1CGGEV(1) LAPACK driver routine (version 3.1) CGGEV(1)
2
6 CGGEV - for a pair of N-by-N complex nonsymmetric matrices (A,B), the
7 generalized eigenvalues, and optionally, the left and/or right general‐
8 ized eigenvectors
9
11 SUBROUTINE CGGEV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA, VL,
12 LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
13 CHARACTER JOBVL, JOBVR
15 INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
17 REAL RWORK( * )
19 COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VL(
21 LDVL, * ), VR( LDVR, * ), WORK( * )
22
24 CGGEV computes for a pair of N-by-N complex nonsymmetric matrices
25 (A,B), the generalized eigenvalues, and optionally, the left and/or
26 right generalized eigenvectors.
27 A generalized eigenvalue for a pair of matrices (A,B) is a scalar
29 lambda or a ratio alpha/beta = lambda, such that A - lambda*B is singu‐
30 lar. It is usually represented as the pair (alpha,beta), as there is a
31 reasonable interpretation for beta=0, and even for both being zero.
32 The right generalized eigenvector v(j) corresponding to the generalized
34 eigenvalue lambda(j) of (A,B) satisfies
35 A * v(j) = lambda(j) * B * v(j).
37 The left generalized eigenvector u(j) corresponding to the generalized
39 eigenvalues lambda(j) of (A,B) satisfies
40 u(j)**H * A = lambda(j) * u(j)**H * B
42 where u(j)**H is the conjugate-transpose of u(j).
44
47 JOBVL (input) CHARACTER*1
48 = 'N': do not compute the left generalized eigenvectors;
49 = 'V': compute the left generalized eigenvectors.
50 JOBVR (input) CHARACTER*1
52 = 'N': do not compute the right generalized eigenvectors;
53 = 'V': compute the right generalized eigenvectors.
54 N (input) INTEGER
56 The order of the matrices A, B, VL, and VR. N >= 0.
57 A (input/output) COMPLEX array, dimension (LDA, N)
59 On entry, the matrix A in the pair (A,B). On exit, A has been
60 overwritten.
61 LDA (input) INTEGER
63 The leading dimension of A. LDA >= max(1,N).
64 B (input/output) COMPLEX array, dimension (LDB, N)
66 On entry, the matrix B in the pair (A,B). On exit, B has been
67 overwritten.
68 LDB (input) INTEGER
70 The leading dimension of B. LDB >= max(1,N).
71 ALPHA (output) COMPLEX array, dimension (N)
73 BETA (output) COMPLEX array, dimension (N) On exit,
74 ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenval‐
75 ues.
76 Note: the quotients ALPHA(j)/BETA(j) may easily over- or under‐
78 flow, and BETA(j) may even be zero. Thus, the user should
79 avoid naively computing the ratio alpha/beta. However, ALPHA
80 will be always less than and usually comparable with norm(A) in
81 magnitude, and BETA always less than and usually comparable
82 with norm(B).
83 VL (output) COMPLEX array, dimension (LDVL,N)
85 If JOBVL = 'V', the left generalized eigenvectors u(j) are
86 stored one after another in the columns of VL, in the same
87 order as their eigenvalues. Each eigenvector is scaled so the
88 largest component has abs(real part) + abs(imag. part) = 1.
89 Not referenced if JOBVL = 'N'.
90 LDVL (input) INTEGER
92 The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL
93 = 'V', LDVL >= N.
94 VR (output) COMPLEX array, dimension (LDVR,N)
96 If JOBVR = 'V', the right generalized eigenvectors v(j) are
97 stored one after another in the columns of VR, in the same
98 order as their eigenvalues. Each eigenvector is scaled so the
99 largest component has abs(real part) + abs(imag. part) = 1.
100 Not referenced if JOBVR = 'N'.
101 LDVR (input) INTEGER
103 The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR
104 = 'V', LDVR >= N.
105 WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
107 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
108 LWORK (input) INTEGER
110 The dimension of the array WORK. LWORK >= max(1,2*N). For
111 good performance, LWORK must generally be larger.
112 If LWORK = -1, then a workspace query is assumed; the routine
114 only calculates the optimal size of the WORK array, returns
115 this value as the first entry of the WORK array, and no error
116 message related to LWORK is issued by XERBLA.
117 RWORK (workspace/output) REAL array, dimension (8*N)
119 INFO (output) INTEGER
121 = 0: successful exit
122 < 0: if INFO = -i, the i-th argument had an illegal value.
123 =1,...,N: The QZ iteration failed. No eigenvectors have been
124 calculated, but ALPHA(j) and BETA(j) should be correct for
125 j=INFO+1,...,N. > N: =N+1: other then QZ iteration failed in
126 SHGEQZ,
127 =N+2: error return from STGEVC.
128 LAPACK driver routine (version 3.N1o)vember 2006 CGGEV(1)