1SGEEVX(1) LAPACK driver routine (version 3.1) SGEEVX(1)
2
6 SGEEVX - for an N-by-N real nonsymmetric matrix A, the eigenvalues and,
7 optionally, the left and/or right eigenvectors
8
10 SUBROUTINE SGEEVX( 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 REAL ABNRM
19 INTEGER IWORK( * )
21 REAL A( LDA, * ), RCONDE( * ), RCONDV( * ), SCALE( * ),
23 VL( LDVL, * ), VR( LDVR, * ), WI( * ), WORK( * ),
24 WR( * )
25
27 SGEEVX computes for an N-by-N real nonsymmetric matrix A, the eigenval‐
28 ues and, optionally, the left and/or right eigenvectors.
29 Optionally also, it computes a balancing transformation to improve the
31 conditioning of the eigenvalues and eigenvectors (ILO, IHI, SCALE, and
32 ABNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and
33 reciprocal condition numbers for the right
34 eigenvectors (RCONDV).
35 The right eigenvector v(j) of A satisfies
37 A * v(j) = lambda(j) * v(j)
38 where lambda(j) is its eigenvalue.
39 The left eigenvector u(j) of A satisfies
40 u(j)**H * A = lambda(j) * u(j)**H
41 where u(j)**H denotes the conjugate transpose of u(j).
42 The computed eigenvectors are normalized to have Euclidean norm equal
44 to 1 and largest component real.
45 Balancing a matrix means permuting the rows and columns to make it more
47 nearly upper triangular, and applying a diagonal similarity transforma‐
48 tion D * A * D**(-1), where D is a diagonal matrix, to make its rows
49 and columns closer in norm and the condition numbers of its eigenvalues
50 and eigenvectors smaller. The computed reciprocal condition numbers
51 correspond to the balanced matrix. Permuting rows and columns will not
52 change the condition numbers (in exact arithmetic) but diagonal scaling
53 will. For further explanation of balancing, see section 4.10.2 of the
54 LAPACK Users' Guide.
55
58 BALANC (input) CHARACTER*1
59 Indicates how the input matrix should be diagonally scaled
60 and/or permuted to improve the conditioning of its eigenvalues.
61 = 'N': Do not diagonally scale or permute;
62 = 'P': Perform permutations to make the matrix more nearly
63 upper triangular. Do not diagonally scale; = 'S': Diagonally
64 scale the matrix, i.e. replace A by D*A*D**(-1), where D is a
65 diagonal matrix chosen to make the rows and columns of A more
66 equal in norm. Do not permute; = 'B': Both diagonally scale and
67 permute A.
68 Computed reciprocal condition numbers will be for the matrix
70 after balancing and/or permuting. Permuting does not change
71 condition numbers (in exact arithmetic), but balancing does.
72 JOBVL (input) CHARACTER*1
74 = 'N': left eigenvectors of A are not computed;
75 = 'V': left eigenvectors of A are computed. If SENSE = 'E' or
76 'B', JOBVL must = 'V'.
77 JOBVR (input) CHARACTER*1
79 = 'N': right eigenvectors of A are not computed;
80 = 'V': right eigenvectors of A are computed. If SENSE = 'E' or
81 'B', JOBVR must = 'V'.
82 SENSE (input) CHARACTER*1
84 Determines which reciprocal condition numbers are computed. =
85 'N': None are computed;
86 = 'E': Computed for eigenvalues only;
87 = 'V': Computed for right eigenvectors only;
88 = 'B': Computed for eigenvalues and right eigenvectors.
89 If SENSE = 'E' or 'B', both left and right eigenvectors must
91 also be computed (JOBVL = 'V' and JOBVR = 'V').
92 N (input) INTEGER
94 The order of the matrix A. N >= 0.
95 A (input/output) REAL array, dimension (LDA,N)
97 On entry, the N-by-N matrix A. On exit, A has been overwrit‐
98 ten. If JOBVL = 'V' or JOBVR = 'V', A contains the real Schur
99 form of the balanced version of the input matrix A.
100 LDA (input) INTEGER
102 The leading dimension of the array A. LDA >= max(1,N).
103 WR (output) REAL array, dimension (N)
105 WI (output) REAL array, dimension (N) WR and WI contain
106 the real and imaginary parts, respectively, of the computed ei‐
107 genvalues. Complex conjugate pairs of eigenvalues will appear
108 consecutively with the eigenvalue having the positive imaginary
109 part first.
110 VL (output) REAL array, dimension (LDVL,N)
112 If JOBVL = 'V', the left eigenvectors u(j) are stored one after
113 another in the columns of VL, in the same order as their eigen‐
114 values. If JOBVL = 'N', VL is not referenced. If the j-th ei‐
115 genvalue is real, then u(j) = VL(:,j), the j-th column of VL.
116 If the j-th and (j+1)-st eigenvalues form a complex conjugate
117 pair, then u(j) = VL(:,j) + i*VL(:,j+1) and
118 u(j+1) = VL(:,j) - i*VL(:,j+1).
119 LDVL (input) INTEGER
121 The leading dimension of the array VL. LDVL >= 1; 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 JOBVR = 'N', VR is not referenced. If the j-
128 th eigenvalue is real, then v(j) = VR(:,j), the j-th column of
129 VR. If the j-th and (j+1)-st eigenvalues form a complex conju‐
130 gate pair, then v(j) = VR(:,j) + i*VR(:,j+1) and
131 v(j+1) = VR(:,j) - i*VR(:,j+1).
132 LDVR (input) INTEGER
134 The leading dimension of the array VR. LDVR >= 1, and if JOBVR
135 = 'V', LDVR >= N.
136 ILO (output) INTEGER
138 IHI (output) INTEGER ILO and IHI are integer values deter‐
139 mined when A was balanced. The balanced A(i,j) = 0 if I > J
140 and J = 1,...,ILO-1 or I = IHI+1,...,N.
141 SCALE (output) REAL array, dimension (N)
143 Details of the permutations and scaling factors applied when
144 balancing A. If P(j) is the index of the row and column inter‐
145 changed with row and column j, and D(j) is the scaling factor
146 applied to row and column j, then SCALE(J) = P(J), for J =
147 1,...,ILO-1 = D(J), for J = ILO,...,IHI = P(J) for J =
148 IHI+1,...,N. The order in which the interchanges are made is N
149 to IHI+1, then 1 to ILO-1.
150 ABNRM (output) REAL
152 The one-norm of the balanced matrix (the maximum of the sum of
153 absolute values of elements of any column).
154 RCONDE (output) REAL array, dimension (N)
156 RCONDE(j) is the reciprocal condition number of the j-th eigen‐
157 value.
158 RCONDV (output) REAL array, dimension (N)
160 RCONDV(j) is the reciprocal condition number of the j-th right
161 eigenvector.
162 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
164 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
165 LWORK (input) INTEGER
167 The dimension of the array WORK. If SENSE = 'N' or 'E', LWORK
168 >= max(1,2*N), and if JOBVL = 'V' or JOBVR = 'V', LWORK >= 3*N.
169 If SENSE = 'V' or 'B', LWORK >= N*(N+6). For good performance,
170 LWORK must generally be larger.
171 If LWORK = -1, then a workspace query is assumed; the routine
173 only calculates the optimal size of the WORK array, returns
174 this value as the first entry of the WORK array, and no error
175 message related to LWORK is issued by XERBLA.
176 IWORK (workspace) INTEGER array, dimension (2*N-2)
178 If SENSE = 'N' or 'E', not referenced.
179 INFO (output) INTEGER
181 = 0: successful exit
182 < 0: if INFO = -i, the i-th argument had an illegal value.
183 > 0: if INFO = i, the QR algorithm failed to compute all the
184 eigenvalues, and no eigenvectors or condition numbers have been
185 computed; elements 1:ILO-1 and i+1:N of WR and WI contain ei‐
186 genvalues which have converged.
187 LAPACK driver routine (version 3.N1o)vember 2006 SGEEVX(1)