1DGEEVX(1) LAPACK driver routine (version 3.2) DGEEVX(1)
2
6 DGEEVX - computes for an N-by-N real nonsymmetric matrix A, the eigen‐
7 values and, optionally, the left and/or right eigenvectors
8
10 SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, VL,
11 LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE,
12 RCONDV, WORK, LWORK, IWORK, INFO )
13 CHARACTER BALANC, JOBVL, JOBVR, SENSE
15 INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
17 DOUBLE PRECISION ABNRM
19 INTEGER IWORK( * )
21 DOUBLE PRECISION A( LDA, * ), RCONDE( * ), RCONDV( * ),
23 SCALE( * ), VL( LDVL, * ), VR( LDVR, * ), WI( * ),
24 WORK( * ), WR( * )
25
27 DGEEVX computes for an N-by-N real nonsymmetric matrix A, the eigenval‐
28 ues and, optionally, the left and/or right eigenvectors. Optionally
29 also, it computes a balancing transformation to improve the condition‐
30 ing of the eigenvalues and eigenvectors (ILO, IHI, SCALE, and ABNRM),
31 reciprocal condition numbers for the eigenvalues (RCONDE), and recipro‐
32 cal condition numbers for the right
33 eigenvectors (RCONDV).
34 The right eigenvector v(j) of A satisfies
35 A * v(j) = lambda(j) * v(j)
36 where lambda(j) is its eigenvalue.
37 The left eigenvector u(j) of A satisfies
38 u(j)**H * A = lambda(j) * u(j)**H
39 where u(j)**H denotes the conjugate transpose of u(j).
40 The computed eigenvectors are normalized to have Euclidean norm equal
41 to 1 and largest component real.
42 Balancing a matrix means permuting the rows and columns to make it more
43 nearly upper triangular, and applying a diagonal similarity transforma‐
44 tion D * A * D**(-1), where D is a diagonal matrix, to make its rows
45 and columns closer in norm and the condition numbers of its eigenvalues
46 and eigenvectors smaller. The computed reciprocal condition numbers
47 correspond to the balanced matrix. Permuting rows and columns will not
48 change the condition numbers (in exact arithmetic) but diagonal scaling
49 will. For further explanation of balancing, see section 4.10.2 of the
50 LAPACK Users' Guide.
51
53 BALANC (input) CHARACTER*1
54 Indicates how the input matrix should be diagonally scaled
55 and/or permuted to improve the conditioning of its eigenvalues.
56 = 'N': Do not diagonally scale or permute;
57 = 'P': Perform permutations to make the matrix more nearly
58 upper triangular. Do not diagonally scale; = 'S': Diagonally
59 scale the matrix, i.e. replace A by D*A*D**(-1), where D is a
60 diagonal matrix chosen to make the rows and columns of A more
61 equal in norm. Do not permute; = 'B': Both diagonally scale and
62 permute A. Computed reciprocal condition numbers will be for
63 the matrix after balancing and/or permuting. Permuting does not
64 change condition numbers (in exact arithmetic), but balancing
65 does.
66 JOBVL (input) CHARACTER*1
68 = 'N': left eigenvectors of A are not computed;
69 = 'V': left eigenvectors of A are computed. If SENSE = 'E' or
70 'B', JOBVL must = 'V'.
71 JOBVR (input) CHARACTER*1
73 = 'N': right eigenvectors of A are not computed;
74 = 'V': right eigenvectors of A are computed. If SENSE = 'E' or
75 'B', JOBVR must = 'V'.
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 right eigenvectors only;
82 = 'B': Computed for eigenvalues and right eigenvectors. If
83 SENSE = 'E' or 'B', both left and right eigenvectors must also
84 be computed (JOBVL = 'V' and JOBVR = 'V').
85 N (input) INTEGER
87 The order of the matrix A. N >= 0.
88 A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
90 On entry, the N-by-N matrix A. On exit, A has been overwrit‐
91 ten. If JOBVL = 'V' or JOBVR = 'V', A contains the real Schur
92 form of the balanced version of the input matrix A.
93 LDA (input) INTEGER
95 The leading dimension of the array A. LDA >= max(1,N).
96 WR (output) DOUBLE PRECISION array, dimension (N)
98 WI (output) DOUBLE PRECISION array, dimension (N) WR and
99 WI contain the real and imaginary parts, respectively, of the
100 computed eigenvalues. Complex conjugate pairs of eigenvalues
101 will appear consecutively with the eigenvalue having the posi‐
102 tive imaginary part first.
103 VL (output) DOUBLE PRECISION array, dimension (LDVL,N)
105 If JOBVL = 'V', the left eigenvectors u(j) are stored one after
106 another in the columns of VL, in the same order as their eigen‐
107 values. If JOBVL = 'N', VL is not referenced. If the j-th ei‐
108 genvalue is real, then u(j) = VL(:,j), the j-th column of VL.
109 If the j-th and (j+1)-st eigenvalues form a complex conjugate
110 pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
111 u(j+1) = VL(:,j) - i*VL(:,j+1).
112 LDVL (input) INTEGER
114 The leading dimension of the array VL. LDVL >= 1; if JOBVL =
115 'V', LDVL >= N.
116 VR (output) DOUBLE PRECISION array, dimension (LDVR,N)
118 If JOBVR = 'V', the right eigenvectors v(j) are stored one
119 after another in the columns of VR, in the same order as their
120 eigenvalues. If JOBVR = 'N', VR is not referenced. If the j-
121 th eigenvalue is real, then v(j) = VR(:,j), the j-th column of
122 VR. If the j-th and (j+1)-st eigenvalues form a complex conju‐
123 gate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
124 v(j+1) = VR(:,j) - i*VR(:,j+1).
125 LDVR (input) INTEGER
127 The leading dimension of the array VR. LDVR >= 1, and if JOBVR
128 = 'V', LDVR >= N.
129 ILO (output) INTEGER
131 IHI (output) INTEGER ILO and IHI are integer values deter‐
132 mined when A was balanced. The balanced A(i,j) = 0 if I > J
133 and J = 1,...,ILO-1 or I = IHI+1,...,N.
134 SCALE (output) DOUBLE PRECISION array, dimension (N)
136 Details of the permutations and scaling factors applied when
137 balancing A. If P(j) is the index of the row and column inter‐
138 changed with row and column j, and D(j) is the scaling factor
139 applied to row and column j, then SCALE(J) = P(J), for J =
140 1,...,ILO-1 = D(J), for J = ILO,...,IHI = P(J) for J =
141 IHI+1,...,N. The order in which the interchanges are made is N
142 to IHI+1, then 1 to ILO-1.
143 ABNRM (output) DOUBLE PRECISION
145 The one-norm of the balanced matrix (the maximum of the sum of
146 absolute values of elements of any column).
147 RCONDE (output) DOUBLE PRECISION array, dimension (N)
149 RCONDE(j) is the reciprocal condition number of the j-th eigen‐
150 value.
151 RCONDV (output) DOUBLE PRECISION array, dimension (N)
153 RCONDV(j) is the reciprocal condition number of the j-th right
154 eigenvector.
155 WORK (workspace/output) DOUBLE PRECISION array, dimension
157 (MAX(1,LWORK))
158 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
159 LWORK (input) INTEGER
161 The dimension of the array WORK. If SENSE = 'N' or 'E', LWORK
162 >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V', LWORK >= 3*N.
163 If SENSE = 'V' or 'B', LWORK >= N*(N+6). For good performance,
164 LWORK must generally be larger. If LWORK = -1, then a
165 workspace query is assumed; the routine only calculates the
166 optimal size of the WORK array, returns this value as the first
167 entry of the WORK array, and no error message related to LWORK
168 is issued by XERBLA.
169 IWORK (workspace) INTEGER array, dimension (2*N-2)
171 If SENSE = 'N' or 'E', not referenced.
172 INFO (output) INTEGER
174 = 0: successful exit
175 < 0: if INFO = -i, the i-th argument had an illegal value.
176 > 0: if INFO = i, the QR algorithm failed to compute all the
177 eigenvalues, and no eigenvectors or condition numbers have been
178 computed; elements 1:ILO-1 and i+1:N of WR and WI contain ei‐
179 genvalues which have converged.
180 LAPACK driver routine (version 3.N2o)vember 2008 DGEEVX(1)