clar1v(l)

1CLAR1V(1)           LAPACK auxiliary routine (version 3.2)           CLAR1V(1)
2
3
4

NAME

6       CLAR1V  - computes the (scaled) r-th column of the inverse of the sumb‐
7       matrix in rows B1 through BN of the tridiagonal matrix L D L^T -  sigma
8       I
9

SYNOPSIS

11       SUBROUTINE CLAR1V( N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z,
12                          WANTNC, NEGCNT,  ZTZ,  MINGMA,  R,  ISUPPZ,  NRMINV,
13                          RESID, RQCORR, WORK )
14
15           LOGICAL        WANTNC
16
17           INTEGER        B1, BN, N, NEGCNT, R
18
19           REAL           GAPTOL,   LAMBDA,  MINGMA,  NRMINV,  PIVMIN,  RESID,
20                          RQCORR, ZTZ
21
22           INTEGER        ISUPPZ( * )
23
24           REAL           D( * ), L( * ), LD( * ), LLD( * ), WORK( * )
25
26           COMPLEX        Z( * )
27

PURPOSE

29       CLAR1V computes the (scaled) r-th column of the inverse of the  sumbma‐
30       trix in rows B1 through BN of the tridiagonal matrix L D L^T - sigma I.
31       When sigma is close to an eigenvalue, the computed vector is  an  accu‐
32       rate  eigenvector. Usually, r corresponds to the index where the eigen‐
33       vector is largest in magnitude.  The following  steps  accomplish  this
34       computation :
35       (a) Stationary qd transform,  L D L^T - sigma I = L(+) D(+) L(+)^T, (b)
36       Progressive qd transform, L D L^T - sigma I =  U(-)  D(-)  U(-)^T,  (c)
37       Computation of the diagonal elements of the inverse of
38           L D L^T - sigma I by combining the above transforms, and choosing
39           r as the index where the diagonal of the inverse is (one of the)
40           largest in magnitude.
41       (d) Computation of the (scaled) r-th column of the inverse using the
42           twisted factorization obtained by combining the top part of the
43           the stationary and the bottom part of the progressive transform.
44

ARGUMENTS

46       N        (input) INTEGER
47                The order of the matrix L D L^T.
48
49       B1       (input) INTEGER
50                First index of the submatrix of L D L^T.
51
52       BN       (input) INTEGER
53                Last index of the submatrix of L D L^T.
54
55       LAMBDA    (input) REAL
56                 The  shift.  In  order  to  compute  an accurate eigenvector,
57                 LAMBDA should be a good approximation to an eigenvalue of L D
58                 L^T.
59
60       L        (input) REAL             array, dimension (N-1)
61                The  (n-1)  subdiagonal elements of the unit bidiagonal matrix
62                L, in elements 1 to N-1.
63
64       D        (input) REAL             array, dimension (N)
65                The n diagonal elements of the diagonal matrix D.
66
67       LD       (input) REAL             array, dimension (N-1)
68                The n-1 elements L(i)*D(i).
69
70       LLD      (input) REAL             array, dimension (N-1)
71                The n-1 elements L(i)*L(i)*D(i).
72
73       PIVMIN   (input) REAL
74                The minimum pivot in the Sturm sequence.
75
76       GAPTOL   (input) REAL
77                Tolerance that indicates when eigenvector entries are negligi‐
78                ble w.r.t. their contribution to the residual.
79
80       Z        (input/output) COMPLEX          array, dimension (N)
81                On  input,  all  entries  of Z must be set to 0.  On output, Z
82                contains the (scaled) r-th column of the inverse. The  scaling
83                is such that Z(R) equals 1.
84
85       WANTNC   (input) LOGICAL
86                Specifies whether NEGCNT has to be computed.
87
88       NEGCNT   (output) INTEGER
89                If  WANTNC  is  .TRUE.  then  NEGCNT  = the number of pivots <
90                pivmin in the  matrix factorization L D L^T, and NEGCNT  =  -1
91                otherwise.
92
93       ZTZ      (output) REAL
94                The square of the 2-norm of Z.
95
96       MINGMA   (output) REAL
97                The  reciprocal of the largest (in magnitude) diagonal element
98                of the inverse of L D L^T - sigma I.
99
100       R        (input/output) INTEGER
101                The twist index for the twisted factorization used to  compute
102                Z.  On input, 0 <= R <= N. If R is input as 0, R is set to the
103                index where (L D L^T - sigma I)^{-1} is largest in  magnitude.
104                If  1  <=  R  <= N, R is unchanged.  On output, R contains the
105                twist index used to compute  Z.   Ideally,  R  designates  the
106                position of the maximum entry in the eigenvector.
107
108       ISUPPZ   (output) INTEGER array, dimension (2)
109                The  support of the vector in Z, i.e., the vector Z is nonzero
110                only in elements ISUPPZ(1) through ISUPPZ( 2 ).
111
112       NRMINV   (output) REAL
113                NRMINV = 1/SQRT( ZTZ )
114
115       RESID    (output) REAL
116                The residual of the FP vector.  RESID =  ABS(  MINGMA  )/SQRT(
117                ZTZ )
118
119       RQCORR   (output) REAL
120                The   Rayleigh   Quotient  correction  to  LAMBDA.   RQCORR  =
121                MINGMA*TMP
122
123       WORK     (workspace) REAL             array, dimension (4*N)
124

FURTHER DETAILS

126       Based on contributions by
127          Beresford Parlett, University of California, Berkeley, USA
128          Jim Demmel, University of California, Berkeley, USA
129          Inderjit Dhillon, University of Texas, Austin, USA
130          Osni Marques, LBNL/NERSC, USA
131          Christof Voemel, University of California, Berkeley, USA
132
133
134
135 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008                       CLAR1V(1)