1DGGES(1) LAPACK driver routine (version 3.1) DGGES(1)
2
6 DGGES - 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 )
34 Optionally, it also orders the eigenvalues so that a selected cluster
36 of eigenvalues appears in the leading diagonal blocks of the upper
37 quasi-triangular matrix S and the upper triangular matrix T.The leading
38 columns of VSL and VSR then form an orthonormal basis for the corre‐
39 sponding left and right eigenspaces (deflating subspaces).
40 (If only the generalized eigenvalues are needed, use the driver DGGEV
42 instead, which is faster.)
43 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or
45 a ratio alpha/beta = w, such that A - w*B is singular. It is usually
46 represented as the pair (alpha,beta), as there is a reasonable inter‐
47 pretation for beta=0 or both being zero.
48 A pair of matrices (S,T) is in generalized real Schur form if T is
50 upper triangular with non-negative diagonal and S is block upper trian‐
51 gular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real
52 generalized eigenvalues, while 2-by-2 blocks of S will be "standard‐
53 ized" by making the corresponding elements of T have the form:
54 [ a 0 ]
55 [ 0 b ]
56 and the pair of corresponding 2-by-2 blocks in S and T will have a com‐
58 plex conjugate pair of generalized eigenvalues.
59
63 JOBVSL (input) CHARACTER*1
64 = 'N': do not compute the left Schur vectors;
65 = 'V': compute the left Schur vectors.
66 JOBVSR (input) CHARACTER*1
68 = 'N': do not compute the right Schur vectors;
69 = 'V': compute the right Schur vectors.
70 SORT (input) CHARACTER*1
72 Specifies whether or not to order the eigenvalues on the diago‐
73 nal of the generalized Schur form. = 'N': Eigenvalues are not
74 ordered;
75 = 'S': Eigenvalues are ordered (see SELCTG);
76 SELCTG (external procedure) LOGICAL FUNCTION of three DOUBLE PRECISION
78 arguments
79 SELCTG must be declared EXTERNAL in the calling subroutine. If
80 SORT = 'N', SELCTG is not referenced. If SORT = 'S', SELCTG is
81 used to select eigenvalues to sort to the top left of the Schur
82 form. An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected
83 if SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
84 one of a complex conjugate pair of eigenvalues is selected,
85 then both complex eigenvalues are selected.
86 Note that in the ill-conditioned case, a selected complex ei‐
88 genvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
89 BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2 in
90 this case.
91 N (input) INTEGER
93 The order of the matrices A, B, VSL, and VSR. N >= 0.
94 A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
96 On entry, the first of the pair of matrices. On exit, A has
97 been overwritten by its generalized Schur form S.
98 LDA (input) INTEGER
100 The leading dimension of A. LDA >= max(1,N).
101 B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
103 On entry, the second of the pair of matrices. On exit, B has
104 been overwritten by its generalized Schur form T.
105 LDB (input) INTEGER
107 The leading dimension of B. LDB >= max(1,N).
108 SDIM (output) INTEGER
110 If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of ei‐
111 genvalues (after sorting) for which SELCTG is true. (Complex
112 conjugate pairs for which SELCTG is true for either eigenvalue
113 count as 2.)
114 ALPHAR (output) DOUBLE PRECISION array, dimension (N)
116 ALPHAI (output) DOUBLE PRECISION array, dimension (N) BETA
117 (output) DOUBLE PRECISION array, dimension (N) On exit,
118 (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the gen‐
119 eralized eigenvalues. ALPHAR(j) + ALPHAI(j)*i, and
120 BETA(j),j=1,...,N are the diagonals of the complex Schur form
121 (S,T) that would result if the 2-by-2 diagonal blocks of the
122 real Schur form of (A,B) were further reduced to triangular
123 form using 2-by-2 complex unitary transformations. If
124 ALPHAI(j) is zero, then the j-th eigenvalue is real; if posi‐
125 tive, then the j-th and (j+1)-st eigenvalues are a complex con‐
126 jugate pair, with ALPHAI(j+1) negative.
127 Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may
129 easily over- or underflow, and BETA(j) may even be zero. Thus,
130 the user should avoid naively computing the ratio. However,
131 ALPHAR and ALPHAI will be always less than and usually compara‐
132 ble with norm(A) in magnitude, and BETA always less than and
133 usually comparable with norm(B).
134 VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N)
136 If JOBVSL = 'V', VSL will contain the left Schur vectors. Not
137 referenced if JOBVSL = 'N'.
138 LDVSL (input) INTEGER
140 The leading dimension of the matrix VSL. LDVSL >=1, and if JOB‐
141 VSL = 'V', LDVSL >= N.
142 VSR (output) DOUBLE PRECISION array, dimension (LDVSR,N)
144 If JOBVSR = 'V', VSR will contain the right Schur vectors. Not
145 referenced if JOBVSR = 'N'.
146 LDVSR (input) INTEGER
148 The leading dimension of the matrix VSR. LDVSR >= 1, and if
149 JOBVSR = 'V', LDVSR >= N.
150 WORK (workspace/output) DOUBLE PRECISION array, dimension
152 (MAX(1,LWORK))
153 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
154 LWORK (input) INTEGER
156 The dimension of the array WORK. If N = 0, LWORK >= 1, else
157 LWORK >= 8*N+16. For good performance , LWORK must generally
158 be larger.
159 If LWORK = -1, then a workspace query is assumed; the routine
161 only calculates the optimal size of the WORK array, returns
162 this value as the first entry of the WORK array, and no error
163 message related to LWORK is issued by XERBLA.
164 BWORK (workspace) LOGICAL array, dimension (N)
166 Not referenced if SORT = 'N'.
167 INFO (output) INTEGER
169 = 0: successful exit
170 < 0: if INFO = -i, the i-th argument had an illegal value.
171 = 1,...,N: The QZ iteration failed. (A,B) are not in Schur
172 form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be correct
173 for j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed
174 in DHGEQZ.
175 =N+2: after reordering, roundoff changed values of some complex
176 eigenvalues so that leading eigenvalues in the Generalized
177 Schur form no longer satisfy SELCTG=.TRUE. This could also be
178 caused due to scaling. =N+3: reordering failed in DTGSEN.
179 LAPACK driver routine (version 3.N1o)vember 2006 DGGES(1)