1SGEES(1) LAPACK driver routine (version 3.2) SGEES(1)
2
6 SGEES - 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 SGEES( 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 REAL A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), WR( *
21 )
22 LOGICAL SELECT
24 EXTERNAL SELECT
26
28 SGEES 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 REAL arguments
52 SELECT must be declared EXTERNAL in the calling subroutine. If
53 SORT = 'S', SELECT is used to select eigenvalues to sort to the
54 top left of the Schur form. If SORT = 'N', SELECT is not ref‐
55 erenced. An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
56 SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
57 conjugate pair of eigenvalues is selected, then both complex
58 eigenvalues are selected. Note that a selected complex eigen‐
59 value may no longer satisfy SELECT(WR(j),WI(j)) = .TRUE. after
60 ordering, since ordering may change the value of complex eigen‐
61 values (especially if the eigenvalue is ill-conditioned); in
62 this case INFO is set to N+2 (see INFO below).
63 N (input) INTEGER
65 The order of the matrix A. N >= 0.
66 A (input/output) REAL array, dimension (LDA,N)
68 On entry, the N-by-N matrix A. On exit, A has been overwritten
69 by its real Schur form T.
70 LDA (input) INTEGER
72 The leading dimension of the array A. LDA >= max(1,N).
73 SDIM (output) INTEGER
75 If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of ei‐
76 genvalues (after sorting) for which SELECT is true. (Complex
77 conjugate pairs for which SELECT is true for either eigenvalue
78 count as 2.)
79 WR (output) REAL array, dimension (N)
81 WI (output) REAL array, dimension (N) WR and WI contain
82 the real and imaginary parts, respectively, of the computed ei‐
83 genvalues in the same order that they appear on the diagonal of
84 the output Schur form T. Complex conjugate pairs of eigenval‐
85 ues will appear consecutively with the eigenvalue having the
86 positive imaginary part first.
87 VS (output) REAL array, dimension (LDVS,N)
89 If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
90 vectors. If JOBVS = 'N', VS is not referenced.
91 LDVS (input) INTEGER
93 The leading dimension of the array VS. LDVS >= 1; if JOBVS =
94 'V', LDVS >= N.
95 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
97 On exit, if INFO = 0, WORK(1) contains the optimal LWORK.
98 LWORK (input) INTEGER
100 The dimension of the array WORK. LWORK >= max(1,3*N). For
101 good performance, LWORK must generally be larger. If LWORK =
102 -1, then a workspace query is assumed; the routine only calcu‐
103 lates the optimal size of the WORK array, returns this value as
104 the first entry of the WORK array, and no error message related
105 to LWORK is issued by XERBLA.
106 BWORK (workspace) LOGICAL array, dimension (N)
108 Not referenced if SORT = 'N'.
109 INFO (output) INTEGER
111 = 0: successful exit
112 < 0: if INFO = -i, the i-th argument had an illegal value.
113 > 0: if INFO = i, and i is
114 <= N: the QR algorithm failed to compute all the
115 eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI contain
116 those eigenvalues which have converged; if JOBVS = 'V', VS con‐
117 tains the matrix which reduces A to its partially converged
118 Schur form. = N+1: the eigenvalues could not be reordered
119 because some eigenvalues were too close to separate (the prob‐
120 lem is very ill-conditioned); = N+2: after reordering, roundoff
121 changed values of some complex eigenvalues so that leading ei‐
122 genvalues in the Schur form no longer satisfy SELECT=.TRUE.
123 This could also be caused by underflow due to scaling.
124 LAPACK driver routine (version 3.N2o)vember 2008 SGEES(1)