1DGGES(1) LAPACK driver routine (version 3.2) DGGES(1)
2
6 DGGES - computes for a pair of N-by-N real nonsymmetric matrices (A,B),
7
9 SUBROUTINE DGGES( JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB,
10 SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
11 WORK, LWORK, BWORK, INFO )
12 CHARACTER JOBVSL, JOBVSR, SORT
14 INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N, SDIM
16 LOGICAL BWORK( * )
18 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B(
20 LDB, * ), BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, *
21 ), WORK( * )
22 LOGICAL SELCTG
24 EXTERNAL SELCTG
26
28 DGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B),
29 the generalized eigenvalues, the generalized real Schur form (S,T),
30 optionally, the left and/or right matrices of Schur vectors (VSL and
31 VSR). This gives the generalized Schur factorization
32 (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )
33 Optionally, it also orders the eigenvalues so that a selected cluster
34 of eigenvalues appears in the leading diagonal blocks of the upper
35 quasi-triangular matrix S and the upper triangular matrix T.The leading
36 columns of VSL and VSR then form an orthonormal basis for the corre‐
37 sponding left and right eigenspaces (deflating subspaces). (If only
38 the generalized eigenvalues are needed, use the driver DGGEV instead,
39 which is faster.)
40 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or
41 a ratio alpha/beta = w, such that A - w*B is singular. It is usually
42 represented as the pair (alpha,beta), as there is a reasonable inter‐
43 pretation for beta=0 or both being zero.
44 A pair of matrices (S,T) is in generalized real Schur form if T is
45 upper triangular with non-negative diagonal and S is block upper trian‐
46 gular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real
47 generalized eigenvalues, while 2-by-2 blocks of S will be "standard‐
48 ized" by making the corresponding elements of T have the form:
49 [ a 0 ]
50 [ 0 b ]
51 and the pair of corresponding 2-by-2 blocks in S and T will have a com‐
52 plex conjugate pair of generalized eigenvalues.
53
55 JOBVSL (input) CHARACTER*1
56 = 'N': do not compute the left Schur vectors;
57 = 'V': compute the left Schur vectors.
58 JOBVSR (input) CHARACTER*1
60 = 'N': do not compute the right Schur vectors;
61 = 'V': compute the right Schur vectors.
62 SORT (input) CHARACTER*1
64 Specifies whether or not to order the eigenvalues on the diago‐
65 nal of the generalized Schur form. = 'N': Eigenvalues are not
66 ordered;
67 = 'S': Eigenvalues are ordered (see SELCTG);
68 SELCTG (external procedure) LOGICAL FUNCTION of three DOUBLE PRECISION
70 arguments
71 SELCTG must be declared EXTERNAL in the calling subroutine. If
72 SORT = 'N', SELCTG is not referenced. If SORT = 'S', SELCTG is
73 used to select eigenvalues to sort to the top left of the Schur
74 form. An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected
75 if SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
76 one of a complex conjugate pair of eigenvalues is selected,
77 then both complex eigenvalues are selected. Note that in the
78 ill-conditioned case, a selected complex eigenvalue may no
79 longer satisfy SELCTG(ALPHAR(j),ALPHAI(j), BETA(j)) = .TRUE.
80 after ordering. INFO is to be set to N+2 in this case.
81 N (input) INTEGER
83 The order of the matrices A, B, VSL, and VSR. N >= 0.
84 A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
86 On entry, the first of the pair of matrices. On exit, A has
87 been overwritten by its generalized Schur form S.
88 LDA (input) INTEGER
90 The leading dimension of A. LDA >= max(1,N).
91 B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
93 On entry, the second of the pair of matrices. On exit, B has
94 been overwritten by its generalized Schur form T.
95 LDB (input) INTEGER
97 The leading dimension of B. LDB >= max(1,N).
98 SDIM (output) INTEGER
100 If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of ei‐
101 genvalues (after sorting) for which SELCTG is true. (Complex
102 conjugate pairs for which SELCTG is true for either eigenvalue
103 count as 2.)
104 ALPHAR (output) DOUBLE PRECISION array, dimension (N)
106 ALPHAI (output) DOUBLE PRECISION array, dimension (N) BETA
107 (output) DOUBLE PRECISION array, dimension (N) On exit,
108 (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the gen‐
109 eralized eigenvalues. ALPHAR(j) + ALPHAI(j)*i, and
110 BETA(j),j=1,...,N are the diagonals of the complex Schur form
111 (S,T) that would result if the 2-by-2 diagonal blocks of the
112 real Schur form of (A,B) were further reduced to triangular
113 form using 2-by-2 complex unitary transformations. If
114 ALPHAI(j) is zero, then the j-th eigenvalue is real; if posi‐
115 tive, then the j-th and (j+1)-st eigenvalues are a complex con‐
116 jugate pair, with ALPHAI(j+1) negative. Note: the quotients
117 ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may easily over- or
118 underflow, and BETA(j) may even be zero. Thus, the user should
119 avoid naively computing the ratio. However, ALPHAR and ALPHAI
120 will be always less than and usually comparable with norm(A) in
121 magnitude, and BETA always less than and usually comparable
122 with norm(B).
123 VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N)
125 If JOBVSL = 'V', VSL will contain the left Schur vectors. Not
126 referenced if JOBVSL = 'N'.
127 LDVSL (input) INTEGER
129 The leading dimension of the matrix VSL. LDVSL >=1, and if JOB‐
130 VSL = 'V', LDVSL >= N.
131 VSR (output) DOUBLE PRECISION array, dimension (LDVSR,N)
133 If JOBVSR = 'V', VSR will contain the right Schur vectors. Not
134 referenced if JOBVSR = 'N'.
135 LDVSR (input) INTEGER
137 The leading dimension of the matrix VSR. LDVSR >= 1, and if
138 JOBVSR = 'V', LDVSR >= N.
139 WORK (workspace/output) DOUBLE PRECISION array, dimension
141 (MAX(1,LWORK))
142 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
143 LWORK (input) INTEGER
145 The dimension of the array WORK. If N = 0, LWORK >= 1, else
146 LWORK >= 8*N+16. For good performance , LWORK must generally
147 be larger. If LWORK = -1, then a workspace query is assumed;
148 the routine only calculates the optimal size of the WORK array,
149 returns this value as the first entry of the WORK array, and no
150 error message related to LWORK is issued by XERBLA.
151 BWORK (workspace) LOGICAL array, dimension (N)
153 Not referenced if SORT = 'N'.
154 INFO (output) INTEGER
156 = 0: successful exit
157 < 0: if INFO = -i, the i-th argument had an illegal value.
158 = 1,...,N: The QZ iteration failed. (A,B) are not in Schur
159 form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be correct
160 for j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed
161 in DHGEQZ.
162 =N+2: after reordering, roundoff changed values of some complex
163 eigenvalues so that leading eigenvalues in the Generalized
164 Schur form no longer satisfy SELCTG=.TRUE. This could also be
165 caused due to scaling. =N+3: reordering failed in DTGSEN.
166 LAPACK driver routine (version 3.N2o)vember 2008 DGGES(1)