1DGEGV(1) LAPACK driver routine (version 3.1) DGEGV(1)
2
6 DGEGV - i deprecated and has been replaced by routine DGGEV
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9 SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
10 BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
11 CHARACTER JOBVL, JOBVR
13 INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
15 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B(
17 LDB, * ), BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
18 WORK( * )
19
21 This routine is deprecated and has been replaced by routine DGGEV.
22 DGEGV computes the eigenvalues and, optionally, the left and/or right
24 eigenvectors of a real 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.
31 An alternate form is to find the eigenvalues mu and corresponding
33 eigenvectors y such that
34 mu*A*y = B*y.
36 These two forms are equivalent with mu = 1/lambda and x = y if neither
38 lambda nor mu is zero. In order to deal with the case that lambda or
39 mu is zero or small, two values alpha and beta are returned for each
40 eigenvalue, such that lambda = alpha/beta and
41 mu = beta/alpha.
42 The vectors x and y in the above equations are right eigenvectors of
44 the matrix pair (A,B). Vectors u and v satisfying
45 u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
47 are left eigenvectors of (A,B).
49 Note: this routine performs "full balancing" on A and B -- see "Further
51 Details", below.
52
54 JOBVL (input) CHARACTER*1
55 = 'N': do not compute the left generalized eigenvectors;
56 = 'V': compute the left generalized eigenvectors (returned in
57 VL).
58 JOBVR (input) CHARACTER*1
60 = 'N': do not compute the right generalized eigenvectors;
61 = 'V': compute the right generalized eigenvectors (returned in
62 VR).
63 N (input) INTEGER
65 The order of the matrices A, B, VL, and VR. N >= 0.
66 A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
68 On entry, the matrix A. If JOBVL = 'V' or JOBVR = 'V', then on
69 exit A contains the real Schur form of A from the generalized
70 Schur factorization of the pair (A,B) after balancing. If no
71 eigenvectors were computed, then only the diagonal blocks from
72 the Schur form will be correct. See DGGHRD and DHGEQZ for
73 details.
74 LDA (input) INTEGER
76 The leading dimension of A. LDA >= max(1,N).
77 B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
79 On entry, the matrix B. If JOBVL = 'V' or JOBVR = 'V', then on
80 exit B contains the upper triangular matrix obtained from B in
81 the generalized Schur factorization of the pair (A,B) after
82 balancing. If no eigenvectors were computed, then only those
83 elements of B corresponding to the diagonal blocks from the
84 Schur form of A will be correct. See DGGHRD and DHGEQZ for
85 details.
86 LDB (input) INTEGER
88 The leading dimension of B. LDB >= max(1,N).
89 ALPHAR (output) DOUBLE PRECISION array, dimension (N)
91 The real parts of each scalar alpha defining an eigenvalue of
92 GNEP.
93 ALPHAI (output) DOUBLE PRECISION array, dimension (N)
95 The imaginary parts of each scalar alpha defining an eigenvalue
96 of GNEP. If ALPHAI(j) is zero, then the j-th eigenvalue is
97 real; if positive, then the j-th and (j+1)-st eigenvalues are a
98 complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
99 BETA (output) DOUBLE PRECISION array, dimension (N)
101 The scalars beta that define the eigenvalues of GNEP.
102 Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and beta
103 = BETA(j) represent the j-th eigenvalue of the matrix pair
104 (A,B), in one of the forms lambda = alpha/beta or mu =
105 beta/alpha. Since either lambda or mu may overflow, they
106 should not, in general, be computed.
107 VL (output) DOUBLE PRECISION array, dimension (LDVL,N)
109 If JOBVL = 'V', the left eigenvectors u(j) are stored in the
110 columns of VL, in the same order as their eigenvalues. If the
111 j-th eigenvalue is real, then u(j) = VL(:,j). If the j-th and
112 (j+1)-st eigenvalues form a complex conjugate pair, then u(j) =
113 VL(:,j) + i*VL(:,j+1) and u(j+1) = VL(:,j) - i*VL(:,j+1).
114 Each eigenvector is scaled so that its largest component has
116 abs(real part) + abs(imag. part) = 1, except for eigenvectors
117 corresponding to an eigenvalue with alpha = beta = 0, which are
118 set to zero. Not referenced if JOBVL = 'N'.
119 LDVL (input) INTEGER
121 The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL
122 = 'V', LDVL >= N.
123 VR (output) DOUBLE PRECISION array, dimension (LDVR,N)
125 If JOBVR = 'V', the right eigenvectors x(j) are stored in the
126 columns of VR, in the same order as their eigenvalues. If the
127 j-th eigenvalue is real, then x(j) = VR(:,j). If the j-th and
128 (j+1)-st eigenvalues form a complex conjugate pair, then x(j) =
129 VR(:,j) + i*VR(:,j+1) and x(j+1) = VR(:,j) - i*VR(:,j+1).
130 Each eigenvector is scaled so that its largest component has
132 abs(real part) + abs(imag. part) = 1, except for eigenvalues
133 corresponding to an eigenvalue with alpha = beta = 0, which are
134 set to zero. Not referenced if JOBVR = 'N'.
135 LDVR (input) INTEGER
137 The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR
138 = 'V', LDVR >= 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. LWORK >= max(1,8*N). For
146 good performance, LWORK must generally be larger. To compute
147 the optimal value of LWORK, call ILAENV to get blocksizes (for
148 DGEQRF, DORMQR, and DORGQR.) Then compute: NB -- MAX of the
149 blocksizes for DGEQRF, DORMQR, and DORGQR; The optimal LWORK
150 is: 2*N + MAX( 6*N, N*(NB+1) ).
151 If LWORK = -1, then a workspace query is assumed; the routine
153 only calculates the optimal size of the WORK array, returns
154 this value as the first entry of the WORK array, and no error
155 message related to LWORK is issued by XERBLA.
156 INFO (output) INTEGER
158 = 0: successful exit
159 < 0: if INFO = -i, the i-th argument had an illegal value.
160 = 1,...,N: The QZ iteration failed. No eigenvectors have been
161 calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) should be
162 correct for j=INFO+1,...,N. > N: errors that usually indicate
163 LAPACK problems:
164 =N+1: error return from DGGBAL
165 =N+2: error return from DGEQRF
166 =N+3: error return from DORMQR
167 =N+4: error return from DORGQR
168 =N+5: error return from DGGHRD
169 =N+6: error return from DHGEQZ (other than failed iteration)
170 =N+7: error return from DTGEVC
171 =N+8: error return from DGGBAK (computing VL)
172 =N+9: error return from DGGBAK (computing VR)
173 =N+10: error return from DLASCL (various calls)
174
176 Balancing
177 ---------
178 This driver calls DGGBAL to both permute and scale rows and columns of
180 A and B. The permutations PL and PR are chosen so that PL*A*PR and
181 PL*B*R will be upper triangular except for the diagonal blocks
182 A(i:j,i:j) and B(i:j,i:j), with i and j as close together as possible.
183 The diagonal scaling matrices DL and DR are chosen so that the pair
184 DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to one (except for the
185 elements that start out zero.)
186 After the eigenvalues and eigenvectors of the balanced matrices have
188 been computed, DGGBAK transforms the eigenvectors back to what they
189 would have been (in perfect arithmetic) if they had not been balanced.
190 Contents of A and B on Exit
192 -------- -- - --- - -- ----
193 If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
195 both), then on exit the arrays A and B will contain the real Schur
196 form[*] of the "balanced" versions of A and B. If no eigenvectors are
197 computed, then only the diagonal blocks will be correct.
198 [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations",
200 by Golub & van Loan, pub. by Johns Hopkins U. Press.
201 LAPACK driver routine (version 3.N1o)vember 2006 DGEGV(1)