1ZGEGV(1) LAPACK driver routine (version 3.2) ZGEGV(1)
2
6 ZGEGV - routine i deprecated and has been replaced by routine ZGGEV
7
9 SUBROUTINE ZGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL,
10 LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
11 CHARACTER JOBVL, JOBVR
13 INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
15 DOUBLE PRECISION RWORK( * )
17 COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VL(
19 LDVL, * ), VR( LDVR, * ), WORK( * )
20
22 This routine is deprecated and has been replaced by routine ZGGEV.
23 ZGEGV computes the eigenvalues and, optionally, the left and/or right
24 eigenvectors of a complex matrix pair (A,B).
25 Given two square matrices A and B,
26 the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
27 eigenvalues lambda and corresponding (non-zero) eigenvectors x such
28 that
29 A*x = lambda*B*x.
30 An alternate form is to find the eigenvalues mu and corresponding
31 eigenvectors y such that
32 mu*A*y = B*y.
33 These two forms are equivalent with mu = 1/lambda and x = y if neither
34 lambda nor mu is zero. In order to deal with the case that lambda or
35 mu is zero or small, two values alpha and beta are returned for each
36 eigenvalue, such that lambda = alpha/beta and
37 mu = beta/alpha.
38 The vectors x and y in the above equations are right eigenvectors of
39 the matrix pair (A,B). Vectors u and v satisfying
40 u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
41 are left eigenvectors of (A,B).
42 Note: this routine performs "full balancing" on A and B -- see "Further
43 Details", below.
44
46 JOBVL (input) CHARACTER*1
47 = 'N': do not compute the left generalized eigenvectors;
48 = 'V': compute the left generalized eigenvectors (returned in
49 VL).
50 JOBVR (input) CHARACTER*1
52 = 'N': do not compute the right generalized eigenvectors;
53 = 'V': compute the right generalized eigenvectors (returned in
54 VR).
55 N (input) INTEGER
57 The order of the matrices A, B, VL, and VR. N >= 0.
58 A (input/output) COMPLEX*16 array, dimension (LDA, N)
60 On entry, the matrix A. If JOBVL = 'V' or JOBVR = 'V', then on
61 exit A contains the Schur form of A from the generalized Schur
62 factorization of the pair (A,B) after balancing. If no eigen‐
63 vectors were computed, then only the diagonal elements of the
64 Schur form will be correct. See ZGGHRD and ZHGEQZ for details.
65 LDA (input) INTEGER
67 The leading dimension of A. LDA >= max(1,N).
68 B (input/output) COMPLEX*16 array, dimension (LDB, N)
70 On entry, the matrix B. If JOBVL = 'V' or JOBVR = 'V', then on
71 exit B contains the upper triangular matrix obtained from B in
72 the generalized Schur factorization of the pair (A,B) after
73 balancing. If no eigenvectors were computed, then only the
74 diagonal elements of B will be correct. See ZGGHRD and ZHGEQZ
75 for details.
76 LDB (input) INTEGER
78 The leading dimension of B. LDB >= max(1,N).
79 ALPHA (output) COMPLEX*16 array, dimension (N)
81 The complex scalars alpha that define the eigenvalues of GNEP.
82 BETA (output) COMPLEX*16 array, dimension (N)
84 The complex scalars beta that define the eigenvalues of GNEP.
85 Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
86 represent the j-th eigenvalue of the matrix pair (A,B), in one
87 of the forms lambda = alpha/beta or mu = beta/alpha. Since
88 either lambda or mu may overflow, they should not, in general,
89 be computed.
90 VL (output) COMPLEX*16 array, dimension (LDVL,N)
92 If JOBVL = 'V', the left eigenvectors u(j) are stored in the
93 columns of VL, in the same order as their eigenvalues. Each
94 eigenvector is scaled so that its largest component has
95 abs(real part) + abs(imag. part) = 1, except for eigenvectors
96 corresponding to an eigenvalue with alpha = beta = 0, which are
97 set to zero. Not referenced if JOBVL = 'N'.
98 LDVL (input) INTEGER
100 The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL
101 = 'V', LDVL >= N.
102 VR (output) COMPLEX*16 array, dimension (LDVR,N)
104 If JOBVR = 'V', the right eigenvectors x(j) are stored in the
105 columns of VR, in the same order as their eigenvalues. Each
106 eigenvector is scaled so that its largest component has
107 abs(real part) + abs(imag. part) = 1, except for eigenvectors
108 corresponding to an eigenvalue with alpha = beta = 0, which are
109 set to zero. Not referenced if JOBVR = 'N'.
110 LDVR (input) INTEGER
112 The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR
113 = 'V', LDVR >= N.
114 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
116 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
117 LWORK (input) INTEGER
119 The dimension of the array WORK. LWORK >= max(1,2*N). For
120 good performance, LWORK must generally be larger. To compute
121 the optimal value of LWORK, call ILAENV to get blocksizes (for
122 ZGEQRF, ZUNMQR, and ZUNGQR.) Then compute: NB -- MAX of the
123 blocksizes for ZGEQRF, ZUNMQR, and ZUNGQR; The optimal LWORK is
124 MAX( 2*N, N*(NB+1) ). If LWORK = -1, then a workspace query is
125 assumed; the routine only calculates the optimal size of the
126 WORK array, returns this value as the first entry of the WORK
127 array, and no error message related to LWORK is issued by
128 XERBLA.
129 RWORK (workspace/output) DOUBLE PRECISION array, dimension (8*N)
131 INFO (output) INTEGER
133 = 0: successful exit
134 < 0: if INFO = -i, the i-th argument had an illegal value.
135 =1,...,N: The QZ iteration failed. No eigenvectors have been
136 calculated, but ALPHA(j) and BETA(j) should be correct for
137 j=INFO+1,...,N. > N: errors that usually indicate LAPACK
138 problems:
139 =N+1: error return from ZGGBAL
140 =N+2: error return from ZGEQRF
141 =N+3: error return from ZUNMQR
142 =N+4: error return from ZUNGQR
143 =N+5: error return from ZGGHRD
144 =N+6: error return from ZHGEQZ (other than failed iteration)
145 =N+7: error return from ZTGEVC
146 =N+8: error return from ZGGBAK (computing VL)
147 =N+9: error return from ZGGBAK (computing VR)
148 =N+10: error return from ZLASCL (various calls)
149
151 Balancing
152 ---------
153 This driver calls ZGGBAL to both permute and scale rows and columns of
154 A and B. The permutations PL and PR are chosen so that PL*A*PR and
155 PL*B*R will be upper triangular except for the diagonal blocks
156 A(i:j,i:j) and B(i:j,i:j), with i and j as close together as possible.
157 The diagonal scaling matrices DL and DR are chosen so that the pair
158 DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to one (except for the
159 elements that start out zero.)
160 After the eigenvalues and eigenvectors of the balanced matrices have
161 been computed, ZGGBAK transforms the eigenvectors back to what they
162 would have been (in perfect arithmetic) if they had not been balanced.
163 Contents of A and B on Exit
164 -------- -- - --- - -- ----
165 If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
166 both), then on exit the arrays A and B will contain the complex Schur
167 form[*] of the "balanced" versions of A and B. If no eigenvectors are
168 computed, then only the diagonal blocks will be correct. [*] In other
169 words, upper triangular form.
170 LAPACK driver routine (version 3.N2o)vember 2008 ZGEGV(1)