1SGGEVX(1) LAPACK driver routine (version 3.2) SGGEVX(1)
2
6 SGGEVX - computes 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
33 conditioning of the eigenvalues and eigenvectors (ILO, IHI, LSCALE,
34 RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for the eigen‐
35 values (RCONDE), and reciprocal condition numbers for the right eigen‐
36 vectors (RCONDV).
37 A generalized eigenvalue for a pair of matrices (A,B) is a scalar
38 lambda or a ratio alpha/beta = lambda, such that A - lambda*B is singu‐
39 lar. It is usually represented as the pair (alpha,beta), as there is a
40 reasonable interpretation for beta=0, and even for both being zero.
41 The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of
42 (A,B) satisfies
43 A * v(j) = lambda(j) * B * v(j) .
44 The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of
45 (A,B) satisfies
46 u(j)**H * A = lambda(j) * u(j)**H * B.
47 where u(j)**H is the conjugate-transpose of u(j).
48
50 BALANC (input) CHARACTER*1
51 Specifies the balance option to be performed. = 'N': do not
52 diagonally scale or permute;
53 = 'P': permute only;
54 = 'S': scale only;
55 = 'B': both permute and scale. Computed reciprocal condition
56 numbers will be for the matrices after permuting and/or balanc‐
57 ing. Permuting does not change condition numbers (in exact
58 arithmetic), but balancing does.
59 JOBVL (input) CHARACTER*1
61 = 'N': do not compute the left generalized eigenvectors;
62 = 'V': compute the left generalized eigenvectors.
63 JOBVR (input) CHARACTER*1
65 = 'N': do not compute the right generalized eigenvectors;
66 = 'V': compute the right generalized eigenvectors.
67 SENSE (input) CHARACTER*1
69 Determines which reciprocal condition numbers are computed. =
70 'N': none are computed;
71 = 'E': computed for eigenvalues only;
72 = 'V': computed for eigenvectors only;
73 = 'B': computed for eigenvalues and eigenvectors.
74 N (input) INTEGER
76 The order of the matrices A, B, VL, and VR. N >= 0.
77 A (input/output) REAL array, dimension (LDA, N)
79 On entry, the matrix A in the pair (A,B). On exit, A has been
80 overwritten. If JOBVL='V' or JOBVR='V' or both, then A contains
81 the first part of the real Schur form of the "balanced" ver‐
82 sions of the input A and B.
83 LDA (input) INTEGER
85 The leading dimension of A. LDA >= max(1,N).
86 B (input/output) REAL array, dimension (LDB, N)
88 On entry, the matrix B in the pair (A,B). On exit, B has been
89 overwritten. If JOBVL='V' or JOBVR='V' or both, then B contains
90 the second part of the real Schur form of the "balanced" ver‐
91 sions of the input A and B.
92 LDB (input) INTEGER
94 The leading dimension of B. LDB >= max(1,N).
95 ALPHAR (output) REAL array, dimension (N)
97 ALPHAI (output) REAL array, dimension (N) BETA (output)
98 REAL array, dimension (N) On exit, (ALPHAR(j) +
99 ALPHAI(j)*i)/BETA(j), j=1,...,N, will be the generalized eigen‐
100 values. If ALPHAI(j) is zero, then the j-th eigenvalue is
101 real; if positive, then the j-th and (j+1)-st eigenvalues are a
102 complex conjugate pair, with ALPHAI(j+1) negative. Note: the
103 quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j) may easily
104 over- or underflow, and BETA(j) may even be zero. Thus, the
105 user should avoid naively computing the ratio ALPHA/BETA. How‐
106 ever, ALPHAR and ALPHAI will be always less than and usually
107 comparable with norm(A) in magnitude, and BETA always less than
108 and usually comparable with norm(B).
109 VL (output) REAL array, dimension (LDVL,N)
111 If JOBVL = 'V', the left eigenvectors u(j) are stored one after
112 another in the columns of VL, in the same order as their eigen‐
113 values. If the j-th eigenvalue is real, then u(j) = VL(:,j),
114 the j-th column of VL. If the j-th and (j+1)-th eigenvalues
115 form a complex conjugate pair, then u(j) = VL(:,j)+i*VL(:,j+1)
116 and u(j+1) = VL(:,j)-i*VL(:,j+1). Each eigenvector will be
117 scaled so the largest component have abs(real part) + abs(imag.
118 part) = 1. Not referenced if JOBVL = 'N'.
119 LDVL (input) INTEGER
121 The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL
122 = 'V', LDVL >= N.
123 VR (output) REAL array, dimension (LDVR,N)
125 If JOBVR = 'V', the right eigenvectors v(j) are stored one
126 after another in the columns of VR, in the same order as their
127 eigenvalues. If the j-th eigenvalue is real, then v(j) =
128 VR(:,j), the j-th column of VR. If the j-th and (j+1)-th eigen‐
129 values form a complex conjugate pair, then v(j) =
130 VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1). Each
131 eigenvector will be scaled so the largest component have
132 abs(real part) + abs(imag. part) = 1. Not referenced if JOBVR
133 = '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) REAL 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) REAL 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) REAL
163 The one-norm of the balanced matrix A.
164 BBNRM (output) REAL
166 The one-norm of the balanced matrix B.
167 RCONDE (output) REAL array, dimension (N)
169 If SENSE = 'E' or 'B', the reciprocal condition numbers of the
170 eigenvalues, stored in consecutive elements of the array. For
171 a complex conjugate pair of eigenvalues two consecutive ele‐
172 ments of RCONDE are set to the same value. Thus RCONDE(j),
173 RCONDV(j), and the j-th columns of VL and VR all correspond to
174 the j-th eigenpair. If SENSE = 'N' or 'V', RCONDE is not ref‐
175 erenced.
176 RCONDV (output) REAL array, dimension (N)
178 If SENSE = 'V' or 'B', the estimated reciprocal condition num‐
179 bers of the eigenvectors, stored in consecutive elements of the
180 array. For a complex eigenvector two consecutive elements of
181 RCONDV are set to the same value. If the eigenvalues cannot be
182 reordered to compute RCONDV(j), RCONDV(j) is set to 0; this can
183 only occur when the true value would be very small anyway. If
184 SENSE = 'N' or 'E', RCONDV is not referenced.
185 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
187 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
188 LWORK (input) INTEGER
190 The dimension of the array WORK. LWORK >= max(1,2*N). If BAL‐
191 ANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V', LWORK >=
192 max(1,6*N). If SENSE = 'E', LWORK >= max(1,10*N). If SENSE =
193 'V' or 'B', LWORK >= 2*N*N+8*N+16. If LWORK = -1, then a
194 workspace query is assumed; the routine only calculates the
195 optimal size of the WORK array, returns this value as the first
196 entry of the WORK array, and no error message related to LWORK
197 is issued by XERBLA.
198 IWORK (workspace) INTEGER array, dimension (N+6)
200 If SENSE = 'E', IWORK is not referenced.
201 BWORK (workspace) LOGICAL array, dimension (N)
203 If SENSE = 'N', BWORK is not referenced.
204 INFO (output) INTEGER
206 = 0: successful exit
207 < 0: if INFO = -i, the i-th argument had an illegal value.
208 = 1,...,N: The QZ iteration failed. No eigenvectors have been
209 calculated, but ALPHAR(j), ALPHAI(j), and BETA(j) should be
210 correct for j=INFO+1,...,N. > N: =N+1: other than QZ itera‐
211 tion failed in SHGEQZ.
212 =N+2: error return from STGEVC.
213
215 Balancing a matrix pair (A,B) includes, first, permuting rows and col‐
216 umns to isolate eigenvalues, second, applying diagonal similarity
217 transformation to the rows and columns to make the rows and columns as
218 close in norm as possible. The computed reciprocal condition numbers
219 correspond to the balanced matrix. Permuting rows and columns will not
220 change the condition numbers (in exact arithmetic) but diagonal scaling
221 will. For further explanation of balancing, see section 4.11.1.2 of
222 LAPACK Users' Guide.
223 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) An
227 approximate error bound for the angle between the i-th computed eigen‐
228 vector VL(i) or VR(i) is given by
229 EPS * norm(ABNRM, BBNRM) / DIF(i).
230 For further explanation of the reciprocal condition numbers RCONDE and
231 RCONDV, see section 4.11 of LAPACK User's Guide.
232 LAPACK driver routine (version 3.N2o)vember 2008 SGGEVX(1)