1SSTEVR(1) LAPACK driver routine (version 3.1) SSTEVR(1)
2
6 SSTEVR - selected eigenvalues and, optionally, eigenvectors of a real
7 symmetric tridiagonal matrix T
8
10 SUBROUTINE SSTEVR( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL, M, W,
11 Z, LDZ, ISUPPZ, WORK, LWORK, IWORK, LIWORK, INFO )
12 CHARACTER JOBZ, RANGE
14 INTEGER IL, INFO, IU, LDZ, LIWORK, LWORK, M, N
16 REAL ABSTOL, VL, VU
18 INTEGER ISUPPZ( * ), IWORK( * )
20 REAL D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
22
24 SSTEVR computes selected eigenvalues and, optionally, eigenvectors of a
25 real symmetric tridiagonal matrix T. 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 Whenever possible, SSTEVR calls SSTEMR to compute the
30 eigenspectrum using Relatively Robust Representations. SSTEMR computes
31 eigenvalues by the dqds algorithm, while orthogonal eigenvectors are
32 computed from various "good" L D L^T representations (also known as
33 Relatively Robust Representations). Gram-Schmidt orthogonalization is
34 avoided as far as possible. More specifically, the various steps of the
35 algorithm are as follows. For the i-th unreduced block of T,
36 (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
37 is a relatively robust representation,
38 (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
39 relative accuracy by the dqds algorithm,
40 (c) If there is a cluster of close eigenvalues, "choose" sigma_i
41 close to the cluster, and go to step (a),
42 (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
43 compute the corresponding eigenvector by forming a
44 rank-revealing twisted factorization.
45 The desired accuracy of the output can be specified by the input param‐
46 eter ABSTOL.
47 For more details, see "A new O(n^2) algorithm for the symmetric tridi‐
49 agonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer
50 Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May
51 1997.
52 Note 1 : SSTEVR calls SSTEMR when the full spectrum is requested on
55 machines which conform to the ieee-754 floating point standard. SSTEVR
56 calls SSTEBZ and SSTEIN on non-ieee machines and
57 when partial spectrum requests are made.
58 Normal execution of SSTEMR may create NaNs and infinities and hence may
60 abort due to a floating point exception in environments which do not
61 handle NaNs and infinities in the ieee standard default manner.
62
65 JOBZ (input) CHARACTER*1
66 = 'N': Compute eigenvalues only;
67 = 'V': Compute eigenvalues and eigenvectors.
68 RANGE (input) CHARACTER*1
70 = 'A': all eigenvalues will be found.
71 = 'V': all eigenvalues in the half-open interval (VL,VU] will
72 be found. = 'I': the IL-th through IU-th eigenvalues will be
73 found.
74 N (input) INTEGER
76 The order of the matrix. N >= 0.
77 D (input/output) REAL array, dimension (N)
79 On entry, the n diagonal elements of the tridiagonal matrix A.
80 On exit, D may be multiplied by a constant factor chosen to
81 avoid over/underflow in computing the eigenvalues.
82 E (input/output) REAL array, dimension (max(1,N-1))
84 On entry, the (n-1) subdiagonal elements of the tridiagonal
85 matrix A in elements 1 to N-1 of E. On exit, E may be multi‐
86 plied by a constant factor chosen to avoid over/underflow in
87 computing the eigenvalues.
88 VL (input) REAL
90 VU (input) REAL If RANGE='V', the lower and upper bounds
91 of the interval to be searched for eigenvalues. VL < VU. Not
92 referenced if RANGE = 'A' or 'I'.
93 IL (input) INTEGER
95 IU (input) INTEGER If RANGE='I', the indices (in ascending
96 order) of the smallest and largest eigenvalues to be returned.
97 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
98 referenced if RANGE = 'A' or 'V'.
99 ABSTOL (input) REAL
101 The absolute error tolerance for the eigenvalues. An approxi‐
102 mate eigenvalue is accepted as converged when it is determined
103 to lie in an interval [a,b] of width less than or equal to
104 ABSTOL + EPS * max( |a|,|b| ) ,
106 where EPS is the machine precision. If ABSTOL is less than or
108 equal to zero, then EPS*|T| will be used in its place, where
109 |T| is the 1-norm of the tridiagonal matrix obtained by reduc‐
110 ing A to tridiagonal form.
111 See "Computing Small Singular Values of Bidiagonal Matrices
113 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
114 LAPACK Working Note #3.
115 If high relative accuracy is important, set ABSTOL to SLAMCH(
117 'Safe minimum' ). Doing so will guarantee that eigenvalues are
118 computed to high relative accuracy when possible in future
119 releases. The current code does not make any guarantees about
120 high relative accuracy, but future releases will. See J. Barlow
121 and J. Demmel, "Computing Accurate Eigensystems of Scaled Diag‐
122 onally Dominant Matrices", LAPACK Working Note #7, for a dis‐
123 cussion of which matrices define their eigenvalues to high rel‐
124 ative accuracy.
125 M (output) INTEGER
127 The total number of eigenvalues found. 0 <= M <= N. If RANGE
128 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
129 W (output) REAL array, dimension (N)
131 The first M elements contain the selected eigenvalues in
132 ascending order.
133 Z (output) REAL array, dimension (LDZ, max(1,M) )
135 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
136 tain the orthonormal eigenvectors of the matrix A corresponding
137 to the selected eigenvalues, with the i-th column of Z holding
138 the eigenvector associated with W(i). Note: the user must
139 ensure that at least max(1,M) columns are supplied in the array
140 Z; if RANGE = 'V', the exact value of M is not known in advance
141 and an upper bound must be used.
142 LDZ (input) INTEGER
144 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
145 'V', LDZ >= max(1,N).
146 ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
148 The support of the eigenvectors in Z, i.e., the indices indi‐
149 cating the nonzero elements in Z. The i-th eigenvector is
150 nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ).
151 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
153 On exit, if INFO = 0, WORK(1) returns the optimal (and minimal)
154 LWORK.
155 LWORK (input) INTEGER
157 The dimension of the array WORK. LWORK >= 20*N.
158 If LWORK = -1, then a workspace query is assumed; the routine
160 only calculates the optimal sizes of the WORK and IWORK arrays,
161 returns these values as the first entries of the WORK and IWORK
162 arrays, and no error message related to LWORK or LIWORK is
163 issued by XERBLA.
164 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
166 On exit, if INFO = 0, IWORK(1) returns the optimal (and mini‐
167 mal) LIWORK.
168 LIWORK (input) INTEGER
170 The dimension of the array IWORK. LIWORK >= 10*N.
171 If LIWORK = -1, then a workspace query is assumed; the routine
173 only calculates the optimal sizes of the WORK and IWORK arrays,
174 returns these values as the first entries of the WORK and IWORK
175 arrays, and no error message related to LWORK or LIWORK is
176 issued by XERBLA.
177 INFO (output) INTEGER
179 = 0: successful exit
180 < 0: if INFO = -i, the i-th argument had an illegal value
181 > 0: Internal error
182
184 Based on contributions by
185 Inderjit Dhillon, IBM Almaden, USA
186 Osni Marques, LBNL/NERSC, USA
187 Ken Stanley, Computer Science Division, University of
188 California at Berkeley, USA
189 Jason Riedy, Computer Science Division, University of
190 California at Berkeley, USA
191 LAPACK driver routine (version 3.N1o)vember 2006 SSTEVR(1)