1CGGEVX(1) LAPACK driver routine (version 3.2) CGGEVX(1)
2
6 CGGEVX - computes for a pair of N-by-N complex nonsymmetric matrices
7 (A,B) the generalized eigenvalues, and optionally, the left and/or
8 right generalized eigenvectors
9
11 SUBROUTINE CGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
12 ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE,
13 RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK,
14 RWORK, IWORK, BWORK, INFO )
15 CHARACTER BALANC, JOBVL, JOBVR, SENSE
17 INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
19 REAL ABNRM, BBNRM
21 LOGICAL BWORK( * )
23 INTEGER IWORK( * )
25 REAL LSCALE( * ), RCONDE( * ), RCONDV( * ), RSCALE( * ),
27 RWORK( * )
28 COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VL(
30 LDVL, * ), VR( LDVR, * ), WORK( * )
31
33 CGGEVX computes for a pair of N-by-N complex nonsymmetric matrices
34 (A,B) the generalized eigenvalues, and optionally, the left and/or
35 right generalized eigenvectors. Optionally, it also computes a balanc‐
36 ing transformation to improve the conditioning of the eigenvalues and
37 eigenvectors (ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal
38 condition numbers for the eigenvalues (RCONDE), and reciprocal condi‐
39 tion numbers for the right eigenvectors (RCONDV).
40 A generalized eigenvalue for a pair of matrices (A,B) is a scalar
41 lambda or a ratio alpha/beta = lambda, such that A - lambda*B is singu‐
42 lar. It is usually represented as the pair (alpha,beta), as there is a
43 reasonable interpretation for beta=0, and even for both being zero.
44 The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of
45 (A,B) satisfies
46 A * v(j) = lambda(j) * B * v(j) .
47 The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of
48 (A,B) satisfies
49 u(j)**H * A = lambda(j) * u(j)**H * B.
50 where u(j)**H is the conjugate-transpose of u(j).
51
53 BALANC (input) CHARACTER*1
54 Specifies the balance option to be performed:
55 = 'N': do not diagonally scale or permute;
56 = 'P': permute only;
57 = 'S': scale only;
58 = 'B': both permute and scale. Computed reciprocal condition
59 numbers will be for the matrices after permuting and/or balanc‐
60 ing. Permuting does not change condition numbers (in exact
61 arithmetic), but balancing does.
62 JOBVL (input) CHARACTER*1
64 = 'N': do not compute the left generalized eigenvectors;
65 = 'V': compute the left generalized eigenvectors.
66 JOBVR (input) CHARACTER*1
68 = 'N': do not compute the right generalized eigenvectors;
69 = 'V': compute the right generalized eigenvectors.
70 SENSE (input) CHARACTER*1
72 Determines which reciprocal condition numbers are computed. =
73 'N': none are computed;
74 = 'E': computed for eigenvalues only;
75 = 'V': computed for eigenvectors only;
76 = 'B': computed for eigenvalues and eigenvectors.
77 N (input) INTEGER
79 The order of the matrices A, B, VL, and VR. N >= 0.
80 A (input/output) COMPLEX array, dimension (LDA, N)
82 On entry, the matrix A in the pair (A,B). On exit, A has been
83 overwritten. If JOBVL='V' or JOBVR='V' or both, then A contains
84 the first part of the complex Schur form of the "balanced" ver‐
85 sions of the input A and B.
86 LDA (input) INTEGER
88 The leading dimension of A. LDA >= max(1,N).
89 B (input/output) COMPLEX array, dimension (LDB, N)
91 On entry, the matrix B in the pair (A,B). On exit, B has been
92 overwritten. If JOBVL='V' or JOBVR='V' or both, then B contains
93 the second part of the complex Schur form of the "balanced"
94 versions of the input A and B.
95 LDB (input) INTEGER
97 The leading dimension of B. LDB >= max(1,N).
98 ALPHA (output) COMPLEX array, dimension (N)
100 BETA (output) COMPLEX array, dimension (N) On exit,
101 ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenval‐
102 ues. Note: the quotient ALPHA(j)/BETA(j) ) may easily over- or
103 underflow, and BETA(j) may even be zero. Thus, the user should
104 avoid naively computing the ratio ALPHA/BETA. However, ALPHA
105 will be always less than and usually comparable with norm(A) in
106 magnitude, and BETA always less than and usually comparable
107 with norm(B).
108 VL (output) COMPLEX array, dimension (LDVL,N)
110 If JOBVL = 'V', the left generalized eigenvectors u(j) are
111 stored one after another in the columns of VL, in the same
112 order as their eigenvalues. Each eigenvector will be scaled so
113 the largest component will have abs(real part) + abs(imag.
114 part) = 1. Not referenced if JOBVL = 'N'.
115 LDVL (input) INTEGER
117 The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL
118 = 'V', LDVL >= N.
119 VR (output) COMPLEX array, dimension (LDVR,N)
121 If JOBVR = 'V', the right generalized eigenvectors v(j) are
122 stored one after another in the columns of VR, in the same
123 order as their eigenvalues. Each eigenvector will be scaled so
124 the largest component will have abs(real part) + abs(imag.
125 part) = 1. Not referenced if JOBVR = 'N'.
126 LDVR (input) INTEGER
128 The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR
129 = 'V', LDVR >= N.
130 ILO (output) INTEGER
132 IHI (output) INTEGER ILO and IHI are integer values such
133 that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j =
134 1,...,ILO-1 or i = IHI+1,...,N. If BALANC = 'N' or 'S', ILO =
135 1 and IHI = N.
136 LSCALE (output) REAL array, dimension (N)
138 Details of the permutations and scaling factors applied to the
139 left side of A and B. If PL(j) is the index of the row inter‐
140 changed with row j, and DL(j) is the scaling factor applied to
141 row j, then LSCALE(j) = PL(j) for j = 1,...,ILO-1 = DL(j) for
142 j = ILO,...,IHI = PL(j) for j = IHI+1,...,N. The order in
143 which the interchanges are made is N to IHI+1, then 1 to ILO-1.
144 RSCALE (output) REAL array, dimension (N)
146 Details of the permutations and scaling factors applied to the
147 right side of A and B. If PR(j) is the index of the column
148 interchanged with column j, and DR(j) is the scaling factor
149 applied to column j, then RSCALE(j) = PR(j) for j =
150 1,...,ILO-1 = DR(j) for j = ILO,...,IHI = PR(j) for j =
151 IHI+1,...,N The order in which the interchanges are made is N
152 to IHI+1, then 1 to ILO-1.
153 ABNRM (output) REAL
155 The one-norm of the balanced matrix A.
156 BBNRM (output) REAL
158 The one-norm of the balanced matrix B.
159 RCONDE (output) REAL array, dimension (N)
161 If SENSE = 'E' or 'B', the reciprocal condition numbers of the
162 eigenvalues, stored in consecutive elements of the array. If
163 SENSE = 'N' or 'V', RCONDE is not referenced.
164 RCONDV (output) REAL array, dimension (N)
166 If SENSE = 'V' or 'B', the estimated reciprocal condition num‐
167 bers of the eigenvectors, stored in consecutive elements of the
168 array. If the eigenvalues cannot be reordered to compute
169 RCONDV(j), RCONDV(j) is set to 0; this can only occur when the
170 true value would be very small anyway. If SENSE = 'N' or 'E',
171 RCONDV is not referenced.
172 WORK (workspace/output) COMPLEX array, dimension (MAX(1,LWORK))
174 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
175 LWORK (input) INTEGER
177 The dimension of the array WORK. LWORK >= max(1,2*N). If SENSE
178 = 'E', LWORK >= max(1,4*N). If SENSE = 'V' or 'B', LWORK >=
179 max(1,2*N*N+2*N). If LWORK = -1, then a workspace query is
180 assumed; the routine only calculates the optimal size of the
181 WORK array, returns this value as the first entry of the WORK
182 array, and no error message related to LWORK is issued by
183 XERBLA.
184 RWORK (workspace) REAL array, dimension (lrwork)
186 lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B', and
187 at least max(1,2*N) otherwise. Real workspace.
188 IWORK (workspace) INTEGER array, dimension (N+2)
190 If SENSE = 'E', IWORK is not referenced.
191 BWORK (workspace) LOGICAL array, dimension (N)
193 If SENSE = 'N', BWORK is not referenced.
194 INFO (output) INTEGER
196 = 0: successful exit
197 < 0: if INFO = -i, the i-th argument had an illegal value.
198 = 1,...,N: The QZ iteration failed. No eigenvectors have been
199 calculated, but ALPHA(j) and BETA(j) should be correct for
200 j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed in
201 CHGEQZ.
202 =N+2: error return from CTGEVC.
203
205 Balancing a matrix pair (A,B) includes, first, permuting rows and col‐
206 umns to isolate eigenvalues, second, applying diagonal similarity
207 transformation to the rows and columns to make the rows and columns as
208 close in norm as possible. The computed reciprocal condition numbers
209 correspond to the balanced matrix. Permuting rows and columns will not
210 change the condition numbers (in exact arithmetic) but diagonal scaling
211 will. For further explanation of balancing, see section 4.11.1.2 of
212 LAPACK Users' Guide.
213 An approximate error bound on the chordal distance between the i-th
214 computed generalized eigenvalue w and the corresponding exact eigenval‐
215 ue lambda is
216 chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I) An
217 approximate error bound for the angle between the i-th computed eigen‐
218 vector VL(i) or VR(i) is given by
219 EPS * norm(ABNRM, BBNRM) / DIF(i).
220 For further explanation of the reciprocal condition numbers RCONDE and
221 RCONDV, see section 4.11 of LAPACK User's Guide.
222 LAPACK driver routine (version 3.N2o)vember 2008 CGGEVX(1)