1DGEES(1) LAPACK driver routine (version 3.2) DGEES(1)
2
6 DGEES - 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 DGEES( JOBVS, SORT, SELECT, N, A, LDA, SDIM, WR, WI, VS,
12 LDVS, WORK, LWORK, BWORK, INFO )
13 CHARACTER JOBVS, SORT
15 INTEGER INFO, LDA, LDVS, LWORK, N, SDIM
17 LOGICAL BWORK( * )
19 DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK(
21 * ), WR( * )
22 LOGICAL SELECT
24 EXTERNAL SELECT
26
28 DGEES computes for an N-by-N real nonsymmetric matrix A, the eigenval‐
29 ues, the real Schur form T, and, optionally, the matrix of Schur vec‐
30 tors Z. This gives the Schur factorization A = Z*T*(Z**T). Option‐
31 ally, it also orders the eigenvalues on the diagonal of the real Schur
32 form so that selected eigenvalues are at the top left. The leading
33 columns of Z then form an orthonormal basis for the invariant subspace
34 corresponding to the selected eigenvalues. A matrix is in real Schur
35 form if it is upper quasi-triangular with 1-by-1 and 2-by-2 blocks.
36 2-by-2 blocks will be standardized in the form
37 [ a b ]
38 [ c a ]
39 where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
40
42 JOBVS (input) CHARACTER*1
43 = 'N': Schur vectors are not computed;
44 = 'V': Schur vectors are computed.
45 SORT (input) CHARACTER*1
47 Specifies whether or not to order the eigenvalues on the diago‐
48 nal of the Schur form. = 'N': Eigenvalues are not ordered;
49 = 'S': Eigenvalues are ordered (see SELECT).
50 SELECT (external procedure) LOGICAL FUNCTION of two DOUBLE PRECISION
52 arguments
53 SELECT must be declared EXTERNAL in the calling subroutine. If
54 SORT = 'S', SELECT is used to select eigenvalues to sort to the
55 top left of the Schur form. If SORT = 'N', SELECT is not ref‐
56 erenced. An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
57 SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
58 conjugate pair of eigenvalues is selected, then both complex
59 eigenvalues are selected. Note that a selected complex eigen‐
60 value may no longer satisfy SELECT(WR(j),WI(j)) = .TRUE. after
61 ordering, since ordering may change the value of complex eigen‐
62 values (especially if the eigenvalue is ill-conditioned); in
63 this case INFO is set to N+2 (see INFO below).
64 N (input) INTEGER
66 The order of the matrix A. N >= 0.
67 A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
69 On entry, the N-by-N matrix A. On exit, A has been overwritten
70 by its real Schur form T.
71 LDA (input) INTEGER
73 The leading dimension of the array A. LDA >= max(1,N).
74 SDIM (output) INTEGER
76 If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of ei‐
77 genvalues (after sorting) for which SELECT is true. (Complex
78 conjugate pairs for which SELECT is true for either eigenvalue
79 count as 2.)
80 WR (output) DOUBLE PRECISION array, dimension (N)
82 WI (output) DOUBLE PRECISION array, dimension (N) WR and
83 WI contain the real and imaginary parts, respectively, of the
84 computed eigenvalues in the same order that they appear on the
85 diagonal of the output Schur form T. Complex conjugate pairs
86 of eigenvalues will appear consecutively with the eigenvalue
87 having the positive imaginary part first.
88 VS (output) DOUBLE PRECISION array, dimension (LDVS,N)
90 If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
91 vectors. If JOBVS = 'N', VS is not referenced.
92 LDVS (input) INTEGER
94 The leading dimension of the array VS. LDVS >= 1; if JOBVS =
95 'V', LDVS >= N.
96 WORK (workspace/output) DOUBLE PRECISION array, dimension
98 (MAX(1,LWORK))
99 On exit, if INFO = 0, WORK(1) contains the optimal LWORK.
100 LWORK (input) INTEGER
102 The dimension of the array WORK. LWORK >= max(1,3*N). For
103 good performance, LWORK must generally be larger. If LWORK =
104 -1, then a workspace query is assumed; the routine only calcu‐
105 lates the optimal size of the WORK array, returns this value as
106 the first entry of the WORK array, and no error message related
107 to LWORK is issued by XERBLA.
108 BWORK (workspace) LOGICAL array, dimension (N)
110 Not referenced if SORT = 'N'.
111 INFO (output) INTEGER
113 = 0: successful exit
114 < 0: if INFO = -i, the i-th argument had an illegal value.
115 > 0: if INFO = i, and i is
116 <= N: the QR algorithm failed to compute all the
117 eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI contain
118 those eigenvalues which have converged; if JOBVS = 'V', VS con‐
119 tains the matrix which reduces A to its partially converged
120 Schur form. = N+1: the eigenvalues could not be reordered
121 because some eigenvalues were too close to separate (the prob‐
122 lem is very ill-conditioned); = N+2: after reordering, roundoff
123 changed values of some complex eigenvalues so that leading ei‐
124 genvalues in the Schur form no longer satisfy SELECT=.TRUE.
125 This could also be caused by underflow due to scaling.
126 LAPACK driver routine (version 3.N2o)vember 2008 DGEES(1)