1SGGEVX(1) LAPACK driver routine (version 3.1) SGGEVX(1)
2
6 SGGEVX - for a pair of N-by-N real nonsymmetric matrices (A,B)
7
9 SUBROUTINE SGGEVX( 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 REAL ABNRM, BBNRM
19 LOGICAL BWORK( * )
21 INTEGER IWORK( * )
23 REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ), B( LDB, * ),
25 BETA( * ), LSCALE( * ), RCONDE( * ), RCONDV( * ),
26 RSCALE( * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )
27
29 SGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
30 the generalized eigenvalues, and optionally, the left and/or right gen‐
31 eralized eigenvectors.
32 Optionally also, it computes a balancing transformation to improve the
34 conditioning of the eigenvalues and eigenvectors (ILO, IHI, LSCALE,
35 RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for the eigen‐
36 values (RCONDE), and reciprocal condition numbers for the right eigen‐
37 vectors (RCONDV).
38 A generalized eigenvalue for a pair of matrices (A,B) is a scalar
40 lambda or a ratio alpha/beta = lambda, such that A - lambda*B is singu‐
41 lar. It is usually represented as the pair (alpha,beta), as there is a
42 reasonable interpretation for beta=0, and even for both being zero.
43 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) .
48 The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of
50 (A,B) satisfies
51 u(j)**H * A = lambda(j) * u(j)**H * B.
53 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. = 'N': do not
61 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) REAL 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 real 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) REAL 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 real Schur form of the "balanced" ver‐
100 sions of the input A and B.
101 LDB (input) INTEGER
103 The leading dimension of B. LDB >= max(1,N).
104 ALPHAR (output) REAL array, dimension (N)
106 ALPHAI (output) REAL array, dimension (N) BETA (output)
107 REAL array, dimension (N) On exit, (ALPHAR(j) +
108 ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigen‐
109 values. If ALPHAI(j) is zero, then the j-th eigenvalue is
110 real; if positive, then the j-th and (j+1)-st eigenvalues are a
111 complex conjugate pair, with ALPHAI(j+1) negative.
112 Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may
114 easily over- or underflow, and BETA(j) may even be zero. Thus,
115 the user should avoid naively computing the ratio ALPHA/BETA.
116 However, ALPHAR and ALPHAI will be always less than and usually
117 comparable with norm(A) in magnitude, and BETA always less than
118 and usually comparable with norm(B).
119 VL (output) REAL array, dimension (LDVL,N)
121 If JOBVL = 'V', the left eigenvectors u(j) are stored one after
122 another in the columns of VL, in the same order as their eigen‐
123 values. If the j-th eigenvalue is real, then u(j) = VL(:,j),
124 the j-th column of VL. If the j-th and (j+1)-th eigenvalues
125 form a complex conjugate pair, then u(j) = VL(:,j)+i*VL(:,j+1)
126 and u(j+1) = VL(:,j)-i*VL(:,j+1). Each eigenvector will be
127 scaled so the largest component have abs(real part) + abs(imag.
128 part) = 1. Not referenced if JOBVL = 'N'.
129 LDVL (input) INTEGER
131 The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL
132 = 'V', LDVL >= N.
133 VR (output) REAL array, dimension (LDVR,N)
135 If JOBVR = 'V', the right eigenvectors v(j) are stored one
136 after another in the columns of VR, in the same order as their
137 eigenvalues. If the j-th eigenvalue is real, then v(j) =
138 VR(:,j), the j-th column of VR. If the j-th and (j+1)-th eigen‐
139 values form a complex conjugate pair, then v(j) =
140 VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). Each
141 eigenvector will be scaled so the largest component have
142 abs(real part) + abs(imag. part) = 1. Not referenced if JOBVR
143 = 'N'.
144 LDVR (input) INTEGER
146 The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR
147 = 'V', LDVR >= N.
148 ILO (output) INTEGER
150 IHI (output) INTEGER ILO and IHI are integer values such
151 that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j =
152 1,...,ILO-1 or i = IHI+1,...,N. If BALANC = 'N' or 'S', ILO =
153 1 and IHI = N.
154 LSCALE (output) REAL array, dimension (N)
156 Details of the permutations and scaling factors applied to the
157 left side of A and B. If PL(j) is the index of the row inter‐
158 changed with row j, and DL(j) is the scaling factor applied to
159 row j, then LSCALE(j) = PL(j) for j = 1,...,ILO-1 = DL(j) for
160 j = ILO,...,IHI = PL(j) for j = IHI+1,...,N. The order in
161 which the interchanges are made is N to IHI+1, then 1 to ILO-1.
162 RSCALE (output) REAL array, dimension (N)
164 Details of the permutations and scaling factors applied to the
165 right side of A and B. If PR(j) is the index of the column
166 interchanged with column j, and DR(j) is the scaling factor
167 applied to column j, then RSCALE(j) = PR(j) for j =
168 1,...,ILO-1 = DR(j) for j = ILO,...,IHI = PR(j) for j =
169 IHI+1,...,N The order in which the interchanges are made is N
170 to IHI+1, then 1 to ILO-1.
171 ABNRM (output) REAL
173 The one-norm of the balanced matrix A.
174 BBNRM (output) REAL
176 The one-norm of the balanced matrix B.
177 RCONDE (output) REAL array, dimension (N)
179 If SENSE = 'E' or 'B', the reciprocal condition numbers of the
180 eigenvalues, stored in consecutive elements of the array. For
181 a complex conjugate pair of eigenvalues two consecutive ele‐
182 ments of RCONDE are set to the same value. Thus RCONDE(j),
183 RCONDV(j), and the j-th columns of VL and VR all correspond to
184 the j-th eigenpair. If SENSE = 'N' or 'V', RCONDE is not ref‐
185 erenced.
186 RCONDV (output) REAL array, dimension (N)
188 If SENSE = 'V' or 'B', the estimated reciprocal condition num‐
189 bers of the eigenvectors, stored in consecutive elements of the
190 array. For a complex eigenvector two consecutive elements of
191 RCONDV are set to the same value. If the eigenvalues cannot be
192 reordered to compute RCONDV(j), RCONDV(j) is set to 0; this can
193 only occur when the true value would be very small anyway. If
194 SENSE = 'N' or 'E', RCONDV is not referenced.
195 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
197 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
198 LWORK (input) INTEGER
200 The dimension of the array WORK. LWORK >= max(1,2*N). If BAL‐
201 ANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V', LWORK >=
202 max(1,6*N). If SENSE = 'E', LWORK >= max(1,10*N). If SENSE =
203 'V' or 'B', LWORK >= 2*N*N+8*N+16.
204 If LWORK = -1, then a workspace query is assumed; the routine
206 only calculates the optimal size of the WORK array, returns
207 this value as the first entry of the WORK array, and no error
208 message related to LWORK is issued by XERBLA.
209 IWORK (workspace) INTEGER array, dimension (N+6)
211 If SENSE = 'E', IWORK is not referenced.
212 BWORK (workspace) LOGICAL array, dimension (N)
214 If SENSE = 'N', BWORK is not referenced.
215 INFO (output) INTEGER
217 = 0: successful exit
218 < 0: if INFO = -i, the i-th argument had an illegal value.
219 = 1,...,N: The QZ iteration failed. No eigenvectors have been
220 calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) should be
221 correct for j=INFO+1,...,N. > N: =N+1: other than QZ itera‐
222 tion failed in SHGEQZ.
223 =N+2: error return from STGEVC.
224
226 Balancing a matrix pair (A,B) includes, first, permuting rows and col‐
227 umns to isolate eigenvalues, second, applying diagonal similarity
228 transformation to the rows and columns to make the rows and columns as
229 close in norm as possible. The computed reciprocal condition numbers
230 correspond to the balanced matrix. Permuting rows and columns will not
231 change the condition numbers (in exact arithmetic) but diagonal scaling
232 will. For further explanation of balancing, see section 4.11.1.2 of
233 LAPACK Users' Guide.
234 An approximate error bound on the chordal distance between the i-th
236 computed generalized eigenvalue w and the corresponding exact eigenval‐
237 ue lambda is
238 chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
240 An approximate error bound for the angle between the i-th computed
242 eigenvector VL(i) or VR(i) is given by
243 EPS * norm(ABNRM, BBNRM) / DIF(i).
245 For further explanation of the reciprocal condition numbers RCONDE and
247 RCONDV, see section 4.11 of LAPACK User's Guide.
248 LAPACK driver routine (version 3.N1o)vember 2006 SGGEVX(1)