Eigenvalues are not ordered;
54 = 'S': Eigenvalues are ordered (see SELECT).
55 SELECT (external procedure) LOGICAL FUNCTION of one COMPLEX argument
57 SELECT must be declared EXTERNAL in the calling subroutine. If
58 SORT = 'S', SELECT is used to select eigenvalues to order to
59 the top left of the Schur form. If SORT = 'N', SELECT is not
60 referenced. An eigenvalue W(j) is selected if SELECT(W(j)) is
61 true.
62 SENSE (input) CHARACTER*1
64 Determines which reciprocal condition numbers are computed. =
65 'N': None are computed;
66 = 'E': Computed for average of selected eigenvalues only;
67 = 'V': Computed for selected right invariant subspace only;
68 = 'B': Computed for both. If SENSE = 'E', 'V' or 'B', SORT
69 must equal 'S'.
70 N (input) INTEGER
72 The order of the matrix A. N >= 0.
73 A (input/output) COMPLEX array, dimension (LDA, N)
75 On entry, the N-by-N matrix A. On exit, A is overwritten by
76 its Schur form T.
77 LDA (input) INTEGER
79 The leading dimension of the array A. LDA >= max(1,N).
80 SDIM (output) INTEGER
82 If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of ei‐
83 genvalues for which SELECT is true.
84 W (output) COMPLEX array, dimension (N)
86 W contains the computed eigenvalues, in the same order that
87 they appear on the diagonal of the output Schur form T.
88 VS (output) COMPLEX array, dimension (LDVS,N)
90 If JOBVS = 'V', VS contains the unitary matrix Z of Schur vec‐
91 tors. If JOBVS = 'N', VS is not referenced.
92 LDVS (input) INTEGER
94 The leading dimension of the array VS. LDVS >= 1, and if JOBVS
95 = 'V', LDVS >= N.
96 RCONDE (output) REAL
98 If SENSE = 'E' or 'B', RCONDE contains the reciprocal condition
99 number for the average of the selected eigenvalues. Not refer‐
100 enced if SENSE = 'N' or 'V'.
101 RCONDV (output) REAL
103 If SENSE = 'V' or 'B', RCONDV contains the reciprocal condition
104 number for the selected right invariant subspace. Not refer‐
105 enced if SENSE = 'N' or 'E'.
106 WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
108 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
109 LWORK (input) INTEGER
111 The dimension of the array WORK. LWORK >= max(1,2*N). Also,
112 if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM), where
113 SDIM is the number of selected eigenvalues computed by this
114 routine. Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also that an
115 error is only returned if LWORK < max(1,2*N), but if SENSE =
116 'E' or 'V' or 'B' this may not be large enough. For good per‐
117 formance, LWORK must generally be larger. If LWORK = -1, then
118 a workspace query is assumed; the routine only calculates upper
119 bound on the optimal size of the array WORK, returns this value
120 as the first entry of the WORK array, and no error message
121 related to LWORK is issued by XERBLA.
122 RWORK (workspace) REAL array, dimension (N)
124 BWORK (workspace) LOGICAL array, dimension (N)
126 Not referenced if SORT = 'N'.
127 INFO (output) INTEGER
129 = 0: successful exit
130 < 0: if INFO = -i, the i-th argument had an illegal value.
131 > 0: if INFO = i, and i is
132 <= N: the QR algorithm failed to compute all the
133 eigenvalues; elements 1:ILO-1 and i+1:N of W contain those ei‐
134 genvalues which have converged; if JOBVS = 'V', VS contains the
135 transformation which reduces A to its partially converged Schur
136 form. = N+1: the eigenvalues could not be reordered because
137 some eigenvalues were too close to separate (the problem is
138 very ill-conditioned); = N+2: after reordering, roundoff
139 changed values of some complex eigenvalues so that leading ei‐
140 genvalues in the Schur form no longer satisfy SELECT=.TRUE.
141 This could also be caused by underflow due to scaling.
142 LAPACK driver routine (version 3.N2o)vember 2008 CGEESX(1)