1DSYEVX(1) LAPACK driver routine (version 3.1) DSYEVX(1)
2
6 DSYEVX - selected eigenvalues and, optionally, eigenvectors of a real
7 symmetric matrix A
8
10 SUBROUTINE DSYEVX( JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU,
11 ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL,
12 INFO )
13 CHARACTER JOBZ, RANGE, UPLO
15 INTEGER IL, INFO, IU, LDA, LDZ, LWORK, M, N
17 DOUBLE PRECISION ABSTOL, VL, VU
19 INTEGER IFAIL( * ), IWORK( * )
21 DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * ), Z( LDZ, *
23 )
24
26 DSYEVX computes selected eigenvalues and, optionally, eigenvectors of a
27 real symmetric matrix A. Eigenvalues and eigenvectors can be selected
28 by specifying either a range of values or a range of indices for the
29 desired eigenvalues.
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33 JOBZ (input) CHARACTER*1
34 = 'N': Compute eigenvalues only;
35 = 'V': Compute eigenvalues and eigenvectors.
36 RANGE (input) CHARACTER*1
38 = 'A': all eigenvalues will be found.
39 = 'V': all eigenvalues in the half-open interval (VL,VU] will
40 be found. = 'I': the IL-th through IU-th eigenvalues will be
41 found.
42 UPLO (input) CHARACTER*1
44 = 'U': Upper triangle of A is stored;
45 = 'L': Lower triangle of A is stored.
46 N (input) INTEGER
48 The order of the matrix A. N >= 0.
49 A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
51 On entry, the symmetric matrix A. If UPLO = 'U', the leading
52 N-by-N upper triangular part of A contains the upper triangular
53 part of the matrix A. If UPLO = 'L', the leading N-by-N lower
54 triangular part of A contains the lower triangular part of the
55 matrix A. On exit, the lower triangle (if UPLO='L') or the
56 upper triangle (if UPLO='U') of A, including the diagonal, is
57 destroyed.
58 LDA (input) INTEGER
60 The leading dimension of the array A. LDA >= max(1,N).
61 VL (input) DOUBLE PRECISION
63 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
64 upper bounds of the interval to be searched for eigenvalues. VL
65 < VU. Not referenced if RANGE = 'A' or 'I'.
66 IL (input) INTEGER
68 IU (input) INTEGER If RANGE='I', the indices (in ascending
69 order) of the smallest and largest eigenvalues to be returned.
70 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
71 referenced if RANGE = 'A' or 'V'.
72 ABSTOL (input) DOUBLE PRECISION
74 The absolute error tolerance for the eigenvalues. An approxi‐
75 mate eigenvalue is accepted as converged when it is determined
76 to lie in an interval [a,b] of width less than or equal to
77 ABSTOL + EPS * max( |a|,|b| ) ,
79 where EPS is the machine precision. If ABSTOL is less than or
81 equal to zero, then EPS*|T| will be used in its place, where
82 |T| is the 1-norm of the tridiagonal matrix obtained by reduc‐
83 ing A to tridiagonal form.
84 Eigenvalues will be computed most accurately when ABSTOL is set
86 to twice the underflow threshold 2*DLAMCH('S'), not zero. If
87 this routine returns with INFO>0, indicating that some eigen‐
88 vectors did not converge, try setting ABSTOL to 2*DLAMCH('S').
89 See "Computing Small Singular Values of Bidiagonal Matrices
91 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
92 LAPACK Working Note #3.
93 M (output) INTEGER
95 The total number of eigenvalues found. 0 <= M <= N. If RANGE
96 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
97 W (output) DOUBLE PRECISION array, dimension (N)
99 On normal exit, the first M elements contain the selected ei‐
100 genvalues in ascending order.
101 Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
103 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
104 tain the orthonormal eigenvectors of the matrix A corresponding
105 to the selected eigenvalues, with the i-th column of Z holding
106 the eigenvector associated with W(i). If an eigenvector fails
107 to converge, then that column of Z contains the latest approxi‐
108 mation to the eigenvector, and the index of the eigenvector is
109 returned in IFAIL. If JOBZ = 'N', then Z is not referenced.
110 Note: the user must ensure that at least max(1,M) columns are
111 supplied in the array Z; if RANGE = 'V', the exact value of M
112 is not known in advance and an upper bound must be used.
113 LDZ (input) INTEGER
115 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
116 'V', LDZ >= max(1,N).
117 WORK (workspace/output) DOUBLE PRECISION array, dimension
119 (MAX(1,LWORK))
120 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
121 LWORK (input) INTEGER
123 The length of the array WORK. LWORK >= 1, when N <= 1; other‐
124 wise 8*N. For optimal efficiency, LWORK >= (NB+3)*N, where NB
125 is the max of the blocksize for DSYTRD and DORMTR returned by
126 ILAENV.
127 If LWORK = -1, then a workspace query is assumed; the routine
129 only calculates the optimal size of the WORK array, returns
130 this value as the first entry of the WORK array, and no error
131 message related to LWORK is issued by XERBLA.
132 IWORK (workspace) INTEGER array, dimension (5*N)
134 IFAIL (output) INTEGER array, dimension (N)
136 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
137 are zero. If INFO > 0, then IFAIL contains the indices of the
138 eigenvectors that failed to converge. If JOBZ = 'N', then
139 IFAIL is not referenced.
140 INFO (output) INTEGER
142 = 0: successful exit
143 < 0: if INFO = -i, the i-th argument had an illegal value
144 > 0: if INFO = i, then i eigenvectors failed to converge.
145 Their indices are stored in array IFAIL.
146 LAPACK driver routine (version 3.N1o)vember 2006 DSYEVX(1)