1DSYGVX(1) LAPACK driver routine (version 3.1) DSYGVX(1)
2
6 DSYGVX - selected eigenvalues, and optionally, eigenvectors of a real
7 generalized symmetric-definite eigenproblem, of the form
8 A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x
9
11 SUBROUTINE DSYGVX( ITYPE, JOBZ, RANGE, UPLO, N, A, LDA, B, LDB, VL, VU,
12 IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK,
13 IFAIL, INFO )
14 CHARACTER JOBZ, RANGE, UPLO
16 INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
18 DOUBLE PRECISION ABSTOL, VL, VU
20 INTEGER IFAIL( * ), IWORK( * )
22 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( *
24 ), Z( LDZ, * )
25
27 DSYGVX computes selected eigenvalues, and optionally, eigenvectors of a
28 real generalized symmetric-definite eigenproblem, of the form
29 A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B
30 are assumed to be symmetric and B is also positive definite. Eigenval‐
31 ues and eigenvectors can be selected by specifying either a range of
32 values or a range of indices for the desired eigenvalues.
33
36 ITYPE (input) INTEGER
37 Specifies the problem type to be solved:
38 = 1: A*x = (lambda)*B*x
39 = 2: A*B*x = (lambda)*x
40 = 3: B*A*x = (lambda)*x
41 JOBZ (input) CHARACTER*1
43 = 'N': Compute eigenvalues only;
44 = 'V': Compute eigenvalues and eigenvectors.
45 RANGE (input) CHARACTER*1
47 = 'A': all eigenvalues will be found.
48 = 'V': all eigenvalues in the half-open interval (VL,VU] will
49 be found. = 'I': the IL-th through IU-th eigenvalues will be
50 found.
51 UPLO (input) CHARACTER*1
53 = 'U': Upper triangle of A and B are stored;
54 = 'L': Lower triangle of A and B are stored.
55 N (input) INTEGER
57 The order of the matrix pencil (A,B). N >= 0.
58 A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
60 On entry, the symmetric matrix A. If UPLO = 'U', the leading
61 N-by-N upper triangular part of A contains the upper triangular
62 part of the matrix A. If UPLO = 'L', the leading N-by-N lower
63 triangular part of A contains the lower triangular part of the
64 matrix A.
65 On exit, the lower triangle (if UPLO='L') or the upper triangle
67 (if UPLO='U') of A, including the diagonal, is destroyed.
68 LDA (input) INTEGER
70 The leading dimension of the array A. LDA >= max(1,N).
71 B (input/output) DOUBLE PRECISION array, dimension (LDA, N)
73 On entry, the symmetric matrix B. If UPLO = 'U', the leading
74 N-by-N upper triangular part of B contains the upper triangular
75 part of the matrix B. If UPLO = 'L', the leading N-by-N lower
76 triangular part of B contains the lower triangular part of the
77 matrix B.
78 On exit, if INFO <= N, the part of B containing the matrix is
80 overwritten by the triangular factor U or L from the Cholesky
81 factorization B = U**T*U or B = L*L**T.
82 LDB (input) INTEGER
84 The leading dimension of the array B. LDB >= max(1,N).
85 VL (input) DOUBLE PRECISION
87 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
88 upper bounds of the interval to be searched for eigenvalues. VL
89 < VU. Not referenced if RANGE = 'A' or 'I'.
90 IL (input) INTEGER
92 IU (input) INTEGER If RANGE='I', the indices (in ascending
93 order) of the smallest and largest eigenvalues to be returned.
94 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
95 referenced if RANGE = 'A' or 'V'.
96 ABSTOL (input) DOUBLE PRECISION
98 The absolute error tolerance for the eigenvalues. An approxi‐
99 mate eigenvalue is accepted as converged when it is determined
100 to lie in an interval [a,b] of width less than or equal to
101 ABSTOL + EPS * max( |a|,|b| ) ,
103 where EPS is the machine precision. If ABSTOL is less than or
105 equal to zero, then EPS*|T| will be used in its place, where
106 |T| is the 1-norm of the tridiagonal matrix obtained by reduc‐
107 ing A to tridiagonal form.
108 Eigenvalues will be computed most accurately when ABSTOL is set
110 to twice the underflow threshold 2*DLAMCH('S'), not zero. If
111 this routine returns with INFO>0, indicating that some eigen‐
112 vectors did not converge, try setting ABSTOL to 2*DLAMCH('S').
113 M (output) INTEGER
115 The total number of eigenvalues found. 0 <= M <= N. If RANGE
116 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
117 W (output) DOUBLE PRECISION array, dimension (N)
119 On normal exit, the first M elements contain the selected ei‐
120 genvalues in ascending order.
121 Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
123 If JOBZ = 'N', then Z is not referenced. If JOBZ = 'V', then
124 if INFO = 0, the first M columns of Z contain the orthonormal
125 eigenvectors of the matrix A corresponding to the selected ei‐
126 genvalues, with the i-th column of Z holding the eigenvector
127 associated with W(i). The eigenvectors are normalized as fol‐
128 lows: if ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3,
129 Z**T*inv(B)*Z = I.
130 If an eigenvector fails to converge, then that column of Z con‐
132 tains the latest approximation to the eigenvector, and the
133 index of the eigenvector is returned in IFAIL. Note: the user
134 must ensure that at least max(1,M) columns are supplied in the
135 array Z; if RANGE = 'V', the exact value of M is not known in
136 advance and an upper bound must be used.
137 LDZ (input) INTEGER
139 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
140 'V', LDZ >= max(1,N).
141 WORK (workspace/output) DOUBLE PRECISION array, dimension
143 (MAX(1,LWORK))
144 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
145 LWORK (input) INTEGER
147 The length of the array WORK. LWORK >= max(1,8*N). For opti‐
148 mal efficiency, LWORK >= (NB+3)*N, where NB is the blocksize
149 for DSYTRD returned by ILAENV.
150 If LWORK = -1, then a workspace query is assumed; the routine
152 only calculates the optimal size of the WORK array, returns
153 this value as the first entry of the WORK array, and no error
154 message related to LWORK is issued by XERBLA.
155 IWORK (workspace) INTEGER array, dimension (5*N)
157 IFAIL (output) INTEGER array, dimension (N)
159 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
160 are zero. If INFO > 0, then IFAIL contains the indices of the
161 eigenvectors that failed to converge. If JOBZ = 'N', then
162 IFAIL is not referenced.
163 INFO (output) INTEGER
165 = 0: successful exit
166 < 0: if INFO = -i, the i-th argument had an illegal value
167 > 0: DPOTRF or DSYEVX returned an error code:
168 <= N: if INFO = i, DSYEVX failed to converge; i eigenvectors
169 failed to converge. Their indices are stored in array IFAIL.
170 > N: if INFO = N + i, for 1 <= i <= N, then the leading minor
171 of order i of B is not positive definite. The factorization of
172 B could not be completed and no eigenvalues or eigenvectors
173 were computed.
174
176 Based on contributions by
177 Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
178 LAPACK driver routine (version 3.N1o)vember 2006 DSYGVX(1)