1SLARRV(1) LAPACK auxiliary routine (version 3.1.1) SLARRV(1)
2
6 SLARRV - the eigenvectors of the tridiagonal matrix T = L D L^T given
7 L, D and APPROXIMATIONS to the eigenvalues of L D L^T
8
10 SUBROUTINE SLARRV( 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 REAL Z( LDZ, * )
25
27 SLARRV 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
33 N (input) INTEGER
34 The order of the matrix. N >= 0.
35 VL (input) REAL
37 VU (input) REAL Lower and upper bounds of the interval
38 that contains the desired eigenvalues. VL < VU. Needed to com‐
39 pute gaps on the left or right end of the extremal eigenvalues
40 in the desired RANGE.
41 D (input/output) REAL array, dimension (N)
43 On entry, the N diagonal elements of the diagonal matrix D. On
44 exit, D may be overwritten.
45 L (input/output) REAL array, dimension (N)
47 On entry, the (N-1) subdiagonal elements of the unit bidiagonal
48 matrix L are in elements 1 to N-1 of L (if the matrix is not
49 splitted.) At the end of each block is stored the corresponding
50 shift as given by SLARRE. On exit, L is overwritten.
51 PIVMIN (in) DOUBLE PRECISION
53 The minimum pivot allowed in the Sturm sequence.
54 ISPLIT (input) INTEGER array, dimension (N)
56 The splitting points, at which T breaks up into blocks. The
57 first block consists of rows/columns 1 to ISPLIT( 1 ), the sec‐
58 ond of rows/columns ISPLIT( 1 )+1 through ISPLIT( 2 ), etc.
59 M (input) INTEGER
61 The total number of input eigenvalues. 0 <= M <= N.
62 DOL (input) INTEGER
64 DOU (input) INTEGER If the user wants to compute only
65 selected eigenvectors from all the eigenvalues supplied, he can
66 specify an index range DOL:DOU. Or else the setting DOL=1,
67 DOU=M should be applied. Note that DOL and DOU refer to the
68 order in which the eigenvalues are stored in W. If the user
69 wants to compute only selected eigenpairs, then the columns
70 DOL-1 to DOU+1 of the eigenvector space Z contain the computed
71 eigenvectors. All other columns of Z are set to zero.
72 MINRGP (input) REAL
74 RTOL1 (input) REAL
76 RTOL2 (input) REAL Parameters for bisection. RIGHT-
77 LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
78 W (input/output) REAL array, dimension (N)
80 The first M elements of W contain the APPROXIMATE eigenvalues
81 for which eigenvectors are to be computed. The eigenvalues
82 should be grouped by split-off block and ordered from smallest
83 to largest within the block ( The output array W from SLARRE is
84 expected here ). Furthermore, they are with respect to the
85 shift of the corresponding root representation for their block.
86 On exit, W holds the eigenvalues of the UNshifted matrix.
87 WERR (input/output) REAL array, dimension (N)
89 The first M elements contain the semiwidth of the uncertainty
90 interval of the corresponding eigenvalue in W
91 WGAP (input/output) REAL array, dimension (N)
93 The separation from the right neighbor eigenvalue in W.
94 IBLOCK (input) INTEGER array, dimension (N)
96 The indices of the blocks (submatrices) associated with the
97 corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i)
98 belongs to the first block from the top, =2 if W(i) belongs to
99 the second block, etc.
100 INDEXW (input) INTEGER array, dimension (N)
102 The indices of the eigenvalues within each block (submatrix);
103 for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the i-th
104 eigenvalue W(i) is the 10-th eigenvalue in the second block.
105 GERS (input) REAL array, dimension (2*N)
107 The N Gerschgorin intervals (the i-th Gerschgorin interval is
108 (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should be
109 computed from the original UNshifted matrix.
110 Z (output) REAL array, dimension (LDZ, max(1,M) )
112 If INFO = 0, the first M columns of Z contain the orthonormal
113 eigenvectors of the matrix T corresponding to the input eigen‐
114 values, with the i-th column of Z holding the eigenvector asso‐
115 ciated with W(i). Note: the user must ensure that at least
116 max(1,M) columns are supplied in the array Z.
117 LDZ (input) INTEGER
119 The leading dimension of the array Z. LDZ >= 1, and if JOBZ =
120 'V', LDZ >= max(1,N).
121 ISUPPZ (output) INTEGER array, dimension ( 2*max(1,M) )
123 The support of the eigenvectors in Z, i.e., the indices indi‐
124 cating the nonzero elements in Z. The I-th eigenvector is
125 nonzero only in elements ISUPPZ( 2*I-1 ) through ISUPPZ( 2*I ).
126 WORK (workspace) REAL array, dimension (12*N)
128 IWORK (workspace) INTEGER array, dimension (7*N)
130 INFO (output) INTEGER
132 = 0: successful exit
133 > 0: A problem occured in SLARRV.
135 < 0: One of the called subroutines signaled an internal prob‐
136 lem. Needs inspection of the corresponding parameter IINFO for
137 further information.
138 =-1: Problem in SLARRB when refining a child's eigenvalues.
140 =-2: Problem in SLARRF when computing the RRR of a child. When
141 a child is inside a tight cluster, it can be difficult to find an
142 RRR. A partial remedy from the user's point of view is to make
143 the parameter MINRGP smaller and recompile. However, as the
144 orthogonality of the computed vectors is proportional to 1/MIN‐
145 RGP, the user should be aware that he might be trading in preci‐
146 sion when he decreases MINRGP. =-3: Problem in SLARRB when
147 refining a single eigenvalue after the Rayleigh correction was
148 rejected. = 5: The Rayleigh Quotient Iteration failed to con‐
149 verge to full accuracy in MAXITR steps.
150
152 Based on contributions by
153 Beresford Parlett, University of California, Berkeley, USA
154 Jim Demmel, University of California, Berkeley, USA
155 Inderjit Dhillon, University of Texas, Austin, USA
156 Osni Marques, LBNL/NERSC, USA
157 Christof Voemel, University of California, Berkeley, USA
158 LAPACK auxiliary routine (versionFe3b.r1u.a1r)y 2007 SLARRV(1)