1SGEGV(1) LAPACK driver routine (version 3.2) SGEGV(1)
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6 SGEGV - routine i deprecated and has been replaced by routine SGGEV
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9 SUBROUTINE SGEGV( 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 REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
17 BETA( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )
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20 This routine is deprecated and has been replaced by routine SGGEV.
21 SGEGV computes the eigenvalues and, optionally, the left and/or right
22 eigenvectors of a real matrix pair (A,B).
23 Given two square matrices A and B,
24 the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
25 eigenvalues lambda and corresponding (non-zero) eigenvectors x such
26 that
27 A*x = lambda*B*x.
28 An alternate form is to find the eigenvalues mu and corresponding
29 eigenvectors y such that
30 mu*A*y = B*y.
31 These two forms are equivalent with mu = 1/lambda and x = y if neither
32 lambda nor mu is zero. In order to deal with the case that lambda or
33 mu is zero or small, two values alpha and beta are returned for each
34 eigenvalue, such that lambda = alpha/beta and
35 mu = beta/alpha.
36 The vectors x and y in the above equations are right eigenvectors of
37 the matrix pair (A,B). Vectors u and v satisfying
38 u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
39 are left eigenvectors of (A,B).
40 Note: this routine performs "full balancing" on A and B -- see "Further
41 Details", below.
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44 JOBVL (input) CHARACTER*1
45 = 'N': do not compute the left generalized eigenvectors;
46 = 'V': compute the left generalized eigenvectors (returned in
47 VL).
48 JOBVR (input) CHARACTER*1
50 = 'N': do not compute the right generalized eigenvectors;
51 = 'V': compute the right generalized eigenvectors (returned in
52 VR).
53 N (input) INTEGER
55 The order of the matrices A, B, VL, and VR. N >= 0.
56 A (input/output) REAL array, dimension (LDA, N)
58 On entry, the matrix A. If JOBVL = 'V' or JOBVR = 'V', then on
59 exit A contains the real Schur form of A from the generalized
60 Schur factorization of the pair (A,B) after balancing. If no
61 eigenvectors were computed, then only the diagonal blocks from
62 the Schur form will be correct. See SGGHRD and SHGEQZ for
63 details.
64 LDA (input) INTEGER
66 The leading dimension of A. LDA >= max(1,N).
67 B (input/output) REAL array, dimension (LDB, N)
69 On entry, the matrix B. If JOBVL = 'V' or JOBVR = 'V', then on
70 exit B contains the upper triangular matrix obtained from B in
71 the generalized Schur factorization of the pair (A,B) after
72 balancing. If no eigenvectors were computed, then only those
73 elements of B corresponding to the diagonal blocks from the
74 Schur form of A will be correct. See SGGHRD and SHGEQZ for
75 details.
76 LDB (input) INTEGER
78 The leading dimension of B. LDB >= max(1,N).
79 ALPHAR (output) REAL array, dimension (N)
81 The real parts of each scalar alpha defining an eigenvalue of
82 GNEP.
83 ALPHAI (output) REAL array, dimension (N)
85 The imaginary parts of each scalar alpha defining an eigenvalue
86 of GNEP. If ALPHAI(j) is zero, then the j-th eigenvalue is
87 real; if positive, then the j-th and (j+1)-st eigenvalues are a
88 complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j).
89 BETA (output) REAL array, dimension (N)
91 The scalars beta that define the eigenvalues of GNEP.
92 Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and beta
93 = BETA(j) represent the j-th eigenvalue of the matrix pair
94 (A,B), in one of the forms lambda = alpha/beta or mu =
95 beta/alpha. Since either lambda or mu may overflow, they
96 should not, in general, be computed.
97 VL (output) REAL array, dimension (LDVL,N)
99 If JOBVL = 'V', the left eigenvectors u(j) are stored in the
100 columns of VL, in the same order as their eigenvalues. If the
101 j-th eigenvalue is real, then u(j) = VL(:,j). If the j-th and
102 (j+1)-st eigenvalues form a complex conjugate pair, then u(j) =
103 VL(:,j) + i*VL(:,j+1) and u(j+1) = VL(:,j) - i*VL(:,j+1). Each
104 eigenvector is scaled so that its largest component has
105 abs(real part) + abs(imag. part) = 1, except for eigenvectors
106 corresponding to an eigenvalue with alpha = beta = 0, which are
107 set to zero. Not referenced if JOBVL = 'N'.
108 LDVL (input) INTEGER
110 The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL
111 = 'V', LDVL >= N.
112 VR (output) REAL array, dimension (LDVR,N)
114 If JOBVR = 'V', the right eigenvectors x(j) are stored in the
115 columns of VR, in the same order as their eigenvalues. If the
116 j-th eigenvalue is real, then x(j) = VR(:,j). If the j-th and
117 (j+1)-st eigenvalues form a complex conjugate pair, then x(j) =
118 VR(:,j) + i*VR(:,j+1) and x(j+1) = VR(:,j) - i*VR(:,j+1). Each
119 eigenvector is scaled so that its largest component has
120 abs(real part) + abs(imag. part) = 1, except for eigenvalues
121 corresponding to an eigenvalue with alpha = beta = 0, which are
122 set to zero. Not referenced if JOBVR = 'N'.
123 LDVR (input) INTEGER
125 The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR
126 = 'V', LDVR >= N.
127 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
129 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
130 LWORK (input) INTEGER
132 The dimension of the array WORK. LWORK >= max(1,8*N). For
133 good performance, LWORK must generally be larger. To compute
134 the optimal value of LWORK, call ILAENV to get blocksizes (for
135 SGEQRF, SORMQR, and SORGQR.) Then compute: NB -- MAX of the
136 blocksizes for SGEQRF, SORMQR, and SORGQR; The optimal LWORK
137 is: 2*N + MAX( 6*N, N*(NB+1) ). If LWORK = -1, then a
138 workspace query is assumed; the routine only calculates the
139 optimal size of the WORK array, returns this value as the first
140 entry of the WORK array, and no error message related to LWORK
141 is issued by XERBLA.
142 INFO (output) INTEGER
144 = 0: successful exit
145 < 0: if INFO = -i, the i-th argument had an illegal value.
146 = 1,...,N: The QZ iteration failed. No eigenvectors have been
147 calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) should be
148 correct for j=INFO+1,...,N. > N: errors that usually indicate
149 LAPACK problems:
150 =N+1: error return from SGGBAL
151 =N+2: error return from SGEQRF
152 =N+3: error return from SORMQR
153 =N+4: error return from SORGQR
154 =N+5: error return from SGGHRD
155 =N+6: error return from SHGEQZ (other than failed iteration)
156 =N+7: error return from STGEVC
157 =N+8: error return from SGGBAK (computing VL)
158 =N+9: error return from SGGBAK (computing VR)
159 =N+10: error return from SLASCL (various calls)
160
162 Balancing
163 ---------
164 This driver calls SGGBAL to both permute and scale rows and columns of
165 A and B. The permutations PL and PR are chosen so that PL*A*PR and
166 PL*B*R will be upper triangular except for the diagonal blocks
167 A(i:j,i:j) and B(i:j,i:j), with i and j as close together as possible.
168 The diagonal scaling matrices DL and DR are chosen so that the pair
169 DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to one (except for the
170 elements that start out zero.)
171 After the eigenvalues and eigenvectors of the balanced matrices have
172 been computed, SGGBAK transforms the eigenvectors back to what they
173 would have been (in perfect arithmetic) if they had not been balanced.
174 Contents of A and B on Exit
175 -------- -- - --- - -- ----
176 If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
177 both), then on exit the arrays A and B will contain the real Schur
178 form[*] of the "balanced" versions of A and B. If no eigenvectors are
179 computed, then only the diagonal blocks will be correct. [*] See
180 SHGEQZ, SGEGS, or read the book "Matrix Computations",
181 by Golub & van Loan, pub. by Johns Hopkins U. Press.
182 LAPACK driver routine (version 3.N2o)vember 2008 SGEGV(1)