1DGGEVX(1) LAPACK driver routine (version 3.1) DGGEVX(1)
2
6 DGGEVX - for a pair of N-by-N real nonsymmetric matrices (A,B)
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9 SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
10 ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO, IHI,
11 LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK,
12 LWORK, IWORK, BWORK, INFO )
13 CHARACTER BALANC, JOBVL, JOBVR, SENSE
15 INTEGER IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
17 DOUBLE PRECISION ABNRM, BBNRM
19 LOGICAL BWORK( * )
21 INTEGER IWORK( * )
23 DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B(
25 LDB, * ), BETA( * ), LSCALE( * ), RCONDE( * ),
26 RCONDV( * ), RSCALE( * ), VL( LDVL, * ), VR( LDVR, *
27 ), WORK( * )
28
30 DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
31 the generalized eigenvalues, and optionally, the left and/or right gen‐
32 eralized eigenvectors.
33 Optionally also, it computes a balancing transformation to improve the
35 conditioning of the eigenvalues and eigenvectors (ILO, IHI, LSCALE,
36 RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for the eigen‐
37 values (RCONDE), and reciprocal condition numbers for the right eigen‐
38 vectors (RCONDV).
39 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
46 (A,B) satisfies
47 A * v(j) = lambda(j) * B * v(j) .
49 The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of
51 (A,B) satisfies
52 u(j)**H * A = lambda(j) * u(j)**H * B.
54 where u(j)**H is the conjugate-transpose of u(j).
56
60 BALANC (input) CHARACTER*1
61 Specifies the balance option to be performed. = 'N': do not
62 diagonally scale or permute;
63 = 'P': permute only;
64 = 'S': scale only;
65 = 'B': both permute and scale. Computed reciprocal condition
66 numbers will be for the matrices after permuting and/or balanc‐
67 ing. Permuting does not change condition numbers (in exact
68 arithmetic), but balancing does.
69 JOBVL (input) CHARACTER*1
71 = 'N': do not compute the left generalized eigenvectors;
72 = 'V': compute the left generalized eigenvectors.
73 JOBVR (input) CHARACTER*1
75 = 'N': do not compute the right generalized eigenvectors;
76 = 'V': compute the right generalized eigenvectors.
77 SENSE (input) CHARACTER*1
79 Determines which reciprocal condition numbers are computed. =
80 'N': none are computed;
81 = 'E': computed for eigenvalues only;
82 = 'V': computed for eigenvectors only;
83 = 'B': computed for eigenvalues and eigenvectors.
84 N (input) INTEGER
86 The order of the matrices A, B, VL, and VR. N >= 0.
87 A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
89 On entry, the matrix A in the pair (A,B). On exit, A has been
90 overwritten. If JOBVL='V' or JOBVR='V' or both, then A contains
91 the first part of the real Schur form of the "balanced" ver‐
92 sions of the input A and B.
93 LDA (input) INTEGER
95 The leading dimension of A. LDA >= max(1,N).
96 B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
98 On entry, the matrix B in the pair (A,B). On exit, B has been
99 overwritten. If JOBVL='V' or JOBVR='V' or both, then B contains
100 the second part of the real Schur form of the "balanced" ver‐
101 sions of the input A and B.
102 LDB (input) INTEGER
104 The leading dimension of B. LDB >= max(1,N).
105 ALPHAR (output) DOUBLE PRECISION array, dimension (N)
107 ALPHAI (output) DOUBLE PRECISION array, dimension (N) BETA
108 (output) DOUBLE PRECISION array, dimension (N) On exit,
109 (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the gen‐
110 eralized eigenvalues. If ALPHAI(j) is zero, then the j-th ei‐
111 genvalue is real; if positive, then the j-th and (j+1)-st ei‐
112 genvalues are a complex conjugate pair, with ALPHAI(j+1) nega‐
113 tive.
114 Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may
116 easily over- or underflow, and BETA(j) may even be zero. Thus,
117 the user should avoid naively computing the ratio ALPHA/BETA.
118 However, ALPHAR and ALPHAI will be always less than and usually
119 comparable with norm(A) in magnitude, and BETA always less than
120 and usually comparable with norm(B).
121 VL (output) DOUBLE PRECISION array, dimension (LDVL,N)
123 If JOBVL = 'V', the left eigenvectors u(j) are stored one after
124 another in the columns of VL, in the same order as their eigen‐
125 values. If the j-th eigenvalue is real, then u(j) = VL(:,j),
126 the j-th column of VL. If the j-th and (j+1)-th eigenvalues
127 form a complex conjugate pair, then u(j) = VL(:,j)+i*VL(:,j+1)
128 and u(j+1) = VL(:,j)-i*VL(:,j+1). Each eigenvector will be
129 scaled so the largest component have abs(real part) + abs(imag.
130 part) = 1. Not referenced if JOBVL = 'N'.
131 LDVL (input) INTEGER
133 The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL
134 = 'V', LDVL >= N.
135 VR (output) DOUBLE PRECISION array, dimension (LDVR,N)
137 If JOBVR = 'V', the right eigenvectors v(j) are stored one
138 after another in the columns of VR, in the same order as their
139 eigenvalues. If the j-th eigenvalue is real, then v(j) =
140 VR(:,j), the j-th column of VR. If the j-th and (j+1)-th eigen‐
141 values form a complex conjugate pair, then v(j) =
142 VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). Each
143 eigenvector will be scaled so the largest component have
144 abs(real part) + abs(imag. part) = 1. Not referenced if JOBVR
145 = 'N'.
146 LDVR (input) INTEGER
148 The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR
149 = 'V', LDVR >= N.
150 ILO (output) INTEGER
152 IHI (output) INTEGER ILO and IHI are integer values such
153 that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j =
154 1,...,ILO-1 or i = IHI+1,...,N. If BALANC = 'N' or 'S', ILO =
155 1 and IHI = N.
156 LSCALE (output) DOUBLE PRECISION array, dimension (N)
158 Details of the permutations and scaling factors applied to the
159 left side of A and B. If PL(j) is the index of the row inter‐
160 changed with row j, and DL(j) is the scaling factor applied to
161 row j, then LSCALE(j) = PL(j) for j = 1,...,ILO-1 = DL(j) for
162 j = ILO,...,IHI = PL(j) for j = IHI+1,...,N. The order in
163 which the interchanges are made is N to IHI+1, then 1 to ILO-1.
164 RSCALE (output) DOUBLE PRECISION array, dimension (N)
166 Details of the permutations and scaling factors applied to the
167 right side of A and B. If PR(j) is the index of the column
168 interchanged with column j, and DR(j) is the scaling factor
169 applied to column j, then RSCALE(j) = PR(j) for j =
170 1,...,ILO-1 = DR(j) for j = ILO,...,IHI = PR(j) for j =
171 IHI+1,...,N The order in which the interchanges are made is N
172 to IHI+1, then 1 to ILO-1.
173 ABNRM (output) DOUBLE PRECISION
175 The one-norm of the balanced matrix A.
176 BBNRM (output) DOUBLE PRECISION
178 The one-norm of the balanced matrix B.
179 RCONDE (output) DOUBLE PRECISION array, dimension (N)
181 If SENSE = 'E' or 'B', the reciprocal condition numbers of the
182 eigenvalues, stored in consecutive elements of the array. For
183 a complex conjugate pair of eigenvalues two consecutive ele‐
184 ments of RCONDE are set to the same value. Thus RCONDE(j),
185 RCONDV(j), and the j-th columns of VL and VR all correspond to
186 the j-th eigenpair. If SENSE = 'N or 'V', RCONDE is not refer‐
187 enced.
188 RCONDV (output) DOUBLE PRECISION array, dimension (N)
190 If SENSE = 'V' or 'B', the estimated reciprocal condition num‐
191 bers of the eigenvectors, stored in consecutive elements of the
192 array. For a complex eigenvector two consecutive elements of
193 RCONDV are set to the same value. If the eigenvalues cannot be
194 reordered to compute RCONDV(j), RCONDV(j) is set to 0; this can
195 only occur when the true value would be very small anyway. If
196 SENSE = 'N' or 'E', RCONDV is not referenced.
197 WORK (workspace/output) DOUBLE PRECISION array, dimension
199 (MAX(1,LWORK))
200 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
201 LWORK (input) INTEGER
203 The dimension of the array WORK. LWORK >= max(1,2*N). If BAL‐
204 ANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V', LWORK >=
205 max(1,6*N). If SENSE = 'E' or 'B', LWORK >= max(1,10*N). If
206 SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
207 If LWORK = -1, then a workspace query is assumed; the routine
209 only calculates the optimal size of the WORK array, returns
210 this value as the first entry of the WORK array, and no error
211 message related to LWORK is issued by XERBLA.
212 IWORK (workspace) INTEGER array, dimension (N+6)
214 If SENSE = 'E', IWORK is not referenced.
215 BWORK (workspace) LOGICAL array, dimension (N)
217 If SENSE = 'N', BWORK is not referenced.
218 INFO (output) INTEGER
220 = 0: successful exit
221 < 0: if INFO = -i, the i-th argument had an illegal value.
222 = 1,...,N: The QZ iteration failed. No eigenvectors have been
223 calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) should be
224 correct for j=INFO+1,...,N. > N: =N+1: other than QZ itera‐
225 tion failed in DHGEQZ.
226 =N+2: error return from DTGEVC.
227
229 Balancing a matrix pair (A,B) includes, first, permuting rows and col‐
230 umns to isolate eigenvalues, second, applying diagonal similarity
231 transformation to the rows and columns to make the rows and columns as
232 close in norm as possible. The computed reciprocal condition numbers
233 correspond to the balanced matrix. Permuting rows and columns will not
234 change the condition numbers (in exact arithmetic) but diagonal scaling
235 will. For further explanation of balancing, see section 4.11.1.2 of
236 LAPACK Users' Guide.
237 An approximate error bound on the chordal distance between the i-th
239 computed generalized eigenvalue w and the corresponding exact eigenval‐
240 ue lambda is
241 chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
243 An approximate error bound for the angle between the i-th computed
245 eigenvector VL(i) or VR(i) is given by
246 EPS * norm(ABNRM, BBNRM) / DIF(i).
248 For further explanation of the reciprocal condition numbers RCONDE and
250 RCONDV, see section 4.11 of LAPACK User's Guide.
251 LAPACK driver routine (version 3.N1o)vember 2006 DGGEVX(1)