1ZGEESX(1) LAPACK driver routine (version 3.2) ZGEESX(1)
2
6 ZGEESX - 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 ZGEESX( 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 DOUBLE PRECISION RCONDE, RCONDV
20 LOGICAL BWORK( * )
22 DOUBLE PRECISION RWORK( * )
24 COMPLEX*16 A( LDA, * ), VS( LDVS, * ), W( * ), WORK( * )
26 LOGICAL SELECT
28 EXTERNAL SELECT
30
32 ZGEESX 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*16 argu‐
57 ment
58 SELECT must be declared EXTERNAL in the calling subroutine. If
59 SORT = 'S', SELECT is used to select eigenvalues to order to
60 the top left of the Schur form. If SORT = 'N', SELECT is not
61 referenced. An eigenvalue W(j) is selected if SELECT(W(j)) is
62 true.
63 SENSE (input) CHARACTER*1
65 Determines which reciprocal condition numbers are computed. =
66 'N': None are computed;
67 = 'E': Computed for average of selected eigenvalues only;
68 = 'V': Computed for selected right invariant subspace only;
69 = 'B': Computed for both. If SENSE = 'E', 'V' or 'B', SORT
70 must equal 'S'.
71 N (input) INTEGER
73 The order of the matrix A. N >= 0.
74 A (input/output) COMPLEX*16 array, dimension (LDA, N)
76 On entry, the N-by-N matrix A. On exit, A is overwritten by
77 its Schur form T.
78 LDA (input) INTEGER
80 The leading dimension of the array A. LDA >= max(1,N).
81 SDIM (output) INTEGER
83 If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of ei‐
84 genvalues for which SELECT is true.
85 W (output) COMPLEX*16 array, dimension (N)
87 W contains the computed eigenvalues, in the same order that
88 they appear on the diagonal of the output Schur form T.
89 VS (output) COMPLEX*16 array, dimension (LDVS,N)
91 If JOBVS = 'V', VS contains the unitary matrix Z of Schur vec‐
92 tors. If JOBVS = 'N', VS is not referenced.
93 LDVS (input) INTEGER
95 The leading dimension of the array VS. LDVS >= 1, and if JOBVS
96 = 'V', LDVS >= N.
97 RCONDE (output) DOUBLE PRECISION
99 If SENSE = 'E' or 'B', RCONDE contains the reciprocal condition
100 number for the average of the selected eigenvalues. Not refer‐
101 enced if SENSE = 'N' or 'V'.
102 RCONDV (output) DOUBLE PRECISION
104 If SENSE = 'V' or 'B', RCONDV contains the reciprocal condition
105 number for the selected right invariant subspace. Not refer‐
106 enced if SENSE = 'N' or 'E'.
107 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
109 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
110 LWORK (input) INTEGER
112 The dimension of the array WORK. LWORK >= max(1,2*N). Also,
113 if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM), where
114 SDIM is the number of selected eigenvalues computed by this
115 routine. Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also that an
116 error is only returned if LWORK < max(1,2*N), but if SENSE =
117 'E' or 'V' or 'B' this may not be large enough. For good per‐
118 formance, LWORK must generally be larger. If LWORK = -1, then
119 a workspace query is assumed; the routine only calculates upper
120 bound on the optimal size of the array WORK, returns this value
121 as the first entry of the WORK array, and no error message
122 related to LWORK is issued by XERBLA.
123 RWORK (workspace) DOUBLE PRECISION array, dimension (N)
125 BWORK (workspace) LOGICAL array, dimension (N)
127 Not referenced if SORT = 'N'.
128 INFO (output) INTEGER
130 = 0: successful exit
131 < 0: if INFO = -i, the i-th argument had an illegal value.
132 > 0: if INFO = i, and i is
133 <= N: the QR algorithm failed to compute all the
134 eigenvalues; elements 1:ILO-1 and i+1:N of W contain those ei‐
135 genvalues which have converged; if JOBVS = 'V', VS contains the
136 transformation which reduces A to its partially converged Schur
137 form. = N+1: the eigenvalues could not be reordered because
138 some eigenvalues were too close to separate (the problem is
139 very ill-conditioned); = N+2: after reordering, roundoff
140 changed values of some complex eigenvalues so that leading ei‐
141 genvalues in the Schur form no longer satisfy SELECT=.TRUE.
142 This could also be caused by underflow due to scaling.
143 LAPACK driver routine (version 3.N2o)vember 2008 ZGEESX(1)