1DSTEVX(1) LAPACK driver routine (version 3.1) DSTEVX(1)
2
6 DSTEVX - selected eigenvalues and, optionally, eigenvectors of a real
7 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
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 N (input) INTEGER
43 The order of the matrix. N >= 0.
44 D (input/output) DOUBLE PRECISION array, dimension (N)
46 On entry, the n diagonal elements of the tridiagonal matrix A.
47 On exit, D may be multiplied by a constant factor chosen to
48 avoid over/underflow in computing the eigenvalues.
49 E (input/output) DOUBLE PRECISION array, dimension (max(1,N-1))
51 On entry, the (n-1) subdiagonal elements of the tridiagonal
52 matrix A in elements 1 to N-1 of E. On exit, E may be multi‐
53 plied by a constant factor chosen to avoid over/underflow in
54 computing the eigenvalues.
55 VL (input) DOUBLE PRECISION
57 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
58 upper bounds of the interval to be searched for eigenvalues. VL
59 < VU. Not referenced if RANGE = 'A' or 'I'.
60 IL (input) INTEGER
62 IU (input) INTEGER If RANGE='I', the indices (in ascending
63 order) of the smallest and largest eigenvalues to be returned.
64 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
65 referenced if RANGE = 'A' or 'V'.
66 ABSTOL (input) DOUBLE PRECISION
68 The absolute error tolerance for the eigenvalues. An approxi‐
69 mate eigenvalue is accepted as converged when it is determined
70 to lie in an interval [a,b] of width less than or equal to
71 ABSTOL + EPS * max( |a|,|b| ) ,
73 where EPS is the machine precision. If ABSTOL is less than or
75 equal to zero, then EPS*|T| will be used in its place, where
76 |T| is the 1-norm of the tridiagonal matrix.
77 Eigenvalues will be computed most accurately when ABSTOL is set
79 to twice the underflow threshold 2*DLAMCH('S'), not zero. If
80 this routine returns with INFO>0, indicating that some eigen‐
81 vectors did not converge, try setting ABSTOL to 2*DLAMCH('S').
82 See "Computing Small Singular Values of Bidiagonal Matrices
84 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
85 LAPACK Working Note #3.
86 M (output) INTEGER
88 The total number of eigenvalues found. 0 <= M <= N. If RANGE
89 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
90 W (output) DOUBLE PRECISION array, dimension (N)
92 The first M elements contain the selected eigenvalues in
93 ascending order.
94 Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
96 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
97 tain the orthonormal eigenvectors of the matrix A corresponding
98 to the selected eigenvalues, with the i-th column of Z holding
99 the eigenvector associated with W(i). If an eigenvector fails
100 to converge (INFO > 0), then that column of Z contains the lat‐
101 est approximation to the eigenvector, and the index of the
102 eigenvector is returned in IFAIL. If JOBZ = 'N', then Z is not
103 referenced. Note: the user must ensure that at least max(1,M)
104 columns are supplied in the array Z; if RANGE = 'V', the exact
105 value of M is not known in advance and an upper bound must be
106 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) DOUBLE PRECISION array, dimension (5*N)
113 IWORK (workspace) INTEGER array, dimension (5*N)
115 IFAIL (output) INTEGER array, dimension (N)
117 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
118 are zero. If INFO > 0, then IFAIL contains the indices of the
119 eigenvectors that failed to converge. If JOBZ = 'N', then
120 IFAIL is not referenced.
121 INFO (output) INTEGER
123 = 0: successful exit
124 < 0: if INFO = -i, the i-th argument had an illegal value
125 > 0: if INFO = i, then i eigenvectors failed to converge.
126 Their indices are stored in array IFAIL.
127 LAPACK driver routine (version 3.N1o)vember 2006 DSTEVX(1)