1SSTEVR(1) LAPACK driver routine (version 3.2) SSTEVR(1)
2
6 SSTEVR - computes selected eigenvalues and, optionally, eigenvectors of
7 a real 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
29 eigenspectrum using Relatively Robust Representations. SSTEMR computes
30 eigenvalues by the dqds algorithm, while orthogonal eigenvectors are
31 computed from various "good" L D L^T representations (also known as
32 Relatively Robust Representations). Gram-Schmidt orthogonalization is
33 avoided as far as possible. More specifically, the various steps of the
34 algorithm are as follows. For the i-th unreduced block of T,
35 (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
36 is a relatively robust representation,
37 (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
38 relative accuracy by the dqds algorithm,
39 (c) If there is a cluster of close eigenvalues, "choose" sigma_i
40 close to the cluster, and go to step (a),
41 (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
42 compute the corresponding eigenvector by forming a
43 rank-revealing twisted factorization.
44 The desired accuracy of the output can be specified by the input param‐
45 eter ABSTOL.
46 For more details, see "A new O(n^2) algorithm for the symmetric tridi‐
47 agonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer
48 Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May
49 1997.
50 Note 1 : SSTEVR calls SSTEMR when the full spectrum is requested on
51 machines which conform to the ieee-754 floating point standard. SSTEVR
52 calls SSTEBZ and SSTEIN on non-ieee machines and
53 when partial spectrum requests are made.
54 Normal execution of SSTEMR may create NaNs and infinities and hence may
55 abort due to a floating point exception in environments which do not
56 handle NaNs and infinities in the ieee standard default manner.
57
59 JOBZ (input) CHARACTER*1
60 = 'N': Compute eigenvalues only;
61 = 'V': Compute eigenvalues and eigenvectors.
62 RANGE (input) CHARACTER*1
64 = 'A': all eigenvalues will be found.
65 = 'V': all eigenvalues in the half-open interval (VL,VU] will
66 be found. = 'I': the IL-th through IU-th eigenvalues will be
67 found.
68 N (input) INTEGER
70 The order of the matrix. N >= 0.
71 D (input/output) REAL array, dimension (N)
73 On entry, the n diagonal elements of the tridiagonal matrix A.
74 On exit, D may be multiplied by a constant factor chosen to
75 avoid over/underflow in computing the eigenvalues.
76 E (input/output) REAL array, dimension (max(1,N-1))
78 On entry, the (n-1) subdiagonal elements of the tridiagonal
79 matrix A in elements 1 to N-1 of E. On exit, E may be multi‐
80 plied by a constant factor chosen to avoid over/underflow in
81 computing the eigenvalues.
82 VL (input) REAL
84 VU (input) REAL If RANGE='V', the lower and upper bounds
85 of the interval to be searched for eigenvalues. VL < VU. Not
86 referenced if RANGE = 'A' or 'I'.
87 IL (input) INTEGER
89 IU (input) INTEGER If RANGE='I', the indices (in ascending
90 order) of the smallest and largest eigenvalues to be returned.
91 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
92 referenced if RANGE = 'A' or 'V'.
93 ABSTOL (input) REAL
95 The absolute error tolerance for the eigenvalues. An approxi‐
96 mate eigenvalue is accepted as converged when it is determined
97 to lie in an interval [a,b] of width less than or equal to
98 ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine pre‐
99 cision. If ABSTOL is less than or equal to zero, then EPS*|T|
100 will be used in its place, where |T| is the 1-norm of the
101 tridiagonal matrix obtained by reducing A to tridiagonal form.
102 See "Computing Small Singular Values of Bidiagonal Matrices
103 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
104 LAPACK Working Note #3. If high relative accuracy is impor‐
105 tant, set ABSTOL to SLAMCH( 'Safe minimum' ). Doing so will
106 guarantee that eigenvalues are computed to high relative accu‐
107 racy when possible in future releases. The current code does
108 not make any guarantees about high relative accuracy, but
109 future releases will. See J. Barlow and J. Demmel, "Computing
110 Accurate Eigensystems of Scaled Diagonally Dominant Matrices",
111 LAPACK Working Note #7, for a discussion of which matrices
112 define their eigenvalues to high relative accuracy.
113 M (output) INTEGER
115 The total number of eigenvalues found. 0 <= M <= N. If RANGE
116 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
117 W (output) REAL array, dimension (N)
119 The first M elements contain the selected eigenvalues in
120 ascending order.
121 Z (output) REAL array, dimension (LDZ, max(1,M) )
123 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
124 tain the orthonormal eigenvectors of the matrix A corresponding
125 to the selected eigenvalues, with the i-th column of Z holding
126 the eigenvector associated with W(i). Note: the user must
127 ensure that at least max(1,M) columns are supplied in the array
128 Z; if RANGE = 'V', the exact value of M is not known in advance
129 and an upper bound must be used.
130 LDZ (input) INTEGER
132 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
133 'V', LDZ >= max(1,N).
134 ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
136 The support of the eigenvectors in Z, i.e., the indices indi‐
137 cating the nonzero elements in Z. The i-th eigenvector is
138 nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ).
139 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
141 On exit, if INFO = 0, WORK(1) returns the optimal (and minimal)
142 LWORK.
143 LWORK (input) INTEGER
145 The dimension of the array WORK. LWORK >= 20*N. If LWORK =
146 -1, then a workspace query is assumed; the routine only calcu‐
147 lates the optimal sizes of the WORK and IWORK arrays, returns
148 these values as the first entries of the WORK and IWORK arrays,
149 and no error message related to LWORK or LIWORK is issued by
150 XERBLA.
151 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
153 On exit, if INFO = 0, IWORK(1) returns the optimal (and mini‐
154 mal) LIWORK.
155 LIWORK (input) INTEGER
157 The dimension of the array IWORK. LIWORK >= 10*N. If LIWORK =
158 -1, then a workspace query is assumed; the routine only calcu‐
159 lates the optimal sizes of the WORK and IWORK arrays, returns
160 these values as the first entries of the WORK and IWORK arrays,
161 and no error message related to LWORK or LIWORK is issued by
162 XERBLA.
163 INFO (output) INTEGER
165 = 0: successful exit
166 < 0: if INFO = -i, the i-th argument had an illegal value
167 > 0: Internal error
168
170 Based on contributions by
171 Inderjit Dhillon, IBM Almaden, USA
172 Osni Marques, LBNL/NERSC, USA
173 Ken Stanley, Computer Science Division, University of
174 California at Berkeley, USA
175 Jason Riedy, Computer Science Division, University of
176 California at Berkeley, USA
177 LAPACK driver routine (version 3.N2o)vember 2008 SSTEVR(1)