1DSTEVR(1) LAPACK driver routine (version 3.2) DSTEVR(1)
2
6 DSTEVR - computes selected eigenvalues and, optionally, eigenvectors of
7 a real 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
30 eigenspectrum using Relatively Robust Representations. DSTEMR 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‐
48 agonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer
49 Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May
50 1997.
51 Note 1 : DSTEVR calls DSTEMR when the full spectrum is requested on
52 machines which conform to the ieee-754 floating point standard. DSTEVR
53 calls DSTEBZ and DSTEIN on non-ieee machines and
54 when partial spectrum requests are made.
55 Normal execution of DSTEMR may create NaNs and infinities and hence may
56 abort due to a floating point exception in environments which do not
57 handle NaNs and infinities in the ieee standard default manner.
58
60 JOBZ (input) CHARACTER*1
61 = 'N': Compute eigenvalues only;
62 = 'V': Compute eigenvalues and eigenvectors.
63 RANGE (input) CHARACTER*1
65 = 'A': all eigenvalues will be found.
66 = 'V': all eigenvalues in the half-open interval (VL,VU] will
67 be found. = 'I': the IL-th through IU-th eigenvalues will be
68 found.
69 N (input) INTEGER
71 The order of the matrix. N >= 0.
72 D (input/output) DOUBLE PRECISION array, dimension (N)
74 On entry, the n diagonal elements of the tridiagonal matrix A.
75 On exit, D may be multiplied by a constant factor chosen to
76 avoid over/underflow in computing the eigenvalues.
77 E (input/output) DOUBLE PRECISION array, dimension (max(1,N-1))
79 On entry, the (n-1) subdiagonal elements of the tridiagonal
80 matrix A in elements 1 to N-1 of E. On exit, E may be multi‐
81 plied by a constant factor chosen to avoid over/underflow in
82 computing the eigenvalues.
83 VL (input) DOUBLE PRECISION
85 VU (input) DOUBLE PRECISION If RANGE='V', the lower and
86 upper bounds of the interval to be searched for eigenvalues. VL
87 < VU. Not referenced if RANGE = 'A' or 'I'.
88 IL (input) INTEGER
90 IU (input) INTEGER If RANGE='I', the indices (in ascending
91 order) of the smallest and largest eigenvalues to be returned.
92 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not
93 referenced if RANGE = 'A' or 'V'.
94 ABSTOL (input) DOUBLE PRECISION
96 The absolute error tolerance for the eigenvalues. An approxi‐
97 mate eigenvalue is accepted as converged when it is determined
98 to lie in an interval [a,b] of width less than or equal to
99 ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine pre‐
100 cision. If ABSTOL is less than or equal to zero, then EPS*|T|
101 will be used in its place, where |T| is the 1-norm of the
102 tridiagonal matrix obtained by reducing A to tridiagonal form.
103 See "Computing Small Singular Values of Bidiagonal Matrices
104 with Guaranteed High Relative Accuracy," by Demmel and Kahan,
105 LAPACK Working Note #3. If high relative accuracy is impor‐
106 tant, set ABSTOL to DLAMCH( 'Safe minimum' ). Doing so will
107 guarantee that eigenvalues are computed to high relative accu‐
108 racy when possible in future releases. The current code does
109 not make any guarantees about high relative accuracy, but
110 future releases will. See J. Barlow and J. Demmel, "Computing
111 Accurate Eigensystems of Scaled Diagonally Dominant Matrices",
112 LAPACK Working Note #7, for a discussion of which matrices
113 define their eigenvalues to high relative accuracy.
114 M (output) INTEGER
116 The total number of eigenvalues found. 0 <= M <= N. If RANGE
117 = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
118 W (output) DOUBLE PRECISION array, dimension (N)
120 The first M elements contain the selected eigenvalues in
121 ascending order.
122 Z (output) DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
124 If JOBZ = 'V', then if INFO = 0, the first M columns of Z con‐
125 tain the orthonormal eigenvectors of the matrix A corresponding
126 to the selected eigenvalues, with the i-th column of Z holding
127 the eigenvector associated with W(i). Note: the user must
128 ensure that at least max(1,M) columns are supplied in the array
129 Z; if RANGE = 'V', the exact value of M is not known in advance
130 and an upper bound must be used.
131 LDZ (input) INTEGER
133 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
134 'V', LDZ >= max(1,N).
135 ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
137 The support of the eigenvectors in Z, i.e., the indices indi‐
138 cating the nonzero elements in Z. The i-th eigenvector is
139 nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ).
140 WORK (workspace/output) DOUBLE PRECISION array, dimension
142 (MAX(1,LWORK))
143 On exit, if INFO = 0, WORK(1) returns the optimal (and minimal)
144 LWORK.
145 LWORK (input) INTEGER
147 The dimension of the array WORK. LWORK >= max(1,20*N). If
148 LWORK = -1, then a workspace query is assumed; the routine only
149 calculates the optimal sizes of the WORK and IWORK arrays,
150 returns these values as the first entries of the WORK and IWORK
151 arrays, and no error message related to LWORK or LIWORK is
152 issued by XERBLA.
153 IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
155 On exit, if INFO = 0, IWORK(1) returns the optimal (and mini‐
156 mal) LIWORK.
157 LIWORK (input) INTEGER
159 The dimension of the array IWORK. LIWORK >= max(1,10*N). If
160 LIWORK = -1, then a workspace query is assumed; the routine
161 only calculates the optimal sizes of the WORK and IWORK arrays,
162 returns these values as the first entries of the WORK and IWORK
163 arrays, and no error message related to LWORK or LIWORK is
164 issued by XERBLA.
165 INFO (output) INTEGER
167 = 0: successful exit
168 < 0: if INFO = -i, the i-th argument had an illegal value
169 > 0: Internal error
170
172 Based on contributions by
173 Inderjit Dhillon, IBM Almaden, USA
174 Osni Marques, LBNL/NERSC, USA
175 Ken Stanley, Computer Science Division, University of
176 California at Berkeley, USA
177 LAPACK driver routine (version 3.N2o)vember 2008 DSTEVR(1)