1ZGEEVX(1) LAPACK driver routine (version 3.2) ZGEEVX(1)
2
6 ZGEEVX - computes for an N-by-N complex nonsymmetric matrix A, the ei‐
7 genvalues and, optionally, the left and/or right eigenvectors
8
10 SUBROUTINE ZGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, LDVL,
11 VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONDE, RCONDV,
12 WORK, LWORK, RWORK, INFO )
13 CHARACTER BALANC, JOBVL, JOBVR, SENSE
15 INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
17 DOUBLE PRECISION ABNRM
19 DOUBLE PRECISION RCONDE( * ), RCONDV( * ), RWORK( * ),
21 SCALE( * )
22 COMPLEX*16 A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), W( * ),
24 WORK( * )
25
27 ZGEEVX computes for an N-by-N complex nonsymmetric matrix A, the eigen‐
28 values 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, ie. 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) COMPLEX*16 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 Schur form
92 of the balanced version of the matrix A.
93 LDA (input) INTEGER
95 The leading dimension of the array A. LDA >= max(1,N).
96 W (output) COMPLEX*16 array, dimension (N)
98 W contains the computed eigenvalues.
99 VL (output) COMPLEX*16 array, dimension (LDVL,N)
101 If JOBVL = 'V', the left eigenvectors u(j) are stored one after
102 another in the columns of VL, in the same order as their eigen‐
103 values. If JOBVL = 'N', VL is not referenced. u(j) = VL(:,j),
104 the j-th column of VL.
105 LDVL (input) INTEGER
107 The leading dimension of the array VL. LDVL >= 1; if JOBVL =
108 'V', LDVL >= N.
109 VR (output) COMPLEX*16 array, dimension (LDVR,N)
111 If JOBVR = 'V', the right eigenvectors v(j) are stored one
112 after another in the columns of VR, in the same order as their
113 eigenvalues. If JOBVR = 'N', VR is not referenced. v(j) =
114 VR(:,j), the j-th column of VR.
115 LDVR (input) INTEGER
117 The leading dimension of the array VR. LDVR >= 1; if JOBVR =
118 'V', LDVR >= N.
119 ILO (output) INTEGER
121 IHI (output) INTEGER ILO and IHI are integer values deter‐
122 mined when A was balanced. The balanced A(i,j) = 0 if I > J
123 and J = 1,...,ILO-1 or I = IHI+1,...,N.
124 SCALE (output) DOUBLE PRECISION array, dimension (N)
126 Details of the permutations and scaling factors applied when
127 balancing A. If P(j) is the index of the row and column inter‐
128 changed with row and column j, and D(j) is the scaling factor
129 applied to row and column j, then SCALE(J) = P(J), for J =
130 1,...,ILO-1 = D(J), for J = ILO,...,IHI = P(J) for J =
131 IHI+1,...,N. The order in which the interchanges are made is N
132 to IHI+1, then 1 to ILO-1.
133 ABNRM (output) DOUBLE PRECISION
135 The one-norm of the balanced matrix (the maximum of the sum of
136 absolute values of elements of any column).
137 RCONDE (output) DOUBLE PRECISION array, dimension (N)
139 RCONDE(j) is the reciprocal condition number of the j-th eigen‐
140 value.
141 RCONDV (output) DOUBLE PRECISION array, dimension (N)
143 RCONDV(j) is the reciprocal condition number of the j-th right
144 eigenvector.
145 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
147 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
148 LWORK (input) INTEGER
150 The dimension of the array WORK. If SENSE = 'N' or 'E', LWORK
151 >= max(1,2*N), and if SENSE = 'V' or 'B', LWORK >= N*N+2*N.
152 For good performance, LWORK must generally be larger. If LWORK
153 = -1, then a workspace query is assumed; the routine only cal‐
154 culates the optimal size of the WORK array, returns this value
155 as the first entry of the WORK array, and no error message
156 related to LWORK is issued by XERBLA.
157 RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
159 INFO (output) INTEGER
161 = 0: successful exit
162 < 0: if INFO = -i, the i-th argument had an illegal value.
163 > 0: if INFO = i, the QR algorithm failed to compute all the
164 eigenvalues, and no eigenvectors or condition numbers have been
165 computed; elements 1:ILO-1 and i+1:N of W contain eigenvalues
166 which have converged.
167 LAPACK driver routine (version 3.N2o)vember 2008 ZGEEVX(1)