1SSTEVX(1) LAPACK driver routine (version 3.2) SSTEVX(1)
2
6 SSTEVX - computes selected eigenvalues and, optionally, eigenvectors of
7 a real symmetric tridiagonal matrix A
8
10 SUBROUTINE SSTEVX( 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 REAL ABSTOL, VL, VU
18 INTEGER IFAIL( * ), IWORK( * )
20 REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
22
24 SSTEVX computes selected eigenvalues and, optionally, eigenvectors of a
25 real symmetric tridiagonal matrix A. Eigenvalues and eigenvectors can
26 be selected by specifying either a range of values or a range of
27 indices for the desired eigenvalues.
28
30 JOBZ (input) CHARACTER*1
31 = 'N': Compute eigenvalues only;
32 = 'V': Compute eigenvalues and eigenvectors.
33 RANGE (input) CHARACTER*1
35 = 'A': all eigenvalues will be found.
36 = 'V': all eigenvalues in the half-open interval (VL,VU] will
37 be found. = 'I': the IL-th through IU-th eigenvalues will be
38 found.
39 N (input) INTEGER
41 The order of the matrix. N >= 0.
42 D (input/output) REAL array, dimension (N)
44 On entry, the n diagonal elements of the tridiagonal matrix A.
45 On exit, D may be multiplied by a constant factor chosen to
46 avoid over/underflow in computing the eigenvalues.
47 E (input/output) REAL array, dimension (max(1,N-1))
49 On entry, the (n-1) subdiagonal elements of the tridiagonal
50 matrix A in elements 1 to N-1 of E. On exit, E may be multi‐
51 plied by a constant factor chosen to avoid over/underflow in
52 computing the eigenvalues.
53 VL (input) REAL
55 VU (input) REAL If RANGE='V', the lower and upper bounds
56 of the interval to be searched for eigenvalues. VL < VU. Not
57 referenced if RANGE = 'A' or 'I'.
58 IL (input) INTEGER
60 IU (input) INTEGER If RANGE='I', the indices (in ascending
61 order) of the smallest and largest eigenvalues to be returned.
62 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
63 referenced if RANGE = 'A' or 'V'.
64 ABSTOL (input) REAL
66 The absolute error tolerance for the eigenvalues. An approxi‐
67 mate eigenvalue is accepted as converged when it is determined
68 to lie in an interval [a,b] of width less than or equal to
69 ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine pre‐
70 cision. If ABSTOL is less than or equal to zero, then EPS*|T|
71 will be used in its place, where |T| is the 1-norm of the
72 tridiagonal matrix. Eigenvalues will be computed most accu‐
73 rately when ABSTOL is set to twice the underflow threshold
74 2*SLAMCH('S'), not zero. If this routine returns with INFO>0,
75 indicating that some eigenvectors did not converge, try setting
76 ABSTOL to 2*SLAMCH('S'). See "Computing Small Singular Values
77 of Bidiagonal Matrices with Guaranteed High Relative Accuracy,"
78 by Demmel and Kahan, LAPACK Working Note #3.
79 M (output) INTEGER
81 The total number of eigenvalues found. 0 <= M <= N. If RANGE
82 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
83 W (output) REAL array, dimension (N)
85 The first M elements contain the selected eigenvalues in
86 ascending order.
87 Z (output) REAL array, dimension (LDZ, max(1,M) )
89 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
90 tain the orthonormal eigenvectors of the matrix A corresponding
91 to the selected eigenvalues, with the i-th column of Z holding
92 the eigenvector associated with W(i). If an eigenvector fails
93 to converge (INFO > 0), then that column of Z contains the lat‐
94 est approximation to the eigenvector, and the index of the
95 eigenvector is returned in IFAIL. If JOBZ = 'N', then Z is not
96 referenced. Note: the user must ensure that at least max(1,M)
97 columns are supplied in the array Z; if RANGE = 'V', the exact
98 value of M is not known in advance and an upper bound must be
99 used.
100 LDZ (input) INTEGER
102 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
103 'V', LDZ >= max(1,N).
104 WORK (workspace) REAL array, dimension (5*N)
106 IWORK (workspace) INTEGER array, dimension (5*N)
108 IFAIL (output) INTEGER array, dimension (N)
110 If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
111 are zero. If INFO > 0, then IFAIL contains the indices of the
112 eigenvectors that failed to converge. If JOBZ = 'N', then
113 IFAIL is not referenced.
114 INFO (output) INTEGER
116 = 0: successful exit
117 < 0: if INFO = -i, the i-th argument had an illegal value
118 > 0: if INFO = i, then i eigenvectors failed to converge.
119 Their indices are stored in array IFAIL.
120 LAPACK driver routine (version 3.N2o)vember 2008 SSTEVX(1)