1DSTEVR(1) LAPACK driver routine (version 3.1) DSTEVR(1)
2
6 DSTEVR - selected eigenvalues and, optionally, eigenvectors of a real
7 symmetric tridiagonal matrix T
8
10 SUBROUTINE DSTEVR( 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 DOUBLE PRECISION ABSTOL, VL, VU
18 INTEGER ISUPPZ( * ), IWORK( * )
20 DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ,
22 * )
23
25 DSTEVR computes selected eigenvalues and, optionally, eigenvectors of a
26 real symmetric tridiagonal matrix T. 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 Whenever possible, DSTEVR calls DSTEMR to compute the
31 eigenspectrum using Relatively Robust Representations. DSTEMR computes
32 eigenvalues by the dqds algorithm, while orthogonal eigenvectors are
33 computed from various "good" L D L^T representations (also known as
34 Relatively Robust Representations). Gram-Schmidt orthogonalization is
35 avoided as far as possible. More specifically, the various steps of the
36 algorithm are as follows. For the i-th unreduced block of T,
37 (a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
38 is a relatively robust representation,
39 (b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
40 relative accuracy by the dqds algorithm,
41 (c) If there is a cluster of close eigenvalues, "choose" sigma_i
42 close to the cluster, and go to step (a),
43 (d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
44 compute the corresponding eigenvector by forming a
45 rank-revealing twisted factorization.
46 The desired accuracy of the output can be specified by the input param‐
47 eter ABSTOL.
48 For more details, see "A new O(n^2) algorithm for the symmetric tridi‐
50 agonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer
51 Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May
52 1997.
53 Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested on
56 machines which conform to the ieee-754 floating point standard. DSTEVR
57 calls DSTEBZ and DSTEIN on non-ieee machines and
58 when partial spectrum requests are made.
59 Normal execution of DSTEMR may create NaNs and infinities and hence may
61 abort due to a floating point exception in environments which do not
62 handle NaNs and infinities in the ieee standard default manner.
63
66 JOBZ (input) CHARACTER*1
67 = 'N': Compute eigenvalues only;
68 = 'V': Compute eigenvalues and eigenvectors.
69 RANGE (input) CHARACTER*1
71 = 'A': all eigenvalues will be found.
72 = 'V': all eigenvalues in the half-open interval (VL,VU] will
73 be found. = 'I': the IL-th through IU-th eigenvalues will be
74 found.
75 N (input) INTEGER
77 The order of the matrix. N >= 0.
78 D (input/output) DOUBLE PRECISION array, dimension (N)
80 On entry, the n diagonal elements of the tridiagonal matrix A.
81 On exit, D may be multiplied by a constant factor chosen to
82 avoid over/underflow in computing the eigenvalues.
83 E (input/output) DOUBLE PRECISION array, dimension (max(1,N-1))
85 On entry, the (n-1) subdiagonal elements of the tridiagonal
86 matrix A in elements 1 to N-1 of E. On exit, E may be multi‐
87 plied by a constant factor chosen to avoid over/underflow in
88 computing the eigenvalues.
89 VL (input) DOUBLE PRECISION
91 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
92 upper bounds of the interval to be searched for eigenvalues. VL
93 < VU. Not referenced if RANGE = 'A' or 'I'.
94 IL (input) INTEGER
96 IU (input) INTEGER If RANGE='I', the indices (in ascending
97 order) of the smallest and largest eigenvalues to be returned.
98 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
99 referenced if RANGE = 'A' or 'V'.
100 ABSTOL (input) DOUBLE PRECISION
102 The absolute error tolerance for the eigenvalues. An approxi‐
103 mate eigenvalue is accepted as converged when it is determined
104 to lie in an interval [a,b] of width less than or equal to
105 ABSTOL + EPS * max( |a|,|b| ) ,
107 where EPS is the machine precision. If ABSTOL is less than or
109 equal to zero, then EPS*|T| will be used in its place, where
110 |T| is the 1-norm of the tridiagonal matrix obtained by reduc‐
111 ing A to tridiagonal form.
112 See "Computing Small Singular Values of Bidiagonal Matrices
114 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
115 LAPACK Working Note #3.
116 If high relative accuracy is important, set ABSTOL to DLAMCH(
118 'Safe minimum' ). Doing so will guarantee that eigenvalues are
119 computed to high relative accuracy when possible in future
120 releases. The current code does not make any guarantees about
121 high relative accuracy, but future releases will. See J. Barlow
122 and J. Demmel, "Computing Accurate Eigensystems of Scaled Diag‐
123 onally Dominant Matrices", LAPACK Working Note #7, for a dis‐
124 cussion of which matrices define their eigenvalues to high rel‐
125 ative accuracy.
126 M (output) INTEGER
128 The total number of eigenvalues found. 0 <= M <= N. If RANGE
129 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
130 W (output) DOUBLE PRECISION array, dimension (N)
132 The first M elements contain the selected eigenvalues in
133 ascending order.
134 Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
136 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
137 tain the orthonormal eigenvectors of the matrix A corresponding
138 to the selected eigenvalues, with the i-th column of Z holding
139 the eigenvector associated with W(i). Note: the user must
140 ensure that at least max(1,M) columns are supplied in the array
141 Z; if RANGE = 'V', the exact value of M is not known in advance
142 and an upper bound must be used.
143 LDZ (input) INTEGER
145 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
146 'V', LDZ >= max(1,N).
147 ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
149 The support of the eigenvectors in Z, i.e., the indices indi‐
150 cating the nonzero elements in Z. The i-th eigenvector is
151 nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ).
152 WORK (workspace/output) DOUBLE PRECISION array, dimension
154 (MAX(1,LWORK))
155 On exit, if INFO = 0, WORK(1) returns the optimal (and minimal)
156 LWORK.
157 LWORK (input) INTEGER
159 The dimension of the array WORK. LWORK >= max(1,20*N).
160 If LWORK = -1, then a workspace query is assumed; the routine
162 only calculates the optimal sizes of the WORK and IWORK arrays,
163 returns these values as the first entries of the WORK and IWORK
164 arrays, and no error message related to LWORK or LIWORK is
165 issued by XERBLA.
166 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
168 On exit, if INFO = 0, IWORK(1) returns the optimal (and mini‐
169 mal) LIWORK.
170 LIWORK (input) INTEGER
172 The dimension of the array IWORK. LIWORK >= max(1,10*N).
173 If LIWORK = -1, then a workspace query is assumed; the routine
175 only calculates the optimal sizes of the WORK and IWORK arrays,
176 returns these values as the first entries of the WORK and IWORK
177 arrays, and no error message related to LWORK or LIWORK is
178 issued by XERBLA.
179 INFO (output) INTEGER
181 = 0: successful exit
182 < 0: if INFO = -i, the i-th argument had an illegal value
183 > 0: Internal error
184
186 Based on contributions by
187 Inderjit Dhillon, IBM Almaden, USA
188 Osni Marques, LBNL/NERSC, USA
189 Ken Stanley, Computer Science Division, University of
190 California at Berkeley, USA
191 LAPACK driver routine (version 3.N1o)vember 2006 DSTEVR(1)