1ZGGEVX(1) LAPACK driver routine (version 3.1) ZGGEVX(1)
2
6 ZGGEVX - 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 ZGGEVX( 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 DOUBLE PRECISION ABNRM, BBNRM
21 LOGICAL BWORK( * )
23 INTEGER IWORK( * )
25 DOUBLE PRECISION LSCALE( * ), RCONDE( * ), RCONDV( * ),
27 RSCALE( * ), RWORK( * )
28 COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ), BETA( * ), VL(
30 LDVL, * ), VR( LDVR, * ), WORK( * )
31
33 ZGGEVX 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.
36 Optionally, it also computes a balancing transformation to improve the
38 conditioning of the eigenvalues and eigenvectors (ILO, IHI, LSCALE,
39 RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for the eigen‐
40 values (RCONDE), and reciprocal condition numbers for the right eigen‐
41 vectors (RCONDV).
42 A generalized eigenvalue for a pair of matrices (A,B) is a scalar
44 lambda or a ratio alpha/beta = lambda, such that A - lambda*B is singu‐
45 lar. It is usually represented as the pair (alpha,beta), as there is a
46 reasonable interpretation for beta=0, and even for both being zero.
47 The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of
49 (A,B) satisfies
50 A * v(j) = lambda(j) * B * v(j) .
51 The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of
52 (A,B) satisfies
53 u(j)**H * A = lambda(j) * u(j)**H * B.
54 where u(j)**H is the conjugate-transpose of u(j).
55
59 BALANC (input) CHARACTER*1
60 Specifies the balance option to be performed:
61 = 'N': do not diagonally scale or permute;
62 = 'P': permute only;
63 = 'S': scale only;
64 = 'B': both permute and scale. Computed reciprocal condition
65 numbers will be for the matrices after permuting and/or balanc‐
66 ing. Permuting does not change condition numbers (in exact
67 arithmetic), but balancing does.
68 JOBVL (input) CHARACTER*1
70 = 'N': do not compute the left generalized eigenvectors;
71 = 'V': compute the left generalized eigenvectors.
72 JOBVR (input) CHARACTER*1
74 = 'N': do not compute the right generalized eigenvectors;
75 = 'V': compute the right generalized eigenvectors.
76 SENSE (input) CHARACTER*1
78 Determines which reciprocal condition numbers are computed. =
79 'N': none are computed;
80 = 'E': computed for eigenvalues only;
81 = 'V': computed for eigenvectors only;
82 = 'B': computed for eigenvalues and eigenvectors.
83 N (input) INTEGER
85 The order of the matrices A, B, VL, and VR. N >= 0.
86 A (input/output) COMPLEX*16 array, dimension (LDA, N)
88 On entry, the matrix A in the pair (A,B). On exit, A has been
89 overwritten. If JOBVL='V' or JOBVR='V' or both, then A contains
90 the first part of the complex Schur form of the "balanced" ver‐
91 sions of the input A and B.
92 LDA (input) INTEGER
94 The leading dimension of A. LDA >= max(1,N).
95 B (input/output) COMPLEX*16 array, dimension (LDB, N)
97 On entry, the matrix B in the pair (A,B). On exit, B has been
98 overwritten. If JOBVL='V' or JOBVR='V' or both, then B contains
99 the second part of the complex Schur form of the "balanced"
100 versions of the input A and B.
101 LDB (input) INTEGER
103 The leading dimension of B. LDB >= max(1,N).
104 ALPHA (output) COMPLEX*16 array, dimension (N)
106 BETA (output) COMPLEX*16 array, dimension (N) On exit,
107 ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenval‐
108 ues.
109 Note: the quotient ALPHA(j)/BETA(j) ) may easily over- or
111 underflow, and BETA(j) may even be zero. Thus, the user should
112 avoid naively computing the ratio ALPHA/BETA. However, ALPHA
113 will be always less than and usually comparable with norm(A) in
114 magnitude, and BETA always less than and usually comparable
115 with norm(B).
116 VL (output) COMPLEX*16 array, dimension (LDVL,N)
118 If JOBVL = 'V', the left generalized eigenvectors u(j) are
119 stored one after another in the columns of VL, in the same
120 order as their eigenvalues. Each eigenvector will be scaled so
121 the largest component will have abs(real part) + abs(imag.
122 part) = 1. Not referenced if JOBVL = 'N'.
123 LDVL (input) INTEGER
125 The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL
126 = 'V', LDVL >= N.
127 VR (output) COMPLEX*16 array, dimension (LDVR,N)
129 If JOBVR = 'V', the right generalized eigenvectors v(j) are
130 stored one after another in the columns of VR, in the same
131 order as their eigenvalues. Each eigenvector will be scaled so
132 the largest component will have abs(real part) + abs(imag.
133 part) = 1. Not referenced if JOBVR = 'N'.
134 LDVR (input) INTEGER
136 The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR
137 = 'V', LDVR >= N.
138 ILO (output) INTEGER
140 IHI (output) INTEGER ILO and IHI are integer values such
141 that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j =
142 1,...,ILO-1 or i = IHI+1,...,N. If BALANC = 'N' or 'S', ILO =
143 1 and IHI = N.
144 LSCALE (output) DOUBLE PRECISION array, dimension (N)
146 Details of the permutations and scaling factors applied to the
147 left side of A and B. If PL(j) is the index of the row inter‐
148 changed with row j, and DL(j) is the scaling factor applied to
149 row j, then LSCALE(j) = PL(j) for j = 1,...,ILO-1 = DL(j) for
150 j = ILO,...,IHI = PL(j) for j = IHI+1,...,N. The order in
151 which the interchanges are made is N to IHI+1, then 1 to ILO-1.
152 RSCALE (output) DOUBLE PRECISION array, dimension (N)
154 Details of the permutations and scaling factors applied to the
155 right side of A and B. If PR(j) is the index of the column
156 interchanged with column j, and DR(j) is the scaling factor
157 applied to column j, then RSCALE(j) = PR(j) for j =
158 1,...,ILO-1 = DR(j) for j = ILO,...,IHI = PR(j) for j =
159 IHI+1,...,N The order in which the interchanges are made is N
160 to IHI+1, then 1 to ILO-1.
161 ABNRM (output) DOUBLE PRECISION
163 The one-norm of the balanced matrix A.
164 BBNRM (output) DOUBLE PRECISION
166 The one-norm of the balanced matrix B.
167 RCONDE (output) DOUBLE PRECISION array, dimension (N)
169 If SENSE = 'E' or 'B', the reciprocal condition numbers of the
170 eigenvalues, stored in consecutive elements of the array. If
171 SENSE = 'N' or 'V', RCONDE is not referenced.
172 RCONDV (output) DOUBLE PRECISION array, dimension (N)
174 If JOB = 'V' or 'B', the estimated reciprocal condition numbers
175 of the eigenvectors, stored in consecutive elements of the
176 array. If the eigenvalues cannot be reordered to compute
177 RCONDV(j), RCONDV(j) is set to 0; this can only occur when the
178 true value would be very small anyway. If SENSE = 'N' or 'E',
179 RCONDV is not referenced.
180 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
182 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
183 LWORK (input) INTEGER
185 The dimension of the array WORK. LWORK >= max(1,2*N). If SENSE
186 = 'E', LWORK >= max(1,4*N). If SENSE = 'V' or 'B', LWORK >=
187 max(1,2*N*N+2*N).
188 If LWORK = -1, then a workspace query is assumed; the routine
190 only calculates the optimal size of the WORK array, returns
191 this value as the first entry of the WORK array, and no error
192 message related to LWORK is issued by XERBLA.
193 RWORK (workspace) REAL array, dimension (lrwork)
195 lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B', and
196 at least max(1,2*N) otherwise. Real workspace.
197 IWORK (workspace) INTEGER array, dimension (N+2)
199 If SENSE = 'E', IWORK is not referenced.
200 BWORK (workspace) LOGICAL array, dimension (N)
202 If SENSE = 'N', BWORK is not referenced.
203 INFO (output) INTEGER
205 = 0: successful exit
206 < 0: if INFO = -i, the i-th argument had an illegal value.
207 = 1,...,N: The QZ iteration failed. No eigenvectors have been
208 calculated, but ALPHA(j) and BETA(j) should be correct for
209 j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed in
210 ZHGEQZ.
211 =N+2: error return from ZTGEVC.
212
214 Balancing a matrix pair (A,B) includes, first, permuting rows and col‐
215 umns to isolate eigenvalues, second, applying diagonal similarity
216 transformation to the rows and columns to make the rows and columns as
217 close in norm as possible. The computed reciprocal condition numbers
218 correspond to the balanced matrix. Permuting rows and columns will not
219 change the condition numbers (in exact arithmetic) but diagonal scaling
220 will. For further explanation of balancing, see section 4.11.1.2 of
221 LAPACK Users' Guide.
222 An approximate error bound on the chordal distance between the i-th
224 computed generalized eigenvalue w and the corresponding exact eigenval‐
225 ue lambda is
226 chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
228 An approximate error bound for the angle between the i-th computed
230 eigenvector VL(i) or VR(i) is given by
231 EPS * norm(ABNRM, BBNRM) / DIF(i).
233 For further explanation of the reciprocal condition numbers RCONDE and
235 RCONDV, see section 4.11 of LAPACK User's Guide.
236 LAPACK driver routine (version 3.N1o)vember 2006 ZGGEVX(1)