1ZGGESX(1) LAPACK driver routine (version 3.1) ZGGESX(1)
2
6 ZGGESX - for a pair of N-by-N complex nonsymmetric matrices (A,B), the
7 generalized eigenvalues, the complex Schur form (S,T),
8
10 SUBROUTINE ZGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B,
11 LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR,
12 RCONDE, RCONDV, WORK, LWORK, RWORK, IWORK, LIWORK,
13 BWORK, INFO )
14 CHARACTER JOBVSL, JOBVSR, SENSE, SORT
16 INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N, SDIM
18 LOGICAL BWORK( * )
20 INTEGER IWORK( * )
22 DOUBLE PRECISION RCONDE( 2 ), RCONDV( 2 ), RWORK( * )
24 COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ),
26 VSL( LDVSL, * ), VSR( LDVSR, * ), WORK( * )
27 LOGICAL SELCTG
29 EXTERNAL SELCTG
31
33 ZGGESX computes for a pair of N-by-N complex nonsymmetric matrices
34 (A,B), the generalized eigenvalues, the complex Schur form (S,T), and,
35 optionally, the left and/or right matrices of Schur vectors (VSL and
36 VSR). This gives the generalized Schur factorization
37 (A,B) = ( (VSL) S (VSR)**H, (VSL) T (VSR)**H )
39 where (VSR)**H is the conjugate-transpose of VSR.
41 Optionally, it also orders the eigenvalues so that a selected cluster
43 of eigenvalues appears in the leading diagonal blocks of the upper tri‐
44 angular matrix S and the upper triangular matrix T; computes a recipro‐
45 cal condition number for the average of the selected eigenvalues
46 (RCONDE); and computes a reciprocal condition number for the right and
47 left deflating subspaces corresponding to the selected eigenvalues
48 (RCONDV). The leading columns of VSL and VSR then form an orthonormal
49 basis for the corresponding left and right eigenspaces (deflating sub‐
50 spaces).
51 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or
53 a ratio alpha/beta = w, such that A - w*B is singular. It is usually
54 represented as the pair (alpha,beta), as there is a reasonable inter‐
55 pretation for beta=0 or for both being zero.
56 A pair of matrices (S,T) is in generalized complex Schur form if T is
58 upper triangular with non-negative diagonal and S is upper triangular.
59
62 JOBVSL (input) CHARACTER*1
63 = 'N': do not compute the left Schur vectors;
64 = 'V': compute the left Schur vectors.
65 JOBVSR (input) CHARACTER*1
67 = 'N': do not compute the right Schur vectors;
68 = 'V': compute the right Schur vectors.
69 SORT (input) CHARACTER*1
71 Specifies whether or not to order the eigenvalues on the diago‐
72 nal of the generalized Schur form. = 'N': Eigenvalues are not
73 ordered;
74 = 'S': Eigenvalues are ordered (see SELCTG).
75 SELCTG (external procedure) LOGICAL FUNCTION of two COMPLEX*16 argu‐
77 ments
78 SELCTG must be declared EXTERNAL in the calling subroutine. If
79 SORT = 'N', SELCTG is not referenced. If SORT = 'S', SELCTG is
80 used to select eigenvalues to sort to the top left of the Schur
81 form. Note that a selected complex eigenvalue may no longer
82 satisfy SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since
83 ordering may change the value of complex eigenvalues (espe‐
84 cially if the eigenvalue is ill-conditioned), in this case INFO
85 is set to N+3 see INFO below).
86 SENSE (input) CHARACTER*1
88 Determines which reciprocal condition numbers are computed. =
89 'N' : None are computed;
90 = 'E' : Computed for average of selected eigenvalues only;
91 = 'V' : Computed for selected deflating subspaces only;
92 = 'B' : Computed for both. If SENSE = 'E', 'V', or 'B', SORT
93 must equal 'S'.
94 N (input) INTEGER
96 The order of the matrices A, B, VSL, and VSR. N >= 0.
97 A (input/output) COMPLEX*16 array, dimension (LDA, N)
99 On entry, the first of the pair of matrices. On exit, A has
100 been overwritten by its generalized Schur form S.
101 LDA (input) INTEGER
103 The leading dimension of A. LDA >= max(1,N).
104 B (input/output) COMPLEX*16 array, dimension (LDB, N)
106 On entry, the second of the pair of matrices. On exit, B has
107 been overwritten by its generalized Schur form T.
108 LDB (input) INTEGER
110 The leading dimension of B. LDB >= max(1,N).
111 SDIM (output) INTEGER
113 If SORT = 'N', SDIM = 0. If SORT = 'S', SDIM = number of ei‐
114 genvalues (after sorting) for which SELCTG is true.
115 ALPHA (output) COMPLEX*16 array, dimension (N)
117 BETA (output) COMPLEX*16 array, dimension (N) On exit,
118 ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenval‐
119 ues. ALPHA(j) and BETA(j),j=1,...,N are the diagonals of the
120 complex Schur form (S,T). BETA(j) will be non-negative real.
121 Note: the quotients ALPHA(j)/BETA(j) may easily over- or under‐
123 flow, and BETA(j) may even be zero. Thus, the user should
124 avoid naively computing the ratio alpha/beta. However, ALPHA
125 will be always less than and usually comparable with norm(A) in
126 magnitude, and BETA always less than and usually comparable
127 with norm(B).
128 VSL (output) COMPLEX*16 array, dimension (LDVSL,N)
130 If JOBVSL = 'V', VSL will contain the left Schur vectors. Not
131 referenced if JOBVSL = 'N'.
132 LDVSL (input) INTEGER
134 The leading dimension of the matrix VSL. LDVSL >=1, and if JOB‐
135 VSL = 'V', LDVSL >= N.
136 VSR (output) COMPLEX*16 array, dimension (LDVSR,N)
138 If JOBVSR = 'V', VSR will contain the right Schur vectors. Not
139 referenced if JOBVSR = 'N'.
140 LDVSR (input) INTEGER
142 The leading dimension of the matrix VSR. LDVSR >= 1, and if
143 JOBVSR = 'V', LDVSR >= N.
144 RCONDE (output) DOUBLE PRECISION array, dimension ( 2 )
146 If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
147 reciprocal condition numbers for the average of the selected
148 eigenvalues. Not referenced if SENSE = 'N' or 'V'.
149 RCONDV (output) DOUBLE PRECISION array, dimension ( 2 )
151 If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
152 reciprocal condition number for the selected deflating sub‐
153 spaces. Not referenced if SENSE = 'N' or 'E'.
154 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
156 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
157 LWORK (input) INTEGER
159 The dimension of the array WORK. If N = 0, LWORK >= 1, else if
160 SENSE = 'E', 'V', or 'B', LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)),
161 else LWORK >= MAX(1,2*N). Note that 2*SDIM*(N-SDIM) <= N*N/2.
162 Note also that an error is only returned if LWORK < MAX(1,2*N),
163 but if SENSE = 'E' or 'V' or 'B' this may not be large enough.
164 If LWORK = -1, then a workspace query is assumed; the routine
166 only calculates the bound on the optimal size of the WORK array
167 and the minimum size of the IWORK array, returns these values
168 as the first entries of the WORK and IWORK arrays, and no error
169 message related to LWORK or LIWORK is issued by XERBLA.
170 RWORK (workspace) DOUBLE PRECISION array, dimension ( 8*N )
172 Real workspace.
173 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
175 On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
176 LIWORK (input) INTEGER
178 The dimension of the array IWORK. If SENSE = 'N' or N = 0,
179 LIWORK >= 1, otherwise LIWORK >= N+2.
180 If LIWORK = -1, then a workspace query is assumed; the routine
182 only calculates the bound on the optimal size of the WORK array
183 and the minimum size of the IWORK array, returns these values
184 as the first entries of the WORK and IWORK arrays, and no error
185 message related to LWORK or LIWORK is issued by XERBLA.
186 BWORK (workspace) LOGICAL array, dimension (N)
188 Not referenced if SORT = 'N'.
189 INFO (output) INTEGER
191 = 0: successful exit
192 < 0: if INFO = -i, the i-th argument had an illegal value.
193 = 1,...,N: The QZ iteration failed. (A,B) are not in Schur
194 form, but ALPHA(j) and BETA(j) should be correct for
195 j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed in
196 ZHGEQZ
197 =N+2: after reordering, roundoff changed values of some complex
198 eigenvalues so that leading eigenvalues in the Generalized
199 Schur form no longer satisfy SELCTG=.TRUE. This could also be
200 caused due to scaling. =N+3: reordering failed in ZTGSEN.
201 LAPACK driver routine (version 3.N1o)vember 2006 ZGGESX(1)