1CLARRV(1) LAPACK auxiliary routine (version 3.2) CLARRV(1)
2
6 CLARRV - computes the eigenvectors of the tridiagonal matrix T = L D
7 L^T given L, D and APPROXIMATIONS to the eigenvalues of L D L^T
8
10 SUBROUTINE CLARRV( N, VL, VU, D, L, PIVMIN, ISPLIT, M, DOL, DOU, MIN‐
11 RGP, RTOL1, RTOL2, W, WERR, WGAP, IBLOCK, INDEXW,
12 GERS, Z, LDZ, ISUPPZ, WORK, IWORK, INFO )
13 INTEGER DOL, DOU, INFO, LDZ, M, N
15 REAL MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU
17 INTEGER IBLOCK( * ), INDEXW( * ), ISPLIT( * ), ISUPPZ( * ),
19 IWORK( * )
20 REAL D( * ), GERS( * ), L( * ), W( * ), WERR( * ), WGAP(
22 * ), WORK( * )
23 COMPLEX Z( LDZ, * )
25
27 CLARRV computes the eigenvectors of the tridiagonal matrix T = L D L^T
28 given L, D and APPROXIMATIONS to the eigenvalues of L D L^T. The input
29 eigenvalues should have been computed by SLARRE.
30
32 N (input) INTEGER
33 The order of the matrix. N >= 0.
34 VL (input) REAL
36 VU (input) REAL Lower and upper bounds of the interval
37 that contains the desired eigenvalues. VL < VU. Needed to com‐
38 pute gaps on the left or right end of the extremal eigenvalues
39 in the desired RANGE.
40 D (input/output) REAL array, dimension (N)
42 On entry, the N diagonal elements of the diagonal matrix D. On
43 exit, D may be overwritten.
44 L (input/output) REAL array, dimension (N)
46 On entry, the (N-1) subdiagonal elements of the unit bidiagonal
47 matrix L are in elements 1 to N-1 of L (if the matrix is not
48 splitted.) At the end of each block is stored the corresponding
49 shift as given by SLARRE. On exit, L is overwritten.
50 PIVMIN (in) DOUBLE PRECISION
52 The minimum pivot allowed in the Sturm sequence.
53 ISPLIT (input) INTEGER array, dimension (N)
55 The splitting points, at which T breaks up into blocks. The
56 first block consists of rows/columns 1 to ISPLIT( 1 ), the sec‐
57 ond of rows/columns ISPLIT( 1 )+1 through ISPLIT( 2 ), etc.
58 M (input) INTEGER
60 The total number of input eigenvalues. 0 <= M <= N.
61 DOL (input) INTEGER
63 DOU (input) INTEGER If the user wants to compute only
64 selected eigenvectors from all the eigenvalues supplied, he can
65 specify an index range DOL:DOU. Or else the setting DOL=1,
66 DOU=M should be applied. Note that DOL and DOU refer to the
67 order in which the eigenvalues are stored in W. If the user
68 wants to compute only selected eigenpairs, then the columns
69 DOL-1 to DOU+1 of the eigenvector space Z contain the computed
70 eigenvectors. All other columns of Z are set to zero.
71 MINRGP (input) REAL
73 RTOL1 (input) REAL
75 RTOL2 (input) REAL Parameters for bisection. RIGHT-
76 LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
77 W (input/output) REAL array, dimension (N)
79 The first M elements of W contain the APPROXIMATE eigenvalues
80 for which eigenvectors are to be computed. The eigenvalues
81 should be grouped by split-off block and ordered from smallest
82 to largest within the block ( The output array W from SLARRE is
83 expected here ). Furthermore, they are with respect to the
84 shift of the corresponding root representation for their block.
85 On exit, W holds the eigenvalues of the UNshifted matrix.
86 WERR (input/output) REAL array, dimension (N)
88 The first M elements contain the semiwidth of the uncertainty
89 interval of the corresponding eigenvalue in W
90 WGAP (input/output) REAL array, dimension (N)
92 The separation from the right neighbor eigenvalue in W.
93 IBLOCK (input) INTEGER array, dimension (N)
95 The indices of the blocks (submatrices) associated with the
96 corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i)
97 belongs to the first block from the top, =2 if W(i) belongs to
98 the second block, etc.
99 INDEXW (input) INTEGER array, dimension (N)
101 The indices of the eigenvalues within each block (submatrix);
102 for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the i-th
103 eigenvalue W(i) is the 10-th eigenvalue in the second block.
104 GERS (input) REAL array, dimension (2*N)
106 The N Gerschgorin intervals (the i-th Gerschgorin interval is
107 (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should be
108 computed from the original UNshifted matrix.
109 Z (output) COMPLEX array, dimension (LDZ, max(1,M) )
111 If INFO = 0, the first M columns of Z contain the orthonormal
112 eigenvectors of the matrix T corresponding to the input eigen‐
113 values, with the i-th column of Z holding the eigenvector asso‐
114 ciated with W(i). Note: the user must ensure that at least
115 max(1,M) columns are supplied in the array Z.
116 LDZ (input) INTEGER
118 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
119 'V', LDZ >= max(1,N).
120 ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
122 The support of the eigenvectors in Z, i.e., the indices indi‐
123 cating the nonzero elements in Z. The I-th eigenvector is
124 nonzero only in elements ISUPPZ( 2*I-1 ) through ISUPPZ( 2*I ).
125 WORK (workspace) REAL array, dimension (12*N)
127 IWORK (workspace) INTEGER array, dimension (7*N)
129 INFO (output) INTEGER
131 = 0: successful exit > 0: A problem occured in CLARRV.
132 < 0: One of the called subroutines signaled an internal prob‐
133 lem. Needs inspection of the corresponding parameter IINFO for
134 further information.
135 =-1: Problem in SLARRB when refining a child's eigenvalues.
137 =-2: Problem in SLARRF when computing the RRR of a child. When
138 a child is inside a tight cluster, it can be difficult to find an
139 RRR. A partial remedy from the user's point of view is to make
140 the parameter MINRGP smaller and recompile. However, as the
141 orthogonality of the computed vectors is proportional to 1/MIN‐
142 RGP, the user should be aware that he might be trading in preci‐
143 sion when he decreases MINRGP. =-3: Problem in SLARRB when
144 refining a single eigenvalue after the Rayleigh correction was
145 rejected. = 5: The Rayleigh Quotient Iteration failed to con‐
146 verge to full accuracy in MAXITR steps.
147
149 Based on contributions by
150 Beresford Parlett, University of California, Berkeley, USA
151 Jim Demmel, University of California, Berkeley, USA
152 Inderjit Dhillon, University of Texas, Austin, USA
153 Osni Marques, LBNL/NERSC, USA
154 Christof Voemel, University of California, Berkeley, USA
155 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008 CLARRV(1)