1DSYEVX(1) LAPACK driver routine (version 3.2) DSYEVX(1)
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6 DSYEVX - computes selected eigenvalues and, optionally, eigenvectors of
7 a real symmetric matrix A
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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 )
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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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32 JOBZ (input) CHARACTER*1
33 = 'N': Compute eigenvalues only;
34 = 'V': Compute eigenvalues and eigenvectors.
35 RANGE (input) CHARACTER*1
37 = 'A': all eigenvalues will be found.
38 = 'V': all eigenvalues in the half-open interval (VL,VU] will
39 be found. = 'I': the IL-th through IU-th eigenvalues will be
40 found.
41 UPLO (input) CHARACTER*1
43 = 'U': Upper triangle of A is stored;
44 = 'L': Lower triangle of A is stored.
45 N (input) INTEGER
47 The order of the matrix A. N >= 0.
48 A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
50 On entry, the symmetric matrix A. If UPLO = 'U', the leading
51 N-by-N upper triangular part of A contains the upper triangular
52 part of the matrix A. If UPLO = 'L', the leading N-by-N lower
53 triangular part of A contains the lower triangular part of the
54 matrix A. On exit, the lower triangle (if UPLO='L') or the
55 upper triangle (if UPLO='U') of A, including the diagonal, is
56 destroyed.
57 LDA (input) INTEGER
59 The leading dimension of the array A. LDA >= max(1,N).
60 VL (input) DOUBLE PRECISION
62 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
63 upper bounds of the interval to be searched for eigenvalues. VL
64 < VU. Not referenced if RANGE = 'A' or 'I'.
65 IL (input) INTEGER
67 IU (input) INTEGER If RANGE='I', the indices (in ascending
68 order) of the smallest and largest eigenvalues to be returned.
69 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
70 referenced if RANGE = 'A' or 'V'.
71 ABSTOL (input) DOUBLE PRECISION
73 The absolute error tolerance for the eigenvalues. An approxi‐
74 mate eigenvalue is accepted as converged when it is determined
75 to lie in an interval [a,b] of width less than or equal to
76 ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine pre‐
77 cision. If ABSTOL is less than or equal to zero, then EPS*|T|
78 will be used in its place, where |T| is the 1-norm of the
79 tridiagonal matrix obtained by reducing A to tridiagonal form.
80 Eigenvalues will be computed most accurately when ABSTOL is set
81 to twice the underflow threshold 2*DLAMCH('S'), not zero. If
82 this routine returns with INFO>0, indicating that some eigen‐
83 vectors did not converge, try setting ABSTOL to 2*DLAMCH('S').
84 See "Computing Small Singular Values of Bidiagonal Matrices
85 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
86 LAPACK Working Note #3.
87 M (output) INTEGER
89 The total number of eigenvalues found. 0 <= M <= N. If RANGE
90 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
91 W (output) DOUBLE PRECISION array, dimension (N)
93 On normal exit, the first M elements contain the selected ei‐
94 genvalues in ascending order.
95 Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M))
97 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
98 tain the orthonormal eigenvectors of the matrix A corresponding
99 to the selected eigenvalues, with the i-th column of Z holding
100 the eigenvector associated with W(i). If an eigenvector fails
101 to converge, then that column of Z contains the latest approxi‐
102 mation to the eigenvector, and the index of the eigenvector is
103 returned in IFAIL. If JOBZ = 'N', then Z is not referenced.
104 Note: the user must ensure that at least max(1,M) columns are
105 supplied in the array Z; if RANGE = 'V', the exact value of M
106 is not known in advance and an upper bound must be used.
107 LDZ (input) INTEGER
109 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
110 'V', LDZ >= max(1,N).
111 WORK (workspace/output) DOUBLE PRECISION array, dimension
113 (MAX(1,LWORK))
114 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
115 LWORK (input) INTEGER
117 The length of the array WORK. LWORK >= 1, when N <= 1; other‐
118 wise 8*N. For optimal efficiency, LWORK >= (NB+3)*N, where NB
119 is the max of the blocksize for DSYTRD and DORMTR returned by
120 ILAENV. If LWORK = -1, then a workspace query is assumed; the
121 routine only calculates the optimal size of the WORK array,
122 returns this value as the first entry of the WORK array, and no
123 error message related to LWORK is issued by XERBLA.
124 IWORK (workspace) INTEGER array, dimension (5*N)
126 IFAIL (output) INTEGER array, dimension (N)
128 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
129 are zero. If INFO > 0, then IFAIL contains the indices of the
130 eigenvectors that failed to converge. If JOBZ = 'N', then
131 IFAIL is not referenced.
132 INFO (output) INTEGER
134 = 0: successful exit
135 < 0: if INFO = -i, the i-th argument had an illegal value
136 > 0: if INFO = i, then i eigenvectors failed to converge.
137 Their indices are stored in array IFAIL.
138 LAPACK driver routine (version 3.N2o)vember 2008 DSYEVX(1)