1DSTEVX(1) LAPACK driver routine (version 3.2) DSTEVX(1)
2
6 DSTEVX - computes selected eigenvalues and, optionally, eigenvectors of
7 a real symmetric tridiagonal matrix A
8
10 SUBROUTINE DSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W,
11 Z, LDZ, WORK, IWORK, IFAIL, INFO )
12 CHARACTER JOBZ, RANGE
14 INTEGER IL, INFO, IU, LDZ, M, N
16 DOUBLE PRECISION ABSTOL, VL, VU
18 INTEGER IFAIL( * ), IWORK( * )
20 DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ,
22 * )
23
25 DSTEVX computes selected eigenvalues and, optionally, eigenvectors of a
26 real symmetric tridiagonal matrix A. Eigenvalues and eigenvectors can
27 be selected by specifying either a range of values or a range of
28 indices for the desired eigenvalues.
29
31 JOBZ (input) CHARACTER*1
32 = 'N': Compute eigenvalues only;
33 = 'V': Compute eigenvalues and eigenvectors.
34 RANGE (input) CHARACTER*1
36 = 'A': all eigenvalues will be found.
37 = 'V': all eigenvalues in the half-open interval (VL,VU] will
38 be found. = 'I': the IL-th through IU-th eigenvalues will be
39 found.
40 N (input) INTEGER
42 The order of the matrix. N >= 0.
43 D (input/output) DOUBLE PRECISION array, dimension (N)
45 On entry, the n diagonal elements of the tridiagonal matrix A.
46 On exit, D may be multiplied by a constant factor chosen to
47 avoid over/underflow in computing the eigenvalues.
48 E (input/output) DOUBLE PRECISION array, dimension (max(1,N-1))
50 On entry, the (n-1) subdiagonal elements of the tridiagonal
51 matrix A in elements 1 to N-1 of E. On exit, E may be multi‐
52 plied by a constant factor chosen to avoid over/underflow in
53 computing the eigenvalues.
54 VL (input) DOUBLE PRECISION
56 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
57 upper bounds of the interval to be searched for eigenvalues. VL
58 < VU. Not referenced if RANGE = 'A' or 'I'.
59 IL (input) INTEGER
61 IU (input) INTEGER If RANGE='I', the indices (in ascending
62 order) of the smallest and largest eigenvalues to be returned.
63 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
64 referenced if RANGE = 'A' or 'V'.
65 ABSTOL (input) DOUBLE PRECISION
67 The absolute error tolerance for the eigenvalues. An approxi‐
68 mate eigenvalue is accepted as converged when it is determined
69 to lie in an interval [a,b] of width less than or equal to
70 ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine pre‐
71 cision. If ABSTOL is less than or equal to zero, then EPS*|T|
72 will be used in its place, where |T| is the 1-norm of the
73 tridiagonal matrix. Eigenvalues will be computed most accu‐
74 rately when ABSTOL is set to twice the underflow threshold
75 2*DLAMCH('S'), not zero. If this routine returns with INFO>0,
76 indicating that some eigenvectors did not converge, try setting
77 ABSTOL to 2*DLAMCH('S'). See "Computing Small Singular Values
78 of Bidiagonal Matrices with Guaranteed High Relative Accuracy,"
79 by Demmel and Kahan, LAPACK Working Note #3.
80 M (output) INTEGER
82 The total number of eigenvalues found. 0 <= M <= N. If RANGE
83 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
84 W (output) DOUBLE PRECISION array, dimension (N)
86 The first M elements contain the selected eigenvalues in
87 ascending order.
88 Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
90 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
91 tain the orthonormal eigenvectors of the matrix A corresponding
92 to the selected eigenvalues, with the i-th column of Z holding
93 the eigenvector associated with W(i). If an eigenvector fails
94 to converge (INFO > 0), then that column of Z contains the lat‐
95 est approximation to the eigenvector, and the index of the
96 eigenvector is returned in IFAIL. If JOBZ = 'N', then Z is not
97 referenced. Note: the user must ensure that at least max(1,M)
98 columns are supplied in the array Z; if RANGE = 'V', the exact
99 value of M is not known in advance and an upper bound must be
100 used.
101 LDZ (input) INTEGER
103 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
104 'V', LDZ >= max(1,N).
105 WORK (workspace) DOUBLE PRECISION array, dimension (5*N)
107 IWORK (workspace) INTEGER array, dimension (5*N)
109 IFAIL (output) INTEGER array, dimension (N)
111 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
112 are zero. If INFO > 0, then IFAIL contains the indices of the
113 eigenvectors that failed to converge. If JOBZ = 'N', then
114 IFAIL is not referenced.
115 INFO (output) INTEGER
117 = 0: successful exit
118 < 0: if INFO = -i, the i-th argument had an illegal value
119 > 0: if INFO = i, then i eigenvectors failed to converge.
120 Their indices are stored in array IFAIL.
121 LAPACK driver routine (version 3.N2o)vember 2008 DSTEVX(1)